From 0ad3603b6761e671b83b301f2d5ec17191c63465 Mon Sep 17 00:00:00 2001
From: Mr_Dwj <explorer-dong@foxmail.com>
Date: Tue, 8 Oct 2024 23:26:15 +0800
Subject: [PATCH] Site updated: 2024-10-08 23:26:06

---
 404.html                                      |   2 +-
 Algorithm/a_template/index.html               |   6 +-
 Algorithm/binary-search/index.html            |  10 +-
 Algorithm/data-structure/index.html           |  10 +-
 Algorithm/dfs-and-similar/index.html          |  10 +-
 Algorithm/divide-and-conquer/index.html       |  10 +-
 Algorithm/dp/index.html                       |   6 +-
 Algorithm/games/index.html                    |  10 +-
 Algorithm/geometry/index.html                 |  10 +-
 Algorithm/graphs/index.html                   |  10 +-
 Algorithm/greedy/index.html                   |  10 +-
 Algorithm/hashing/index.html                  |  10 +-
 Algorithm/number-theory/index.html            |  10 +-
 Algorithm/prefix-and-difference/index.html    |   6 +-
 Algorithm/z_logic/index.html                  |  10 +-
 BackEnd/CPlusPlus/Crow/crow-basic/index.html  |   2 +-
 BackEnd/CPlusPlus/cpp-basic/index.html        |   2 +-
 BackEnd/Python/Flask/deploy-flask/index.html  |   6 +-
 BackEnd/Python/numpy/index.html               |   2 +-
 BackEnd/Python/python-basic/index.html        |   2 +-
 BackEnd/back-end-guide/index.html             |  10 +-
 .../elastic-search-basic/index.html           |   2 +-
 DataBase/MySQL/mysql-basic/index.html         |   2 +-
 .../MySQL/solve-mysql-problems/index.html     |   2 +-
 DataBase/Redis/redis-basic/index.html         |   2 +-
 DataBase/data-base-guide/index.html           |   6 +-
 DataBase/openGauss/opengauss-basic/index.html |   2 +-
 DevTools/CLion/config-clion/index.html        |   2 +-
 .../index.html                                |   6 +-
 .../solve-clion-decoding-error/index.html     |  10 +-
 DevTools/DevCpp/devc-self-config/index.html   |  10 +-
 DevTools/Git/git-basic/index.html             |   2 +-
 DevTools/Git/git-pull-request/index.html      |   2 +-
 .../Git/git-self-define-command/index.html    |   6 +-
 DevTools/Git/solve-git-problems/index.html    |   2 +-
 DevTools/IDLE/idle-self-config/index.html     |   2 +-
 DevTools/jetbrains-license/index.html         |   2 +-
 FrontEnd/CSS/bootstrap/index.html             |   2 +-
 FrontEnd/CSS/css-basic/index.html             |   2 +-
 FrontEnd/HTML/html-basic/index.html           |   2 +-
 .../hexo-auto-deploy-to-server/index.html     |   2 +-
 FrontEnd/Hexo/hexo-config-mermaid/index.html  |   2 +-
 .../Hexo/hexo-githubpages-domain/index.html   |   6 +-
 FrontEnd/Hexo/hexo-learning-record/index.html |  10 +-
 FrontEnd/Hexo/hexo-migrate/index.html         |  10 +-
 FrontEnd/Hexo/why-hexo/index.html             |   2 +-
 .../JavaScript/javascript-basic/index.html    |   2 +-
 FrontEnd/JavaScript/jquery/index.html         |   2 +-
 FrontEnd/JavaScript/react/index.html          |   2 +-
 FrontEnd/JavaScript/vue/index.html            |   2 +-
 FrontEnd/front-end-guide/index.html           |   6 +-
 GPA/1st-term/AdvancedMath/index.html          |   2 +-
 .../ObjectOrientedClassDesign/index.html      |   2 +-
 .../ObjectOrientedProgramming/index.html      |   2 +-
 GPA/3rd-term/CollegePhysics_2/index.html      |   6 +-
 GPA/3rd-term/DataStructure/index.html         |   6 +-
 .../DataStructureClassDesign/index.html       |   6 +-
 GPA/3rd-term/DigitalLogicCircuit/index.html   |  10 +-
 GPA/3rd-term/LinearAlgebra/index.html         |   6 +-
 GPA/4th-term/MachineLearning/index.html       |  10 +-
 .../MachineLearningClassDesign/index.html     |   2 +-
 GPA/4th-term/OptMethod/index.html             |  10 +-
 GPA/4th-term/ProbAndStat/index.html           |  10 +-
 GPA/4th-term/PyAlgo/index.html                |  10 +-
 GPA/4th-term/PyApply/index.html               |  10 +-
 GPA/4th-term/SysBasic/index.html              |   6 +-
 GPA/5th-term/ComputerOrganization/index.html  |  11 +-
 GPA/5th-term/DataBase/index.html              |   2 +-
 GPA/5th-term/DataMining/index.html            |   2 +-
 .../DigitalImageProcessing/index.html         | 113 ++++++---
 .../NeuralNetworkAndDeepLearning/index.html   |  18 +-
 GPA/5th-term/OperatingSystem/index.html       |  37 ++-
 Operation/Nginx/nginx-basic/index.html        |   2 +-
 Operation/Nginx/nginx-command/index.html      |   2 +-
 .../Nginx/solve-nginx-reload-error/index.html |   2 +-
 Operation/Nginx/whereis-nginx/index.html      |   2 +-
 Operation/Safe/solve-server-invade/index.html |   2 +-
 Operation/Shell/shell-basic/index.html        |  84 +++++--
 .../Shell/solve-shell-problems/index.html     |   2 +-
 Operation/Shell/ubuntu-guide/index.html       |   2 +-
 Operation/operation-guide/index.html          |  10 +-
 README/index.html                             |   2 +-
 RoadMap/road-map-guide/index.html             |  10 +-
 about/index.html                              |  12 +-
 archives/2024/03/index.html                   |  18 +-
 archives/2024/03/page/2/index.html            |  22 +-
 archives/2024/03/page/3/index.html            |  22 +-
 archives/2024/03/page/4/index.html            |  30 +--
 archives/2024/03/page/5/index.html            |  10 +-
 archives/2024/04/index.html                   |   2 +-
 archives/2024/05/index.html                   |   2 +-
 archives/2024/06/index.html                   |   2 +-
 archives/2024/07/index.html                   |   2 +-
 archives/2024/08/index.html                   |   2 +-
 archives/2024/08/page/2/index.html            |   2 +-
 archives/2024/08/page/3/index.html            |   2 +-
 archives/2024/09/index.html                   |   2 +-
 archives/2024/index.html                      |   2 +-
 archives/2024/page/2/index.html               |   2 +-
 archives/2024/page/3/index.html               |   2 +-
 archives/2024/page/4/index.html               |   2 +-
 archives/2024/page/5/index.html               |  22 +-
 archives/2024/page/6/index.html               |  22 +-
 archives/2024/page/7/index.html               |  26 +--
 archives/2024/page/8/index.html               |  26 +--
 archives/2024/page/9/index.html               |   6 +-
 archives/index.html                           |   2 +-
 archives/page/2/index.html                    |   2 +-
 archives/page/3/index.html                    |   2 +-
 archives/page/4/index.html                    |   2 +-
 archives/page/5/index.html                    |  22 +-
 archives/page/6/index.html                    |  22 +-
 archives/page/7/index.html                    |  26 +--
 archives/page/8/index.html                    |  26 +--
 archives/page/9/index.html                    |   6 +-
 categories/Algorithm/index.html               |  30 +--
 categories/Algorithm/page/2/index.html        |  10 +-
 categories/BackEnd/CPlusPlus/Crow/index.html  |   2 +-
 categories/BackEnd/CPlusPlus/index.html       |   2 +-
 categories/BackEnd/Python/Flask/index.html    |   2 +-
 categories/BackEnd/Python/index.html          |   2 +-
 categories/BackEnd/index.html                 |   2 +-
 categories/DataBase/ElasticSearch/index.html  |   2 +-
 categories/DataBase/MySQL/index.html          |   2 +-
 categories/DataBase/Redis/index.html          |   2 +-
 categories/DataBase/index.html                |   2 +-
 categories/DataBase/openGauss/index.html      |   2 +-
 categories/DevTools/CLion/index.html          |   2 +-
 categories/DevTools/DevCpp/index.html         |   2 +-
 categories/DevTools/Git/index.html            |   2 +-
 categories/DevTools/IDLE/index.html           |   2 +-
 categories/DevTools/index.html                |  10 +-
 categories/FrontEnd/CSS/index.html            |   2 +-
 categories/FrontEnd/HTML/index.html           |   2 +-
 categories/FrontEnd/Hexo/index.html           |  10 +-
 categories/FrontEnd/JavaScript/index.html     |   2 +-
 categories/FrontEnd/index.html                |   2 +-
 categories/FrontEnd/page/2/index.html         |  10 +-
 categories/GPA/1st-term/index.html            |   2 +-
 categories/GPA/2nd-term/index.html            |   2 +-
 categories/GPA/3rd-term/index.html            |  14 +-
 categories/GPA/4th-term/index.html            |  14 +-
 categories/GPA/5th-term/index.html            |   2 +-
 categories/GPA/index.html                     |   2 +-
 categories/GPA/page/2/index.html              |  26 +--
 categories/GPA/page/3/index.html              |   2 +-
 categories/Operation/Nginx/index.html         |   2 +-
 categories/Operation/Safe/index.html          |   2 +-
 categories/Operation/Shell/index.html         |   2 +-
 categories/Operation/index.html               |   2 +-
 categories/RoadMap/index.html                 |   2 +-
 categories/index.html                         |   2 +-
 index.html                                    |   2 +-
 inner/index.html                              |   2 +-
 links/index.html                              |   2 +-
 local-search.xml                              | 218 +++++++++---------
 page/10/index.html                            |   2 +-
 page/11/index.html                            |   2 +-
 page/12/index.html                            |   2 +-
 page/13/index.html                            |   2 +-
 page/14/index.html                            |   2 +-
 page/15/index.html                            |  22 +-
 page/16/index.html                            |  10 +-
 page/17/index.html                            |  18 +-
 page/18/index.html                            |  18 +-
 page/19/index.html                            |  10 +-
 page/2/index.html                             |   2 +-
 page/20/index.html                            |  18 +-
 page/21/index.html                            |   2 +-
 page/22/index.html                            |  22 +-
 page/23/index.html                            |  22 +-
 page/24/index.html                            |  26 +--
 page/25/index.html                            |  26 +--
 page/26/index.html                            |  10 +-
 page/27/index.html                            |  18 +-
 page/28/index.html                            |   2 +-
 page/3/index.html                             |   2 +-
 page/4/index.html                             |   2 +-
 page/5/index.html                             |   2 +-
 page/6/index.html                             |   2 +-
 page/7/index.html                             |   2 +-
 page/8/index.html                             |   2 +-
 page/9/index.html                             |   4 +-
 tags/index.html                               |   2 +-
 184 files changed, 898 insertions(+), 779 deletions(-)

diff --git a/404.html b/404.html
index ced6e692b..62e1f8e6a 100644
--- a/404.html
+++ b/404.html
@@ -70,7 +70,7 @@
   <script id="fluid-configs">
     var Fluid = window.Fluid || {};
     Fluid.ctx = Object.assign({}, Fluid.ctx)
-    var CONFIG = {"hostname":"blog.dwj601.cn","root":"/","version":"1.9.8","typing":{"enable":true,"typeSpeed":70,"cursorChar":"","loop":false,"scope":[]},"anchorjs":{"enable":true,"element":"h1,h2,h3,h4,h5,h6","placement":"left","visible":"hover","icon":""},"progressbar":{"enable":true,"height_px":3,"color":"#29d","options":{"showSpinner":false,"trickleSpeed":100}},"code_language":{"enable":true,"default":"TEXT"},"copy_btn":true,"image_caption":{"enable":true},"image_zoom":{"enable":true,"img_url_replace":["",""]},"toc":{"enable":true,"placement":"left","headingSelector":"h1,h2,h3","collapseDepth":3},"lazyload":{"enable":true,"loading_img":"https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif","onlypost":false,"offset_factor":2},"web_analytics":{"enable":true,"follow_dnt":true,"baidu":"d08d23389220ad9b8f711cfb85757f4b","google":{"measurement_id":null},"tencent":{"sid":null,"cid":null},"leancloud":{"app_id":null,"app_key":null,"server_url":null,"path":"window.location.pathname","ignore_local":false},"umami":{"src":null,"website_id":null,"domains":null,"start_time":"2024-01-01T00:00:00.000Z","token":null,"api_server":null},"woyaola":null,"cnzz":null},"search_path":"/local-search.xml","include_content_in_search":true};
+    var CONFIG = {"hostname":"blog.dwj601.cn","root":"/","version":"1.9.8","typing":{"enable":true,"typeSpeed":70,"cursorChar":"_","loop":false,"scope":[]},"anchorjs":{"enable":true,"element":"h1,h2,h3,h4,h5,h6","placement":"left","visible":"hover","icon":""},"progressbar":{"enable":true,"height_px":3,"color":"#29d","options":{"showSpinner":false,"trickleSpeed":100}},"code_language":{"enable":true,"default":"TEXT"},"copy_btn":true,"image_caption":{"enable":true},"image_zoom":{"enable":true,"img_url_replace":["",""]},"toc":{"enable":true,"placement":"left","headingSelector":"h1,h2,h3","collapseDepth":3},"lazyload":{"enable":true,"loading_img":"https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif","onlypost":false,"offset_factor":2},"web_analytics":{"enable":true,"follow_dnt":true,"baidu":"d08d23389220ad9b8f711cfb85757f4b","google":{"measurement_id":null},"tencent":{"sid":null,"cid":null},"leancloud":{"app_id":null,"app_key":null,"server_url":null,"path":"window.location.pathname","ignore_local":false},"umami":{"src":null,"website_id":null,"domains":null,"start_time":"2024-01-01T00:00:00.000Z","token":null,"api_server":null},"woyaola":null,"cnzz":null},"search_path":"/local-search.xml","include_content_in_search":true};
 
     if (CONFIG.web_analytics.follow_dnt) {
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diff --git a/Algorithm/a_template/index.html b/Algorithm/a_template/index.html
index f5779475e..c245096d2 100644
--- a/Algorithm/a_template/index.html
+++ b/Algorithm/a_template/index.html
@@ -80,7 +80,7 @@
   <script id="fluid-configs">
     var Fluid = window.Fluid || {};
     Fluid.ctx = Object.assign({}, Fluid.ctx)
-    var CONFIG = {"hostname":"blog.dwj601.cn","root":"/","version":"1.9.8","typing":{"enable":true,"typeSpeed":70,"cursorChar":"","loop":false,"scope":[]},"anchorjs":{"enable":true,"element":"h1,h2,h3,h4,h5,h6","placement":"left","visible":"hover","icon":""},"progressbar":{"enable":true,"height_px":3,"color":"#29d","options":{"showSpinner":false,"trickleSpeed":100}},"code_language":{"enable":true,"default":"TEXT"},"copy_btn":true,"image_caption":{"enable":true},"image_zoom":{"enable":true,"img_url_replace":["",""]},"toc":{"enable":true,"placement":"left","headingSelector":"h1,h2,h3","collapseDepth":3},"lazyload":{"enable":true,"loading_img":"https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif","onlypost":false,"offset_factor":2},"web_analytics":{"enable":true,"follow_dnt":true,"baidu":"d08d23389220ad9b8f711cfb85757f4b","google":{"measurement_id":null},"tencent":{"sid":null,"cid":null},"leancloud":{"app_id":null,"app_key":null,"server_url":null,"path":"window.location.pathname","ignore_local":false},"umami":{"src":null,"website_id":null,"domains":null,"start_time":"2024-01-01T00:00:00.000Z","token":null,"api_server":null},"woyaola":null,"cnzz":null},"search_path":"/local-search.xml","include_content_in_search":true};
+    var CONFIG = {"hostname":"blog.dwj601.cn","root":"/","version":"1.9.8","typing":{"enable":true,"typeSpeed":70,"cursorChar":"_","loop":false,"scope":[]},"anchorjs":{"enable":true,"element":"h1,h2,h3,h4,h5,h6","placement":"left","visible":"hover","icon":""},"progressbar":{"enable":true,"height_px":3,"color":"#29d","options":{"showSpinner":false,"trickleSpeed":100}},"code_language":{"enable":true,"default":"TEXT"},"copy_btn":true,"image_caption":{"enable":true},"image_zoom":{"enable":true,"img_url_replace":["",""]},"toc":{"enable":true,"placement":"left","headingSelector":"h1,h2,h3","collapseDepth":3},"lazyload":{"enable":true,"loading_img":"https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif","onlypost":false,"offset_factor":2},"web_analytics":{"enable":true,"follow_dnt":true,"baidu":"d08d23389220ad9b8f711cfb85757f4b","google":{"measurement_id":null},"tencent":{"sid":null,"cid":null},"leancloud":{"app_id":null,"app_key":null,"server_url":null,"path":"window.location.pathname","ignore_local":false},"umami":{"src":null,"website_id":null,"domains":null,"start_time":"2024-01-01T00:00:00.000Z","token":null,"api_server":null},"woyaola":null,"cnzz":null},"search_path":"/local-search.xml","include_content_in_search":true};
 
     if (CONFIG.web_analytics.follow_dnt) {
       var dntVal = navigator.doNotTrack || window.doNotTrack || navigator.msDoNotTrack;
@@ -622,8 +622,8 @@ <h2 id="计算几何">计算几何</h2>
                   <article class="post-next col-6">
                     
                     
-                      <a href="/Algorithm/binary-search/" title="binary-search">
-                        <span class="hidden-mobile">binary-search</span>
+                      <a href="/Algorithm/data-structure/" title="data-structure">
+                        <span class="hidden-mobile">data-structure</span>
                         <span class="visible-mobile">下一篇</span>
                         <i class="iconfont icon-arrowright"></i>
                       </a>
diff --git a/Algorithm/binary-search/index.html b/Algorithm/binary-search/index.html
index b639f8bee..9928ebd44 100644
--- a/Algorithm/binary-search/index.html
+++ b/Algorithm/binary-search/index.html
@@ -84,7 +84,7 @@
   <script id="fluid-configs">
     var Fluid = window.Fluid || {};
     Fluid.ctx = Object.assign({}, Fluid.ctx)
-    var CONFIG = {"hostname":"blog.dwj601.cn","root":"/","version":"1.9.8","typing":{"enable":true,"typeSpeed":70,"cursorChar":"","loop":false,"scope":[]},"anchorjs":{"enable":true,"element":"h1,h2,h3,h4,h5,h6","placement":"left","visible":"hover","icon":""},"progressbar":{"enable":true,"height_px":3,"color":"#29d","options":{"showSpinner":false,"trickleSpeed":100}},"code_language":{"enable":true,"default":"TEXT"},"copy_btn":true,"image_caption":{"enable":true},"image_zoom":{"enable":true,"img_url_replace":["",""]},"toc":{"enable":true,"placement":"left","headingSelector":"h1,h2,h3","collapseDepth":3},"lazyload":{"enable":true,"loading_img":"https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif","onlypost":false,"offset_factor":2},"web_analytics":{"enable":true,"follow_dnt":true,"baidu":"d08d23389220ad9b8f711cfb85757f4b","google":{"measurement_id":null},"tencent":{"sid":null,"cid":null},"leancloud":{"app_id":null,"app_key":null,"server_url":null,"path":"window.location.pathname","ignore_local":false},"umami":{"src":null,"website_id":null,"domains":null,"start_time":"2024-01-01T00:00:00.000Z","token":null,"api_server":null},"woyaola":null,"cnzz":null},"search_path":"/local-search.xml","include_content_in_search":true};
+    var CONFIG = {"hostname":"blog.dwj601.cn","root":"/","version":"1.9.8","typing":{"enable":true,"typeSpeed":70,"cursorChar":"_","loop":false,"scope":[]},"anchorjs":{"enable":true,"element":"h1,h2,h3,h4,h5,h6","placement":"left","visible":"hover","icon":""},"progressbar":{"enable":true,"height_px":3,"color":"#29d","options":{"showSpinner":false,"trickleSpeed":100}},"code_language":{"enable":true,"default":"TEXT"},"copy_btn":true,"image_caption":{"enable":true},"image_zoom":{"enable":true,"img_url_replace":["",""]},"toc":{"enable":true,"placement":"left","headingSelector":"h1,h2,h3","collapseDepth":3},"lazyload":{"enable":true,"loading_img":"https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif","onlypost":false,"offset_factor":2},"web_analytics":{"enable":true,"follow_dnt":true,"baidu":"d08d23389220ad9b8f711cfb85757f4b","google":{"measurement_id":null},"tencent":{"sid":null,"cid":null},"leancloud":{"app_id":null,"app_key":null,"server_url":null,"path":"window.location.pathname","ignore_local":false},"umami":{"src":null,"website_id":null,"domains":null,"start_time":"2024-01-01T00:00:00.000Z","token":null,"api_server":null},"woyaola":null,"cnzz":null},"search_path":"/local-search.xml","include_content_in_search":true};
 
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@@ -580,9 +580,9 @@ <h3 id="【二分查找】Bomb">【二分查找】Bomb</h3>
                   <article class="post-prev col-6">
                     
                     
-                      <a href="/Algorithm/a_template/" title="a_template">
+                      <a href="/Algorithm/data-structure/" title="data-structure">
                         <i class="iconfont icon-arrowleft"></i>
-                        <span class="hidden-mobile">a_template</span>
+                        <span class="hidden-mobile">data-structure</span>
                         <span class="visible-mobile">上一篇</span>
                       </a>
                     
@@ -590,8 +590,8 @@ <h3 id="【二分查找】Bomb">【二分查找】Bomb</h3>
                   <article class="post-next col-6">
                     
                     
-                      <a href="/Algorithm/divide-and-conquer/" title="divide-and-conquer">
-                        <span class="hidden-mobile">divide-and-conquer</span>
+                      <a href="/Algorithm/dfs-and-similar/" title="dfs-and-similar">
+                        <span class="hidden-mobile">dfs-and-similar</span>
                         <span class="visible-mobile">下一篇</span>
                         <i class="iconfont icon-arrowright"></i>
                       </a>
diff --git a/Algorithm/data-structure/index.html b/Algorithm/data-structure/index.html
index da9a8fa64..59e113c70 100644
--- a/Algorithm/data-structure/index.html
+++ b/Algorithm/data-structure/index.html
@@ -86,7 +86,7 @@
   <script id="fluid-configs">
     var Fluid = window.Fluid || {};
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@@ -792,9 +792,9 @@ <h2 id="【线段树-二分】以组为单位订音乐会的门票-fire">【线
                   <article class="post-prev col-6">
                     
                     
-                      <a href="/Algorithm/dfs-and-similar/" title="dfs-and-similar">
+                      <a href="/Algorithm/a_template/" title="a_template">
                         <i class="iconfont icon-arrowleft"></i>
-                        <span class="hidden-mobile">dfs-and-similar</span>
+                        <span class="hidden-mobile">a_template</span>
                         <span class="visible-mobile">上一篇</span>
                       </a>
                     
@@ -802,8 +802,8 @@ <h2 id="【线段树-二分】以组为单位订音乐会的门票-fire">【线
                   <article class="post-next col-6">
                     
                     
-                      <a href="/Algorithm/dp/" title="dp">
-                        <span class="hidden-mobile">dp</span>
+                      <a href="/Algorithm/binary-search/" title="binary-search">
+                        <span class="hidden-mobile">binary-search</span>
                         <span class="visible-mobile">下一篇</span>
                         <i class="iconfont icon-arrowright"></i>
                       </a>
diff --git a/Algorithm/dfs-and-similar/index.html b/Algorithm/dfs-and-similar/index.html
index 99121d25c..c8484b043 100644
--- a/Algorithm/dfs-and-similar/index.html
+++ b/Algorithm/dfs-and-similar/index.html
@@ -85,7 +85,7 @@
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@@ -857,9 +857,9 @@ <h3 id="【dfs】将石头分散到网格图的最少移动次数">【dfs】将
                   <article class="post-prev col-6">
                     
                     
-                      <a href="/Algorithm/divide-and-conquer/" title="divide-and-conquer">
+                      <a href="/Algorithm/binary-search/" title="binary-search">
                         <i class="iconfont icon-arrowleft"></i>
-                        <span class="hidden-mobile">divide-and-conquer</span>
+                        <span class="hidden-mobile">binary-search</span>
                         <span class="visible-mobile">上一篇</span>
                       </a>
                     
@@ -867,8 +867,8 @@ <h3 id="【dfs】将石头分散到网格图的最少移动次数">【dfs】将
                   <article class="post-next col-6">
                     
                     
-                      <a href="/Algorithm/data-structure/" title="data-structure">
-                        <span class="hidden-mobile">data-structure</span>
+                      <a href="/Algorithm/divide-and-conquer/" title="divide-and-conquer">
+                        <span class="hidden-mobile">divide-and-conquer</span>
                         <span class="visible-mobile">下一篇</span>
                         <i class="iconfont icon-arrowright"></i>
                       </a>
diff --git a/Algorithm/divide-and-conquer/index.html b/Algorithm/divide-and-conquer/index.html
index 291a6f71d..2f3819536 100644
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+++ b/Algorithm/divide-and-conquer/index.html
@@ -80,7 +80,7 @@
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@@ -425,9 +425,9 @@ <h3 id="【分治】随机排列">【分治】随机排列</h3>
                   <article class="post-prev col-6">
                     
                     
-                      <a href="/Algorithm/binary-search/" title="binary-search">
+                      <a href="/Algorithm/dfs-and-similar/" title="dfs-and-similar">
                         <i class="iconfont icon-arrowleft"></i>
-                        <span class="hidden-mobile">binary-search</span>
+                        <span class="hidden-mobile">dfs-and-similar</span>
                         <span class="visible-mobile">上一篇</span>
                       </a>
                     
@@ -435,8 +435,8 @@ <h3 id="【分治】随机排列">【分治】随机排列</h3>
                   <article class="post-next col-6">
                     
                     
-                      <a href="/Algorithm/dfs-and-similar/" title="dfs-and-similar">
-                        <span class="hidden-mobile">dfs-and-similar</span>
+                      <a href="/Algorithm/dp/" title="dp">
+                        <span class="hidden-mobile">dp</span>
                         <span class="visible-mobile">下一篇</span>
                         <i class="iconfont icon-arrowright"></i>
                       </a>
diff --git a/Algorithm/dp/index.html b/Algorithm/dp/index.html
index d2def6fc5..2494759eb 100644
--- a/Algorithm/dp/index.html
+++ b/Algorithm/dp/index.html
@@ -93,7 +93,7 @@
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       var dntVal = navigator.doNotTrack || window.doNotTrack || navigator.msDoNotTrack;
@@ -979,9 +979,9 @@ <h3 id="【状压dp】Avoid-K-Palindrome-fire">【状压dp】Avoid K Palindrome
                   <article class="post-prev col-6">
                     
                     
-                      <a href="/Algorithm/data-structure/" title="data-structure">
+                      <a href="/Algorithm/divide-and-conquer/" title="divide-and-conquer">
                         <i class="iconfont icon-arrowleft"></i>
-                        <span class="hidden-mobile">data-structure</span>
+                        <span class="hidden-mobile">divide-and-conquer</span>
                         <span class="visible-mobile">上一篇</span>
                       </a>
                     
diff --git a/Algorithm/games/index.html b/Algorithm/games/index.html
index 104bd6080..d2e465f23 100644
--- a/Algorithm/games/index.html
+++ b/Algorithm/games/index.html
@@ -80,7 +80,7 @@
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@@ -426,9 +426,9 @@ <h3 id="【博弈-贪心-交互】Salyg1n-and-the-MEX-Game">【博弈/贪心/交
                   <article class="post-prev col-6">
                     
                     
-                      <a href="/Algorithm/hashing/" title="hashing">
+                      <a href="/BackEnd/back-end-guide/" title="back-end-guide">
                         <i class="iconfont icon-arrowleft"></i>
-                        <span class="hidden-mobile">hashing</span>
+                        <span class="hidden-mobile">back-end-guide</span>
                         <span class="visible-mobile">上一篇</span>
                       </a>
                     
@@ -436,8 +436,8 @@ <h3 id="【博弈-贪心-交互】Salyg1n-and-the-MEX-Game">【博弈/贪心/交
                   <article class="post-next col-6">
                     
                     
-                      <a href="/Algorithm/graphs/" title="graphs">
-                        <span class="hidden-mobile">graphs</span>
+                      <a href="/Algorithm/geometry/" title="geometry">
+                        <span class="hidden-mobile">geometry</span>
                         <span class="visible-mobile">下一篇</span>
                         <i class="iconfont icon-arrowright"></i>
                       </a>
diff --git a/Algorithm/geometry/index.html b/Algorithm/geometry/index.html
index 1eb0bf30b..946a7cbce 100644
--- a/Algorithm/geometry/index.html
+++ b/Algorithm/geometry/index.html
@@ -82,7 +82,7 @@
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@@ -472,9 +472,9 @@ <h3 id="【凸包】奶牛过马路">【凸包】奶牛过马路</h3>
                   <article class="post-prev col-6">
                     
                     
-                      <a href="/Algorithm/z_logic/" title="z_logic">
+                      <a href="/Algorithm/games/" title="games">
                         <i class="iconfont icon-arrowleft"></i>
-                        <span class="hidden-mobile">z_logic</span>
+                        <span class="hidden-mobile">games</span>
                         <span class="visible-mobile">上一篇</span>
                       </a>
                     
@@ -482,8 +482,8 @@ <h3 id="【凸包】奶牛过马路">【凸包】奶牛过马路</h3>
                   <article class="post-next col-6">
                     
                     
-                      <a href="/Algorithm/hashing/" title="hashing">
-                        <span class="hidden-mobile">hashing</span>
+                      <a href="/Algorithm/graphs/" title="graphs">
+                        <span class="hidden-mobile">graphs</span>
                         <span class="visible-mobile">下一篇</span>
                         <i class="iconfont icon-arrowright"></i>
                       </a>
diff --git a/Algorithm/graphs/index.html b/Algorithm/graphs/index.html
index fe08e4ecb..f0bff5238 100644
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@@ -694,9 +694,9 @@ <h3 id="【LCA】树的直径">【LCA】树的直径</h3>
                   <article class="post-prev col-6">
                     
                     
-                      <a href="/Algorithm/games/" title="games">
+                      <a href="/Algorithm/geometry/" title="geometry">
                         <i class="iconfont icon-arrowleft"></i>
-                        <span class="hidden-mobile">games</span>
+                        <span class="hidden-mobile">geometry</span>
                         <span class="visible-mobile">上一篇</span>
                       </a>
                     
@@ -704,8 +704,8 @@ <h3 id="【LCA】树的直径">【LCA】树的直径</h3>
                   <article class="post-next col-6">
                     
                     
-                      <a href="/Algorithm/number-theory/" title="number-theory">
-                        <span class="hidden-mobile">number-theory</span>
+                      <a href="/Algorithm/greedy/" title="greedy">
+                        <span class="hidden-mobile">greedy</span>
                         <span class="visible-mobile">下一篇</span>
                         <i class="iconfont icon-arrowright"></i>
                       </a>
diff --git a/Algorithm/greedy/index.html b/Algorithm/greedy/index.html
index 6acc64e0a..b1c88b80f 100644
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@@ -809,9 +809,9 @@ <h3 id="【按位贪心-分类讨论】">【按位贪心/分类讨论】</h3>
                   <article class="post-prev col-6">
                     
                     
-                      <a href="/Algorithm/number-theory/" title="number-theory">
+                      <a href="/Algorithm/graphs/" title="graphs">
                         <i class="iconfont icon-arrowleft"></i>
-                        <span class="hidden-mobile">number-theory</span>
+                        <span class="hidden-mobile">graphs</span>
                         <span class="visible-mobile">上一篇</span>
                       </a>
                     
@@ -819,8 +819,8 @@ <h3 id="【按位贪心-分类讨论】">【按位贪心/分类讨论】</h3>
                   <article class="post-next col-6">
                     
                     
-                      <a href="/Algorithm/prefix-and-difference/" title="prefix-and-difference">
-                        <span class="hidden-mobile">prefix-and-difference</span>
+                      <a href="/Algorithm/hashing/" title="hashing">
+                        <span class="hidden-mobile">hashing</span>
                         <span class="visible-mobile">下一篇</span>
                         <i class="iconfont icon-arrowright"></i>
                       </a>
diff --git a/Algorithm/hashing/index.html b/Algorithm/hashing/index.html
index 179ac83b7..fe8fba200 100644
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@@ -466,9 +466,9 @@ <h3 id="【哈希-枚举-思维】Torn-Lucky-Ticket">【哈希/枚举/思维】T
                   <article class="post-prev col-6">
                     
                     
-                      <a href="/Algorithm/geometry/" title="geometry">
+                      <a href="/Algorithm/greedy/" title="greedy">
                         <i class="iconfont icon-arrowleft"></i>
-                        <span class="hidden-mobile">geometry</span>
+                        <span class="hidden-mobile">greedy</span>
                         <span class="visible-mobile">上一篇</span>
                       </a>
                     
@@ -476,8 +476,8 @@ <h3 id="【哈希-枚举-思维】Torn-Lucky-Ticket">【哈希/枚举/思维】T
                   <article class="post-next col-6">
                     
                     
-                      <a href="/Algorithm/games/" title="games">
-                        <span class="hidden-mobile">games</span>
+                      <a href="/Algorithm/number-theory/" title="number-theory">
+                        <span class="hidden-mobile">number-theory</span>
                         <span class="visible-mobile">下一篇</span>
                         <i class="iconfont icon-arrowright"></i>
                       </a>
diff --git a/Algorithm/number-theory/index.html b/Algorithm/number-theory/index.html
index f510346b2..f6a3e8491 100644
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@@ -474,9 +474,9 @@ <h3 id="【组合数学】序列数量">【组合数学】序列数量</h3>
                   <article class="post-prev col-6">
                     
                     
-                      <a href="/Algorithm/graphs/" title="graphs">
+                      <a href="/Algorithm/hashing/" title="hashing">
                         <i class="iconfont icon-arrowleft"></i>
-                        <span class="hidden-mobile">graphs</span>
+                        <span class="hidden-mobile">hashing</span>
                         <span class="visible-mobile">上一篇</span>
                       </a>
                     
@@ -484,8 +484,8 @@ <h3 id="【组合数学】序列数量">【组合数学】序列数量</h3>
                   <article class="post-next col-6">
                     
                     
-                      <a href="/Algorithm/greedy/" title="greedy">
-                        <span class="hidden-mobile">greedy</span>
+                      <a href="/Algorithm/prefix-and-difference/" title="prefix-and-difference">
+                        <span class="hidden-mobile">prefix-and-difference</span>
                         <span class="visible-mobile">下一篇</span>
                         <i class="iconfont icon-arrowright"></i>
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diff --git a/Algorithm/prefix-and-difference/index.html b/Algorithm/prefix-and-difference/index.html
index 0f3679228..2841c9275 100644
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@@ -463,9 +463,9 @@ <h3 id="【差分-贪心】增减序列">【差分/贪心】增减序列</h3>
                   <article class="post-prev col-6">
                     
                     
-                      <a href="/Algorithm/greedy/" title="greedy">
+                      <a href="/Algorithm/number-theory/" title="number-theory">
                         <i class="iconfont icon-arrowleft"></i>
-                        <span class="hidden-mobile">greedy</span>
+                        <span class="hidden-mobile">number-theory</span>
                         <span class="visible-mobile">上一篇</span>
                       </a>
                     
diff --git a/Algorithm/z_logic/index.html b/Algorithm/z_logic/index.html
index f7ffb7cc9..6779d9196 100644
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                   <article class="post-prev col-6">
                     
                     
-                      <a href="/BackEnd/back-end-guide/" title="back-end-guide">
+                      <a href="/DevTools/CLion/solve-clion-cannot-open-relative-file/" title="solve-clion-cannot-open-relative-file">
                         <i class="iconfont icon-arrowleft"></i>
-                        <span class="hidden-mobile">back-end-guide</span>
+                        <span class="hidden-mobile">solve-clion-cannot-open-relative-file</span>
                         <span class="visible-mobile">上一篇</span>
                       </a>
                     
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                   <article class="post-next col-6">
                     
                     
-                      <a href="/Algorithm/geometry/" title="geometry">
-                        <span class="hidden-mobile">geometry</span>
+                      <a href="/BackEnd/back-end-guide/" title="back-end-guide">
+                        <span class="hidden-mobile">back-end-guide</span>
                         <span class="visible-mobile">下一篇</span>
                         <i class="iconfont icon-arrowright"></i>
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                   <article class="post-next col-6">
                     
                     
-                      <a href="/RoadMap/road-map-guide/" title="road-map-guide">
-                        <span class="hidden-mobile">road-map-guide</span>
+                      <a href="/Operation/operation-guide/" title="operation-guide">
+                        <span class="hidden-mobile">operation-guide</span>
                         <span class="visible-mobile">下一篇</span>
                         <i class="iconfont icon-arrowright"></i>
                       </a>
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index 7dc2eabdb..6d093759d 100644
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                   <article class="post-prev col-6">
                     
                     
-                      <a href="/DevTools/CLion/solve-clion-cannot-open-relative-file/" title="solve-clion-cannot-open-relative-file">
+                      <a href="/Algorithm/z_logic/" title="z_logic">
                         <i class="iconfont icon-arrowleft"></i>
-                        <span class="hidden-mobile">solve-clion-cannot-open-relative-file</span>
+                        <span class="hidden-mobile">z_logic</span>
                         <span class="visible-mobile">上一篇</span>
                       </a>
                     
@@ -433,8 +433,8 @@ <h2 id="参考">参考</h2>
                   <article class="post-next col-6">
                     
                     
-                      <a href="/Algorithm/z_logic/" title="z_logic">
-                        <span class="hidden-mobile">z_logic</span>
+                      <a href="/Algorithm/games/" title="games">
+                        <span class="hidden-mobile">games</span>
                         <span class="visible-mobile">下一篇</span>
                         <i class="iconfont icon-arrowright"></i>
                       </a>
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index e2890990d..1e3169a85 100644
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index 521fe5823..e8ad45941 100644
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@@ -421,9 +421,9 @@ <h2 id="参考">参考</h2>
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-                      <a href="/DevTools/CLion/solve-clion-decoding-error/" title="solve-clion-decoding-error">
+                      <a href="/DevTools/DevCpp/devc-self-config/" title="self-config">
                         <i class="iconfont icon-arrowleft"></i>
-                        <span class="hidden-mobile">solve-clion-decoding-error</span>
+                        <span class="hidden-mobile">self-config</span>
                         <span class="visible-mobile">上一篇</span>
                       </a>
                     
diff --git a/DataBase/openGauss/opengauss-basic/index.html b/DataBase/openGauss/opengauss-basic/index.html
index c9c7bf0ac..d7cf9cace 100644
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-                      <a href="/DevTools/Git/git-self-define-command/" title="git-self-define-command">
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+                      <a href="/DevTools/CLion/solve-clion-decoding-error/" title="solve-clion-decoding-error">
+                        <span class="hidden-mobile">solve-clion-decoding-error</span>
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-                      <a href="/FrontEnd/Hexo/hexo-migrate/" title="hexo-migrate">
-                        <span class="hidden-mobile">hexo-migrate</span>
+                      <a href="/FrontEnd/Hexo/hexo-learning-record/" title="hexo-learning-record">
+                        <span class="hidden-mobile">hexo-learning-record</span>
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-                      <a href="/FrontEnd/Hexo/hexo-migrate/" title="hexo-migrate">
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-                        <span class="hidden-mobile">hexo-migrate</span>
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-                        <span class="hidden-mobile">front-end-guide</span>
+                      <a href="/FrontEnd/Hexo/hexo-migrate/" title="hexo-migrate">
+                        <span class="hidden-mobile">hexo-migrate</span>
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-                      <a href="/FrontEnd/Hexo/hexo-githubpages-domain/" title="hexo-githubpages-domain">
+                      <a href="/FrontEnd/Hexo/hexo-learning-record/" title="hexo-learning-record">
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-                        <span class="hidden-mobile">hexo-githubpages-domain</span>
+                        <span class="hidden-mobile">hexo-learning-record</span>
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+                      <a href="/FrontEnd/front-end-guide/" title="front-end-guide">
+                        <span class="hidden-mobile">front-end-guide</span>
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@@ -443,9 +443,9 @@ <h2 id="参考">参考</h2>
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-                      <a href="/FrontEnd/Hexo/hexo-learning-record/" title="hexo-learning-record">
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-                        <span class="hidden-mobile">hexo-learning-record</span>
+                        <span class="hidden-mobile">hexo-migrate</span>
                         <span class="visible-mobile">上一篇</span>
                       </a>
                     
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                   <article class="post-next col-6">
                     
                     
-                      <a href="/GPA/3rd-term/DataStructure/" title="DataStructure">
-                        <span class="hidden-mobile">DataStructure</span>
+                      <a href="/GPA/3rd-term/DigitalLogicCircuit/" title="DigitalLogicCircuit">
+                        <span class="hidden-mobile">DigitalLogicCircuit</span>
                         <span class="visible-mobile">下一篇</span>
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-                      <a href="/GPA/3rd-term/CollegePhysics_2/" title="CollegePhysics_2">
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-                        <span class="hidden-mobile">CollegePhysics_2</span>
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-                      <a href="/GPA/3rd-term/DigitalLogicCircuit/" title="DigitalLogicCircuit">
-                        <span class="hidden-mobile">DigitalLogicCircuit</span>
+                      <a href="/GPA/3rd-term/LinearAlgebra/" title="LinearAlgebra">
+                        <span class="hidden-mobile">LinearAlgebra</span>
                         <span class="visible-mobile">下一篇</span>
                         <i class="iconfont icon-arrowright"></i>
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-                      <a href="/GPA/3rd-term/DataStructureClassDesign/" title="DataStructureClassDesign">
+                      <a href="/GPA/3rd-term/CollegePhysics_2/" title="CollegePhysics_2">
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-                        <span class="hidden-mobile">DataStructureClassDesign</span>
+                        <span class="hidden-mobile">CollegePhysics_2</span>
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-                      <a href="/GPA/3rd-term/LinearAlgebra/" title="LinearAlgebra">
-                        <span class="hidden-mobile">LinearAlgebra</span>
+                      <a href="/GPA/3rd-term/DataStructure/" title="DataStructure">
+                        <span class="hidden-mobile">DataStructure</span>
                         <span class="visible-mobile">下一篇</span>
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@@ -1177,9 +1177,9 @@ <h4 id="行列式角度">行列式角度</h4>
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-                      <a href="/GPA/3rd-term/DigitalLogicCircuit/" title="DigitalLogicCircuit">
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-                        <span class="hidden-mobile">DigitalLogicCircuit</span>
+                        <span class="hidden-mobile">DataStructureClassDesign</span>
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index 388ccf784..c8d858cfd 100644
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@@ -1969,9 +1969,9 @@ <h2 id="考试大纲">考试大纲</h2>
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-                      <a href="/GPA/4th-term/PyAlgo/" title="PyAlgo">
+                      <a href="/GPA/4th-term/OptMethod/" title="OptMethod">
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-                        <span class="hidden-mobile">PyAlgo</span>
+                        <span class="hidden-mobile">OptMethod</span>
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@@ -1979,8 +1979,8 @@ <h2 id="考试大纲">考试大纲</h2>
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-                        <span class="hidden-mobile">ProbAndStat</span>
+                      <a href="/GPA/4th-term/PyAlgo/" title="PyAlgo">
+                        <span class="hidden-mobile">PyAlgo</span>
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-                        <span class="hidden-mobile">ProbAndStat</span>
+                        <span class="hidden-mobile">PyApply</span>
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-                      <a href="/GPA/4th-term/SysBasic/" title="SysBasic">
-                        <span class="hidden-mobile">SysBasic</span>
+                      <a href="/GPA/4th-term/MachineLearning/" title="MachineLearning">
+                        <span class="hidden-mobile">MachineLearning</span>
                         <span class="visible-mobile">下一篇</span>
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-                      <a href="/GPA/4th-term/MachineLearning/" title="MachineLearning">
+                      <a href="/GPA/4th-term/PyAlgo/" title="PyAlgo">
                         <i class="iconfont icon-arrowleft"></i>
-                        <span class="hidden-mobile">MachineLearning</span>
+                        <span class="hidden-mobile">PyAlgo</span>
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                   <article class="post-next col-6">
                     
                     
-                      <a href="/GPA/4th-term/OptMethod/" title="OptMethod">
-                        <span class="hidden-mobile">OptMethod</span>
+                      <a href="/GPA/4th-term/SysBasic/" title="SysBasic">
+                        <span class="hidden-mobile">SysBasic</span>
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-                      <a href="/GPA/4th-term/PyApply/" title="PyApply">
+                      <a href="/GPA/4th-term/MachineLearning/" title="MachineLearning">
                         <i class="iconfont icon-arrowleft"></i>
-                        <span class="hidden-mobile">PyApply</span>
+                        <span class="hidden-mobile">MachineLearning</span>
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-                      <a href="/GPA/4th-term/MachineLearning/" title="MachineLearning">
-                        <span class="hidden-mobile">MachineLearning</span>
+                      <a href="/GPA/4th-term/ProbAndStat/" title="ProbAndStat">
+                        <span class="hidden-mobile">ProbAndStat</span>
                         <span class="visible-mobile">下一篇</span>
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                   <article class="post-prev col-6">
                     
                     
-                      <a href="/Operation/operation-guide/" title="operation-guide">
+                      <a href="/RoadMap/road-map-guide/" title="road-map-guide">
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-                        <span class="hidden-mobile">operation-guide</span>
+                        <span class="hidden-mobile">road-map-guide</span>
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                       </a>
                     
@@ -503,8 +503,8 @@ <h2 id="项目三">项目三</h2>
                   <article class="post-next col-6">
                     
                     
-                      <a href="/GPA/4th-term/PyAlgo/" title="PyAlgo">
-                        <span class="hidden-mobile">PyAlgo</span>
+                      <a href="/GPA/4th-term/OptMethod/" title="OptMethod">
+                        <span class="hidden-mobile">OptMethod</span>
                         <span class="visible-mobile">下一篇</span>
                         <i class="iconfont icon-arrowright"></i>
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@@ -1272,9 +1272,9 @@ <h4 id="4-4-2-重定位过程">4.4.2 重定位过程</h4>
                   <article class="post-prev col-6">
                     
                     
-                      <a href="/GPA/4th-term/OptMethod/" title="OptMethod">
+                      <a href="/GPA/4th-term/ProbAndStat/" title="ProbAndStat">
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-                        <span class="hidden-mobile">OptMethod</span>
+                        <span class="hidden-mobile">ProbAndStat</span>
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 <meta property="article:published_time" content="2024-08-27T12:14:48.819Z">
-<meta property="article:modified_time" content="2024-09-29T03:38:12.614Z">
+<meta property="article:modified_time" content="2024-10-08T15:03:58.061Z">
 <meta property="article:author" content="Mr_Dwj">
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@@ -338,7 +338,7 @@ <h1 id="seo-header">ComputerOrganization</h1>
               <p id="updated-time" class="note note-info" style="">
                 
                   
-                    本文最后更新于 2024年9月29日 中午
+                    本文最后更新于 2024年10月8日 晚上
                   
                 
               </p>
@@ -422,8 +422,7 @@ <h2 id="前言">前言</h2>
 </blockquote>
 <p>会收获什么？</p>
 <blockquote>
-<p>对计算机整体有一个宏观的把握，并且能够知道计算机各个模块之间是如何关联与调度的，以及每一个模块内部的工作原理。</p>
-<p>最后能够从逻辑上实现一个单核 CPU。</p>
+<p>对计算机整体有一个宏观的把握，了解 CPU、存储和 I/O 设备的工作逻辑，尤其要熟悉单核 CPU 的工作逻辑。最后能够利用 verilog 硬件描述语言从逻辑上实现一个单核 CPU。</p>
 </blockquote>
 <p>注：</p>
 <blockquote>
@@ -629,7 +628,7 @@ <h2 id="九、并行处理系统">九、并行处理系统 *</h2>
       
         <div class="license-meta-item license-meta-date">
           <div>更新于</div>
-          <div>2024年9月29日</div>
+          <div>2024年10月8日</div>
         </div>
       
       
diff --git a/GPA/5th-term/DataBase/index.html b/GPA/5th-term/DataBase/index.html
index 3f5f659bc..138b1568f 100644
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diff --git a/GPA/5th-term/DigitalImageProcessing/index.html b/GPA/5th-term/DigitalImageProcessing/index.html
index 61acc7301..079d43543 100644
--- a/GPA/5th-term/DigitalImageProcessing/index.html
+++ b/GPA/5th-term/DigitalImageProcessing/index.html
@@ -26,11 +26,19 @@
 <meta property="og:site_name" content="Dwj の Blog">
 <meta property="og:description" content="数字图像处理 前言 学科地位：    主讲教师 学分配额 学科类别     李峻 3 专业课    成绩组成：    作业 实验 期中 期末     10% 10% 30% 50%    教材情况：    课程名称 选用教材 版次 作者 出版社 ISBN 号     数字图像处理 《数字图像处理 使用 MATLAB 分析与实现》 2 蔡利梅 王利娟 清华大学出版社 9787302654100">
 <meta property="og:locale" content="zh_CN">
+<meta property="og:image" content="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410082035267.png">
+<meta property="og:image" content="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410082044689.png">
+<meta property="og:image" content="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410082043712.png">
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+<meta property="og:image" content="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410082105689.jpg">
+<meta property="og:image" content="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410082115300.png">
 <meta property="article:published_time" content="2024-08-27T12:17:29.191Z">
-<meta property="article:modified_time" content="2024-09-30T08:08:47.940Z">
+<meta property="article:modified_time" content="2024-10-08T13:17:36.851Z">
 <meta property="article:author" content="Mr_Dwj">
 <meta property="article:tag" content="人工智能 机器学习 深度学习 算法 数据结构 计算机科学 编程">
 <meta name="twitter:card" content="summary_large_image">
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@@ -245,7 +253,7 @@
       <span class="post-meta mr-2">
         <i class="iconfont icon-chart"></i>
         
-          1.1k 字
+          3.7k 字
         
       </span>
     
@@ -256,7 +264,7 @@
         
         
         
-          8 分钟
+          23 分钟
         
       </span>
     
@@ -320,7 +328,7 @@ <h1 id="seo-header">DigitalImageProcessing</h1>
               <p id="updated-time" class="note note-info" style="">
                 
                   
-                    本文最后更新于 2024年9月30日 下午
+                    本文最后更新于 2024年10月8日 晚上
                   
                 
               </p>
@@ -399,17 +407,14 @@ <h2 id="前言">前言</h2>
 </table>
 <p>为什么要学这门课？</p>
 <blockquote>
-<p>说实在的我也不知道这门课的意义到底有多大，感觉与目前为止主流的炼丹或者 sys 的关系不大，更像是某个分支，比如平面图形学。但是转念一想，图像其实也是一种数据，也可以用来炼丹；同样的，二维图像的各种处理技术也对应了很多的数学原理和算法策略。这么一想好像也就和其他的课程串起来了。</p>
+<p>图像是数据的一种，图像的应用场景极为广泛，因此如何获取、处理、存储数据也就显得十分重要。在本课程中我们将主要学习图像的处理技术。当然，如果非要说数字图像处理和别的课程的关联，愚以为既然图像是数据，那么就可以用到数据算法那一套上，也就与别的课程有了关联。</p>
 </blockquote>
 <p>会收获什么？</p>
 <blockquote>
-<p><strong>先从编程说起</strong>。终于有机会可以正式的任务驱动式的学习 MATLAB 编程和 OpenCV (Python) 编程了。</p>
-<p><strong>当然，没有理论何谈编程</strong>？图像处理的数学理论和算法策略，也会是一个全新且不小的收获，且学且更新。目前看来与炼丹并不是一回事，与传统的数据结构也没有很大的交集。更多的是一种矩阵论、线性代数论。</p>
+<p>图像简单来说就是一个矩阵。如何在这种数据结构上进行理论研究与算法应用显得十分有学习意义。本课程也将使用 MATLAB 和 OpenCV-Python 进行编码实现。</p>
 </blockquote>
 <h2 id="图像获取">图像获取</h2>
-<p>TODO</p>
-<h2 id="图像处理">图像处理</h2>
-<p>图像是数据存储和表达的一种方式。然而模拟图像无法进行更深入的研究，于是乎，数字图像应运而生。数字图像处理技术也就有了更多的研究意义与应用价值。</p>
+<p><strong>如何从现实世界中的模拟图像转化为计算机世界的数字图像</strong>？由于现实世界中的模拟图像是连续的，而计算机无法处理连续数据，因此我们只能对模拟图像进行离散化采样。进而引出了下面的一些概念。</p>
 <p><strong>像素是什么</strong>？一张模拟图像被离散化后的 <strong>最小单元</strong>。</p>
 <p><strong>分辨率是什么</strong>？一张模拟图像被离散化后的 <strong>像素数量</strong>。例如，如果一张模拟图像被划分为由 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2560</mn><mo>×</mo><mn>1920</mn></mrow><annotation encoding="application/x-tex">2560\times1920</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2560</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1920</span></span></span></span> 个像素点组成的数字图像，那么这张数字图像的分辨率就是像素点的数量，约 500 万。</p>
 <p><strong>在单一变量原则的情况下，分辨率越高，数字图像就越清晰吗</strong>？显然不是。由分辨率的定义，一张图像的分辨率表示图像中像素点的个数。如果 A, B 两张图的分辨率一致，但是 A 图的面积远超过 B，那么显然的 B 比 A 更清晰。因为 A 的一像素对应的信息量太多了，不够精细，也就导致 A 更模糊。因此我们引出了图像清晰度的度量理念。</p>
@@ -436,31 +441,69 @@ <h2 id="图像处理">图像处理</h2>
 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord">DPI</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0463em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">每英寸的距离</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">打印的点数</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
 </li>
 </ul>
-<h3 id="图像运算">图像运算</h3>
-<p>首先需要掌握一些最基本的概念：齐次坐标表示、变换矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span>、图像插值策略。</p>
-<h4 id="几何变换">几何变换</h4>
-<p>图像的几何变换基本步骤我们称之为 <strong>后向映射法</strong>。逻辑很简单，根据原图像的四个顶点计算出新图像的尺寸后，逐个填充新图像的像素值即可。那么问题来了：如何计算新图像的尺寸？如何填充新图像的像素值？</p>
-<ul>
-<li>平移</li>
-<li>镜像</li>
-<li>旋转</li>
-<li>缩放</li>
-<li>错切</li>
-</ul>
-<h4 id="代数运算">代数运算</h4>
-<h4 id="领域及模版运算">领域及模版运算</h4>
-<h3 id="图像正交变换">图像正交变换</h3>
-<p>本章我们重点学习傅里叶变换。</p>
-<h3 id="图像增强">图像增强</h3>
-<h3 id="图像平滑">图像平滑</h3>
-<h3 id="图像锐化">图像锐化</h3>
-<h3 id="图像复原">图像复原</h3>
-<h3 id="图像的数学形态学处理">图像的数学形态学处理</h3>
-<h2 id="图像编码">图像编码</h2>
+<p><strong>离散化采样出来的数字图像是如何量化表示模拟图像的</strong>？介绍三种，分别是二值图像、灰度图像和彩色图像。其中二值图像中，每一个像素点就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 或 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，灰度图像中，每一个像素点取值在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mn>255</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0,255]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">255</span><span class="mclose">]</span></span></span></span> 之间，彩色图像中每一个像素点是一个三通道向量，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi><mo separator="true">,</mo><mi>G</mi><mo separator="true">,</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">R,G,B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">G</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> 的取值均在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mn>255</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0,255]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">255</span><span class="mclose">]</span></span></span></span> 之间。</p>
+<h2 id="图像编码">图像编码 *</h2>
+<p>这一部分本课程不作要求，但愚以为这是十分重要的一个部分。</p>
+<p>承接上一部分的「图像获取」技术。现在我们有了图像数据，就需要对其进行传输和存储，但是图像的数据量极大，如果不进行压缩编码，必将在传输和存储的过程中造成极大的消耗，因此图像的压缩编码是很重要的。目前主要有两种压缩编码策略，分别是 <strong>无损压缩编码</strong> 和 <strong>有损压缩编码</strong>。</p>
+<h2 id="图像处理">图像处理</h2>
+<p>有了图像数据，我们就可以进行最基本、最底层语义的图像处理，进而才能有后续更高语义上的 <strong>图像分析</strong> 和 <strong>图像理解</strong>。这一部分我们将从图像的各个处理策略入手，分别进行学习。</p>
+<h3 id="1-图像基本运算">1. 图像基本运算</h3>
+<p>首先需要掌握一些最基本的概念：齐次坐标表示与变换矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span>、图像插值策略、图像映射策略。</p>
+<p><strong>关于齐次坐标表示和变换矩阵</strong>。就是将图像中的点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><msup><mo stretchy="false">)</mo><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">(x,y)^T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0913em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span></span></span> 用 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">(x,y,1)^T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0913em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span></span></span> 来表示。之所以要这样表示，从 <a target="_blank" rel="noopener" href="https://blog.csdn.net/wangmj_hdu/article/details/119143771">这篇博客</a> 可以看出是为了「统一图像变换的运算形式」，但应该没这么简单，可能还有很多别的意义。至于所谓的变换矩阵，其实就是利用矩阵乘法对图像的像素点转换进行格式上的统一运算，所有的变换操作都可以统一为对像素点进行矩阵乘法，例如：对于当前的一个像素点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">p=(x,y,1)^T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0913em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span></span></span> 和一个变换矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>k</mi><mi>x</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>k</mi><mi>y</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><mi>y</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>p</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>q</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>s</mi></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \begin{bmatrix} k_x &amp; b &amp; \Delta x \\ c &amp; k_y &amp;\Delta y \\ p &amp; q &amp; s \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.6em;vertical-align:-1.55em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M403 1759 V84 H666 V0 H319 V1759 v0 v1759 h347 v-84
+H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Δ</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M347 1759 V0 H0 V84 H263 V1759 v0 v1759 H0 v84 H347z
+M347 1759 V0 H263 V1759 v0 v1759 h84z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span></span></span></span>。新像素点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>p</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mo stretchy="false">(</mo><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo separator="true">,</mo><msup><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo separator="true">,</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">p'=(x',y',1)^T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9463em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0913em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span></span></span> 可以通过矩阵乘法 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>p</mi><mrow><mn>3</mn><mo>×</mo><mn>1</mn></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>=</mo><msub><mi>T</mi><mrow><mn>3</mn><mo>×</mo><mn>3</mn></mrow></msub><mo>×</mo><msub><mi>p</mi><mrow><mn>3</mn><mo>×</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">p'_{3\times 1}=T_{3\times 3} \times p_{3\times 1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0583em;vertical-align:-0.3064em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-2.4519em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span><span class="mbin mtight">×</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3064em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span><span class="mbin mtight">×</span><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span><span class="mbin mtight">×</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span> 得到，即：</p>
+<p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msup><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>k</mi><mi>x</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>k</mi><mi>y</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><mi>y</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>p</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>q</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>s</mi></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>x</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>y</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} =
+\begin{bmatrix} k_x &amp; b &amp; \Delta x \\ c &amp; k_y &amp;\Delta y \\ p &amp; q &amp; s \end{bmatrix}
+\begin{bmatrix} x \\ y\\ 1 \end{bmatrix}
+</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.6em;vertical-align:-1.55em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M403 1759 V84 H666 V0 H319 V1759 v0 v1759 h347 v-84
+H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M347 1759 V0 H0 V84 H263 V1759 v0 v1759 H0 v84 H347z
+M347 1759 V0 H263 V1759 v0 v1759 h84z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.6em;vertical-align:-1.55em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M403 1759 V84 H666 V0 H319 V1759 v0 v1759 h347 v-84
+H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Δ</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M347 1759 V0 H0 V84 H263 V1759 v0 v1759 H0 v84 H347z
+M347 1759 V0 H263 V1759 v0 v1759 h84z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M403 1759 V84 H666 V0 H319 V1759 v0 v1759 h347 v-84
+H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M347 1759 V0 H0 V84 H263 V1759 v0 v1759 H0 v84 H347z
+M347 1759 V0 H263 V1759 v0 v1759 h84z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
+<p><strong>关于图像插值策略</strong>。常用的有三种，分别是最邻近差值、双线性插值、双三次插值。其中最邻近插值很显然，双三次插值最精准但是计算量很大，因此最常用的还是双线性插值。</p>
+<p><strong>关于图像映射策略</strong>。有前向映射法和后向映射法，常用的是后向映射法，因此讲讲后向映射法的流程。逻辑很简单，根据原图像的「四个顶点」与「几何变换逻辑」确定新图像的位置后，基于几何变换的 <strong>逆变换</strong> 逐个填充新图像的像素值即可。填充策略也很显然，当我们对新图像的像素点使用逆变换寻找在原图像的对应点时，会发生以下两种情况：</p>
+<ol>
+<li>新图像在原图像的对应点超过了原图像的范围。说明这个像素点是由于几何变换而凭空出现的，直接赋背景色即可。</li>
+<li>新图像在原图像的对应点没超过原图像的范围。说明这个像素点一定是有实际意义的，此时又有两种情况：
+<ol>
+<li>对应点存在的。说明找到了「最佳配对」，这是最完美的对应情况，直接赋原图的像素值即可；</li>
+<li>对应点不存在。说明没有最佳配对，此时我们只能退而求其次给这个像素点生成一个配对的像素值，怎么生成呢？就可以用到上面介绍的插值策略了。</li>
+</ol>
+</li>
+</ol>
+<h4 id="1-1-几何变换">1.1 几何变换</h4>
+<p>有了上述的基础知识之后，就可以对图像进行下面 5 种常见的几何变换了。</p>
+<p><strong>平移</strong></p>
+<p><strong>镜像</strong></p>
+<p><strong>旋转</strong></p>
+<p><strong>缩放</strong></p>
+<p><strong>错切</strong></p>
+
+    <div class="fold">
+      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-1d81f2f5" role="button" aria-expanded="false" aria-controls="collapse-1d81f2f5">
+        <div class="fold-arrow">▶</div>实战演练
+      </div>
+      <div class="fold-collapse collapse" id="collapse-1d81f2f5">
+        <div class="fold-content">
+          <p>一、放大+双线性插值</p><p>图像为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>4</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>7</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>5</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>8</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>3</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>6</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>9</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mstyle></mrow><annotation encoding="application/x-tex">f=\displaystyle \begin{bmatrix} 1 &amp; 4 &amp; 7 \\ 2 &amp; 5 &amp; 8 \\ 3 &amp; 6 &amp; 9 \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.6em;vertical-align:-1.55em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M403 1759 V84 H666 V0 H319 V1759 v0 v1759 h347 v-84H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">5</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">7</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">8</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">9</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M347 1759 V0 H0 V84 H263 V1759 v0 v1759 H0 v84 H347zM347 1759 V0 H263 V1759 v0 v1759 h84z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span></span></span></span>，放大比例为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mi>x</mi></msub><mo>=</mo><mn>2.3</mn><mo separator="true">,</mo><msub><mi>k</mi><mi>y</mi></msub><mo>=</mo><mn>1.6</mn></mrow><annotation encoding="application/x-tex">k_x=2.3,k_y=1.6</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord">2.3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1.6</span></span></span></span>。</p><p>MATLAB 代码：</p><figure class="highlight matlab"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br></pre></td><td class="code"><pre><code class="hljs matlab">f = [<span class="hljs-number">1</span> <span class="hljs-number">4</span> <span class="hljs-number">7</span>; <span class="hljs-number">2</span> <span class="hljs-number">5</span> <span class="hljs-number">8</span>; <span class="hljs-number">3</span> <span class="hljs-number">6</span> <span class="hljs-number">9</span>];<br>k_x = <span class="hljs-number">2.3</span>;<br>k_y = <span class="hljs-number">1.6</span>;<br><br><span class="hljs-comment">% 计算新图像的尺寸</span><br>[original_height, original_width] = <span class="hljs-built_in">size</span>(f);<br>new_height = <span class="hljs-built_in">round</span>(original_height * k_y);<br>new_width = <span class="hljs-built_in">round</span>(original_width * k_x);<br><br><span class="hljs-comment">% 后向映射法填充所有像素</span><br>g = <span class="hljs-built_in">zeros</span>(new_height, new_width);<br><span class="hljs-keyword">for</span> <span class="hljs-built_in">i</span> = <span class="hljs-number">1</span>:new_height<br>    <span class="hljs-keyword">for</span> <span class="hljs-built_in">j</span> = <span class="hljs-number">1</span>:new_width<br>        <span class="hljs-comment">% 计算原图像中的坐标</span><br>        orig_x = (<span class="hljs-built_in">j</span> - <span class="hljs-number">0.5</span>) / k_x + <span class="hljs-number">0.5</span>;<br>        orig_y = (<span class="hljs-built_in">i</span> - <span class="hljs-number">0.5</span>) / k_y + <span class="hljs-number">0.5</span>;<br><br>        <span class="hljs-comment">% 找到原图像中的邻近像素</span><br>        x1 = <span class="hljs-built_in">floor</span>(orig_x);<br>        x2 = <span class="hljs-built_in">ceil</span>(orig_x);<br>        y1 = <span class="hljs-built_in">floor</span>(orig_y);<br>        y2 = <span class="hljs-built_in">ceil</span>(orig_y);<br><br>        <span class="hljs-comment">% 限制边界</span><br>        x1 = <span class="hljs-built_in">max</span>(<span class="hljs-built_in">min</span>(x1, original_width), <span class="hljs-number">1</span>);<br>        x2 = <span class="hljs-built_in">max</span>(<span class="hljs-built_in">min</span>(x2, original_width), <span class="hljs-number">1</span>);<br>        y1 = <span class="hljs-built_in">max</span>(<span class="hljs-built_in">min</span>(y1, original_height), <span class="hljs-number">1</span>);<br>        y2 = <span class="hljs-built_in">max</span>(<span class="hljs-built_in">min</span>(y2, original_height), <span class="hljs-number">1</span>);<br><br>        <span class="hljs-comment">% 双线性插值</span><br>        Q11 = f(y1, x1);<br>        Q12 = f(y1, x2);<br>        Q21 = f(y2, x1);<br>        Q22 = f(y2, x2);<br>        <br>        <span class="hljs-comment">% 插值计算</span><br>        g(<span class="hljs-built_in">i</span>, <span class="hljs-built_in">j</span>) = (Q11 * (x2 - orig_x) * (y2 - orig_y) + ...<br>                   Q12 * (orig_x - x1) * (y2 - orig_y) + ...<br>                   Q21 * (x2 - orig_x) * (orig_y - y1) + ...<br>                   Q22 * (orig_x - x1) * (orig_y - y1));<br>    <span class="hljs-keyword">end</span><br><span class="hljs-keyword">end</span><br><br><span class="hljs-comment">% 显示结果</span><br><span class="hljs-built_in">disp</span>(g)<br></code></pre></td></tr></tbody></table></figure><p>运行结果：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410082035267.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="运行结果"></p><p>OpenCV-Python 代码：</p><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">import</span> numpy <span class="hljs-keyword">as</span> np<br><br>f = np.array([[<span class="hljs-number">1</span>, <span class="hljs-number">4</span>, <span class="hljs-number">7</span>], [<span class="hljs-number">2</span>, <span class="hljs-number">5</span>, <span class="hljs-number">8</span>], [<span class="hljs-number">3</span>, <span class="hljs-number">6</span>, <span class="hljs-number">9</span>]], dtype=np.float32)<br>k_x = <span class="hljs-number">2.3</span><br>k_y = <span class="hljs-number">1.6</span><br><br><span class="hljs-comment"># 计算新图像的尺寸</span><br>original_height, original_width = f.shape<br>new_height = <span class="hljs-built_in">round</span>(original_height * k_y)<br>new_width = <span class="hljs-built_in">round</span>(original_width * k_x)<br><br><span class="hljs-comment"># 后向映射法填充所有像素</span><br>g = np.zeros((new_height, new_width), dtype=np.float32)<br><span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(new_height):<br>    <span class="hljs-keyword">for</span> j <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(new_width):<br>        <span class="hljs-comment"># 计算原图像中的坐标</span><br>        orig_x = (j + <span class="hljs-number">0.5</span>) / k_x - <span class="hljs-number">0.5</span><br>        orig_y = (i + <span class="hljs-number">0.5</span>) / k_y - <span class="hljs-number">0.5</span><br><br>        <span class="hljs-comment"># 找到原图像中的邻近像素</span><br>        x1 = <span class="hljs-built_in">int</span>(np.floor(orig_x))<br>        x2 = <span class="hljs-built_in">int</span>(np.ceil(orig_x))<br>        y1 = <span class="hljs-built_in">int</span>(np.floor(orig_y))<br>        y2 = <span class="hljs-built_in">int</span>(np.ceil(orig_y))<br><br>        <span class="hljs-comment"># 限制边界</span><br>        x1 = <span class="hljs-built_in">max</span>(<span class="hljs-built_in">min</span>(x1, original_width - <span class="hljs-number">1</span>), <span class="hljs-number">0</span>)<br>        x2 = <span class="hljs-built_in">max</span>(<span class="hljs-built_in">min</span>(x2, original_width - <span class="hljs-number">1</span>), <span class="hljs-number">0</span>)<br>        y1 = <span class="hljs-built_in">max</span>(<span class="hljs-built_in">min</span>(y1, original_height - <span class="hljs-number">1</span>), <span class="hljs-number">0</span>)<br>        y2 = <span class="hljs-built_in">max</span>(<span class="hljs-built_in">min</span>(y2, original_height - <span class="hljs-number">1</span>), <span class="hljs-number">0</span>)<br><br>        <span class="hljs-comment"># 双线性插值</span><br>        Q11 = f[y1, x1]<br>        Q12 = f[y1, x2]<br>        Q21 = f[y2, x1]<br>        Q22 = f[y2, x2]<br><br>        <span class="hljs-comment"># 插值计算</span><br>        g[i, j] = (Q11 * (x2 - orig_x) * (y2 - orig_y) +<br>                   Q12 * (orig_x - x1) * (y2 - orig_y) +<br>                   Q21 * (x2 - orig_x) * (orig_y - y1) +<br>                   Q22 * (orig_x - x1) * (orig_y - y1))<br><br><span class="hljs-comment"># 显示结果</span><br><span class="hljs-keyword">for</span> line <span class="hljs-keyword">in</span> g:<br>    <span class="hljs-keyword">for</span> d <span class="hljs-keyword">in</span> line:<br>        <span class="hljs-built_in">print</span>(<span class="hljs-string">f'<span class="hljs-subst">{d:<span class="hljs-number">.4</span>f}</span>'</span>, end=<span class="hljs-string">' '</span>)<br>    <span class="hljs-built_in">print</span>()<br></code></pre></td></tr></tbody></table></figure><p>运行结果：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410082044689.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="运行结果"></p><p>二、旋转+双线性插值</p><p>图像为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>4</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>7</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>5</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>8</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>3</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>6</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>9</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mstyle></mrow><annotation encoding="application/x-tex">f=\displaystyle \begin{bmatrix} 1 &amp; 4 &amp; 7 \\ 2 &amp; 5 &amp; 8 \\ 3 &amp; 6 &amp; 9 \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.6em;vertical-align:-1.55em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M403 1759 V84 H666 V0 H319 V1759 v0 v1759 h347 v-84H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">5</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">7</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">8</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">9</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M347 1759 V0 H0 V84 H263 V1759 v0 v1759 H0 v84 H347zM347 1759 V0 H263 V1759 v0 v1759 h84z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span></span></span></span>，旋转角度为逆时针 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>60</mn></mrow><annotation encoding="application/x-tex">60</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">60</span></span></span></span> 度。</p><p>MATLAB 代码：</p><figure class="highlight matlab"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br></pre></td><td class="code"><pre><code class="hljs matlab">f = [<span class="hljs-number">1</span> <span class="hljs-number">4</span> <span class="hljs-number">7</span>; <span class="hljs-number">2</span> <span class="hljs-number">5</span> <span class="hljs-number">8</span>; <span class="hljs-number">3</span> <span class="hljs-number">6</span> <span class="hljs-number">9</span>];<br><br><span class="hljs-comment">% 计算新图像尺寸</span><br>theta = <span class="hljs-number">60</span>; <span class="hljs-comment">% 角度</span><br>theta_rad = deg2rad(theta); <span class="hljs-comment">% 转换为弧度</span><br>[original_height, original_width] = <span class="hljs-built_in">size</span>(f);<br>new_height = <span class="hljs-built_in">round</span>(<span class="hljs-built_in">abs</span>(original_height * <span class="hljs-built_in">cos</span>(theta_rad)) + <span class="hljs-built_in">abs</span>(original_width * <span class="hljs-built_in">sin</span>(theta_rad)));<br>new_width = <span class="hljs-built_in">round</span>(<span class="hljs-built_in">abs</span>(original_height * <span class="hljs-built_in">sin</span>(theta_rad)) + <span class="hljs-built_in">abs</span>(original_width * <span class="hljs-built_in">cos</span>(theta_rad)));<br><br><span class="hljs-comment">% 后向映射法</span><br>g = <span class="hljs-built_in">zeros</span>(new_height, new_width);<br>center_x = original_width / <span class="hljs-number">2</span>;<br>center_y = original_height / <span class="hljs-number">2</span>;<br><span class="hljs-keyword">for</span> <span class="hljs-built_in">i</span> = <span class="hljs-number">1</span>:new_height<br>    <span class="hljs-keyword">for</span> <span class="hljs-built_in">j</span> = <span class="hljs-number">1</span>:new_width<br>        <span class="hljs-comment">% 计算原图像中的坐标</span><br>        orig_x = (<span class="hljs-built_in">j</span> - new_width / <span class="hljs-number">2</span>) * <span class="hljs-built_in">cos</span>(theta_rad) + (<span class="hljs-built_in">i</span> - new_height / <span class="hljs-number">2</span>) * <span class="hljs-built_in">sin</span>(theta_rad) + center_x;<br>        orig_y = -(<span class="hljs-built_in">j</span> - new_width / <span class="hljs-number">2</span>) * <span class="hljs-built_in">sin</span>(theta_rad) + (<span class="hljs-built_in">i</span> - new_height / <span class="hljs-number">2</span>) * <span class="hljs-built_in">cos</span>(theta_rad) + center_y;<br><br>        <span class="hljs-comment">% 找到原图像中的邻近像素</span><br>        x1 = <span class="hljs-built_in">floor</span>(orig_x);<br>        x2 = <span class="hljs-built_in">ceil</span>(orig_x);<br>        y1 = <span class="hljs-built_in">floor</span>(orig_y);<br>        y2 = <span class="hljs-built_in">ceil</span>(orig_y);<br><br>        <span class="hljs-comment">% 限制边界</span><br>        x1 = <span class="hljs-built_in">max</span>(<span class="hljs-built_in">min</span>(x1, original_width), <span class="hljs-number">1</span>);<br>        x2 = <span class="hljs-built_in">max</span>(<span class="hljs-built_in">min</span>(x2, original_width), <span class="hljs-number">1</span>);<br>        y1 = <span class="hljs-built_in">max</span>(<span class="hljs-built_in">min</span>(y1, original_height), <span class="hljs-number">1</span>);<br>        y2 = <span class="hljs-built_in">max</span>(<span class="hljs-built_in">min</span>(y2, original_height), <span class="hljs-number">1</span>);<br><br>        <span class="hljs-comment">% 双线性插值</span><br>        Q11 = f(y1, x1);<br>        Q12 = f(y1, x2);<br>        Q21 = f(y2, x1);<br>        Q22 = f(y2, x2);<br>        <br>        <span class="hljs-comment">% 插值计算</span><br>        g(<span class="hljs-built_in">i</span>, <span class="hljs-built_in">j</span>) = (Q11 * (x2 - orig_x) * (y2 - orig_y) + ...<br>                   Q12 * (orig_x - x1) * (y2 - orig_y) + ...<br>                   Q21 * (x2 - orig_x) * (orig_y - y1) + ...<br>                   Q22 * (orig_x - x1) * (orig_y - y1));<br>    <span class="hljs-keyword">end</span><br><span class="hljs-keyword">end</span><br><br><span class="hljs-comment">% 显示结果</span><br><span class="hljs-built_in">disp</span>(g)<br></code></pre></td></tr></tbody></table></figure><p>运行结果：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410082043712.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="运行结果"></p><p>OpenCV-Python 代码：</p><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">import</span> numpy <span class="hljs-keyword">as</span> np<br><br>f = np.array([[<span class="hljs-number">1</span>, <span class="hljs-number">4</span>, <span class="hljs-number">7</span>], [<span class="hljs-number">2</span>, <span class="hljs-number">5</span>, <span class="hljs-number">8</span>], [<span class="hljs-number">3</span>, <span class="hljs-number">6</span>, <span class="hljs-number">9</span>]], dtype=np.float32)<br><br><span class="hljs-comment"># 计算新图像尺寸</span><br>theta = <span class="hljs-number">60</span>  <span class="hljs-comment"># 角度</span><br>theta_rad = np.radians(theta)  <span class="hljs-comment"># 转换为弧度</span><br>original_height, original_width = f.shape<br>new_height = <span class="hljs-built_in">int</span>(<span class="hljs-built_in">round</span>(<span class="hljs-built_in">abs</span>(original_height * np.cos(theta_rad)) + <span class="hljs-built_in">abs</span>(original_width * np.sin(theta_rad))))<br>new_width = <span class="hljs-built_in">int</span>(<span class="hljs-built_in">round</span>(<span class="hljs-built_in">abs</span>(original_height * np.sin(theta_rad)) + <span class="hljs-built_in">abs</span>(original_width * np.cos(theta_rad))))<br><br><span class="hljs-comment"># 后向映射法</span><br>g = np.zeros((new_height, new_width), dtype=np.float32)<br>center_x = original_width / <span class="hljs-number">2</span><br>center_y = original_height / <span class="hljs-number">2</span><br><span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(new_height):<br>    <span class="hljs-keyword">for</span> j <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(new_width):<br>        <span class="hljs-comment"># 计算原图像中的坐标</span><br>        orig_x = (j - new_width / <span class="hljs-number">2</span>) * np.cos(theta_rad) + (i - new_height / <span class="hljs-number">2</span>) * np.sin(theta_rad) + center_x<br>        orig_y = -(j - new_width / <span class="hljs-number">2</span>) * np.sin(theta_rad) + (i - new_height / <span class="hljs-number">2</span>) * np.cos(theta_rad) + center_y<br><br>        <span class="hljs-comment"># 找到原图像中的邻近像素</span><br>        x1 = <span class="hljs-built_in">int</span>(np.floor(orig_x))<br>        x2 = <span class="hljs-built_in">int</span>(np.ceil(orig_x))<br>        y1 = <span class="hljs-built_in">int</span>(np.floor(orig_y))<br>        y2 = <span class="hljs-built_in">int</span>(np.ceil(orig_y))<br><br>        <span class="hljs-comment"># 限制边界</span><br>        x1 = <span class="hljs-built_in">max</span>(<span class="hljs-built_in">min</span>(x1, original_width - <span class="hljs-number">1</span>), <span class="hljs-number">0</span>)<br>        x2 = <span class="hljs-built_in">max</span>(<span class="hljs-built_in">min</span>(x2, original_width - <span class="hljs-number">1</span>), <span class="hljs-number">0</span>)<br>        y1 = <span class="hljs-built_in">max</span>(<span class="hljs-built_in">min</span>(y1, original_height - <span class="hljs-number">1</span>), <span class="hljs-number">0</span>)<br>        y2 = <span class="hljs-built_in">max</span>(<span class="hljs-built_in">min</span>(y2, original_height - <span class="hljs-number">1</span>), <span class="hljs-number">0</span>)<br><br>        <span class="hljs-comment"># 双线性插值</span><br>        Q11 = f[y1, x1]<br>        Q12 = f[y1, x2]<br>        Q21 = f[y2, x1]<br>        Q22 = f[y2, x2]<br><br>        <span class="hljs-comment"># 插值计算</span><br>        g[i, j] = (Q11 * (x2 - orig_x) * (y2 - orig_y) +<br>                   Q12 * (orig_x - x1) * (y2 - orig_y) +<br>                   Q21 * (x2 - orig_x) * (orig_y - y1) +<br>                   Q22 * (orig_x - x1) * (orig_y - y1))<br><br><span class="hljs-comment"># 显示结果</span><br><span class="hljs-keyword">for</span> line <span class="hljs-keyword">in</span> g:<br>    <span class="hljs-keyword">for</span> d <span class="hljs-keyword">in</span> line:<br>        <span class="hljs-built_in">print</span>(<span class="hljs-string">f'<span class="hljs-subst">{d:<span class="hljs-number">.4</span>f}</span>'</span>, end=<span class="hljs-string">' '</span>)<br>    <span class="hljs-built_in">print</span>()<br></code></pre></td></tr></tbody></table></figure><p>运行结果：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410082043419.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="运行结果"></p><p>三、旋转+镜像+错切+缩小+双线性插值</p><p>选择一张图片，顺时针旋转 20 度，水平镜像，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mi>x</mi></msub><mo>=</mo><mn>0.3</mn><mo separator="true">,</mo><msub><mi>k</mi><mi>y</mi></msub><mo>=</mo><mn>0.5</mn></mrow><annotation encoding="application/x-tex">k_x=0.3,k_y=0.5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord">0.3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.5</span></span></span></span> 做错切变换，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mi>x</mi></msub><mo>=</mo><mn>0.6</mn><mo separator="true">,</mo><msub><mi>k</mi><mi>y</mi></msub><mo>=</mo><mn>0.6</mn></mrow><annotation encoding="application/x-tex">k_x=0.6,k_y=0.6</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord">0.6</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.6</span></span></span></span> 缩小图片。以下图为例：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410082054211.jpg" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="bird"></p><p>MATLAB 代码：</p><figure class="highlight matlab"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br></pre></td><td class="code"><pre><code class="hljs matlab"><span class="hljs-comment">% 读取图片</span><br>img = imread(<span class="hljs-string">'bird.jpg'</span>);<br>[h, w, ~] = <span class="hljs-built_in">size</span>(img);<br><br><span class="hljs-built_in">figure</span>;<br>subplot(<span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">1</span>), imshow(img), title(<span class="hljs-string">'原始图像'</span>);<br><br><span class="hljs-comment">% 1. 顺时针旋转 20 度</span><br>rotated_img = imrotate(img, <span class="hljs-number">-20</span>, <span class="hljs-string">'bilinear'</span>, <span class="hljs-string">'crop'</span>);<br>subplot(<span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">2</span>), imshow(rotated_img), title(<span class="hljs-string">'顺时针旋转 20 度后'</span>);<br><br><span class="hljs-comment">% 2. 水平镜像</span><br>mirrored_img = flip(rotated_img, <span class="hljs-number">2</span>);<br>subplot(<span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">3</span>), imshow(mirrored_img), title(<span class="hljs-string">'水平镜像后'</span>);<br><br><span class="hljs-comment">% 3. 错切变换</span><br>tform = affine2d([<span class="hljs-number">1</span> <span class="hljs-number">0.3</span> <span class="hljs-number">0</span>; <span class="hljs-number">0.5</span> <span class="hljs-number">1</span> <span class="hljs-number">0</span>; <span class="hljs-number">0</span> <span class="hljs-number">0</span> <span class="hljs-number">1</span>]); <br>sheared_img = imwarp(mirrored_img, tform, <span class="hljs-string">'Interp'</span>, <span class="hljs-string">'bilinear'</span>);<br>subplot(<span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">4</span>), imshow(sheared_img), title(<span class="hljs-string">'错切变换后'</span>);<br><br><span class="hljs-comment">% 4. 缩放并用白色背景填充</span><br>scale_x = <span class="hljs-number">0.6</span>;<br>scale_y = <span class="hljs-number">0.6</span>;<br>scaled_img = imresize(sheared_img, [<span class="hljs-built_in">round</span>(h*scale_y), <span class="hljs-built_in">round</span>(w*scale_x)], <span class="hljs-string">'bilinear'</span>);<br>background = uint8(<span class="hljs-number">255</span> * <span class="hljs-built_in">ones</span>(h, w, <span class="hljs-number">3</span>));<br>x_offset = <span class="hljs-built_in">round</span>((w - <span class="hljs-built_in">size</span>(scaled_img, <span class="hljs-number">2</span>)) / <span class="hljs-number">2</span>);<br>y_offset = <span class="hljs-built_in">round</span>((h - <span class="hljs-built_in">size</span>(scaled_img, <span class="hljs-number">1</span>)) / <span class="hljs-number">2</span>);<br>background(y_offset:(y_offset + <span class="hljs-built_in">size</span>(scaled_img, <span class="hljs-number">1</span>) - <span class="hljs-number">1</span>), x_offset:(x_offset + <span class="hljs-built_in">size</span>(scaled_img, <span class="hljs-number">2</span>) - <span class="hljs-number">1</span>), :) = scaled_img;<br><br>subplot(<span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">5</span>), imshow(background), title(<span class="hljs-string">'缩小后'</span>);<br></code></pre></td></tr></tbody></table></figure><p>运行结果：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410082105689.jpg" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="运行结果"></p><p>OpenCV-Python 代码：</p><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">import</span> cv2<br><span class="hljs-keyword">import</span> numpy <span class="hljs-keyword">as</span> np<br><span class="hljs-keyword">import</span> matplotlib.pyplot <span class="hljs-keyword">as</span> plt<br><br>plt.rcParams[<span class="hljs-string">'font.sans-serif'</span>] = [<span class="hljs-string">'SimHei'</span>]<br>plt.rcParams[<span class="hljs-string">'axes.unicode_minus'</span>] = <span class="hljs-literal">False</span><br><br>img = cv2.imread(<span class="hljs-string">'E:/python/OpenCV/Homework_1/bird.png'</span>)<br>img = cv2.cvtColor(img, cv2.COLOR_BGR2RGB)<br>(h, w) = img.shape[:<span class="hljs-number">2</span>]<br>plt.figure(figsize=(<span class="hljs-number">12</span>, <span class="hljs-number">8</span>))<br><br><span class="hljs-comment"># 显示原始图像</span><br>plt.subplot(<span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">1</span>)<br>plt.imshow(img)<br>plt.title(<span class="hljs-string">'原始图像'</span>)<br><br><span class="hljs-comment"># 1. 顺时针旋转 20 度</span><br>center = (w // <span class="hljs-number">2</span>, h // <span class="hljs-number">2</span>)<br>M = cv2.getRotationMatrix2D(center, -<span class="hljs-number">20</span>, <span class="hljs-number">1.0</span>)<br>rotated_img = cv2.warpAffine(img, M, (w, h), flags=cv2.INTER_LINEAR)<br>plt.subplot(<span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">2</span>)<br>plt.imshow(rotated_img)<br>plt.title(<span class="hljs-string">'旋转 20 度后'</span>)<br><br><span class="hljs-comment"># 2. 水平镜像</span><br>mirrored_img = cv2.flip(rotated_img, <span class="hljs-number">1</span>)<br>plt.subplot(<span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">3</span>)<br>plt.imshow(mirrored_img)<br>plt.title(<span class="hljs-string">'水平镜像后'</span>)<br><br><span class="hljs-comment"># 3. 错切变换</span><br>kx, ky = <span class="hljs-number">0.3</span>, <span class="hljs-number">0.5</span><br>M_shear = np.array([[<span class="hljs-number">1</span>, kx, <span class="hljs-number">0</span>], [ky, <span class="hljs-number">1</span>, <span class="hljs-number">0</span>]], dtype=np.float32)<br><span class="hljs-comment"># sheared_img = cv2.warpAffine(mirrored_img, M_shear, (w, h), flags=cv2.INTER_LINEAR)</span><br>sheared_img = cv2.warpAffine(mirrored_img, M_shear, (<span class="hljs-built_in">int</span>(w + h * kx), <span class="hljs-built_in">int</span>(h + w * ky)), flags=cv2.INTER_LINEAR)<br>plt.subplot(<span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">4</span>)<br>plt.imshow(sheared_img)<br>plt.title(<span class="hljs-string">'错切变换后'</span>)<br><br><span class="hljs-comment"># 4. 缩放并添加白色背景</span><br>kx, ky = <span class="hljs-number">0.6</span>, <span class="hljs-number">0.6</span><br>scaled_img = cv2.resize(sheared_img, (<span class="hljs-number">0</span>, <span class="hljs-number">0</span>), fx=kx, fy=ky, interpolation=cv2.INTER_LINEAR)<br>background = np.ones_like(sheared_img, dtype=np.uint8) * <span class="hljs-number">255</span><br>x_offset = (background.shape[<span class="hljs-number">1</span>] - scaled_img.shape[<span class="hljs-number">1</span>]) // <span class="hljs-number">2</span><br>y_offset = (background.shape[<span class="hljs-number">0</span>] - scaled_img.shape[<span class="hljs-number">0</span>]) // <span class="hljs-number">2</span><br>background[y_offset:y_offset + scaled_img.shape[<span class="hljs-number">0</span>], x_offset:x_offset + scaled_img.shape[<span class="hljs-number">1</span>]] = scaled_img<br>plt.subplot(<span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">5</span>)<br>plt.imshow(background)<br>plt.title(<span class="hljs-string">'缩小后'</span>)<br><br><span class="hljs-comment"># 显示所有子图</span><br>plt.tight_layout()<br>plt.show()<br></code></pre></td></tr></tbody></table></figure><p>运行结果：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410082115300.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="运行结果"></p>
+        </div>
+      </div>
+    </div>
+<h4 id="1-2-代数运算">1.2 代数运算</h4>
+<h4 id="1-3-模版运算">1.3 模版运算</h4>
+<h3 id="2-图像正交变换">2. 图像正交变换</h3>
+<h3 id="3-图像增强">3. 图像增强</h3>
+<h3 id="4-图像平滑">4. 图像平滑</h3>
+<h3 id="5-图像锐化">5. 图像锐化</h3>
+<h3 id="6-图像复原">6. 图像复原</h3>
+<h3 id="7-图像的数学形态学处理">7. 图像的数学形态学处理</h3>
+<h3 id="8-图像分割">8. 图像分割</h3>
 <h2 id="图像分析">图像分析</h2>
-<h3 id="图像分割">图像分割</h3>
-<h3 id="图像描述与分析">图像描述与分析</h3>
+<p>基于上一部分最底层语义的「图像处理」技术，本部分将展开更高一层语义「图像分析」技术的学习。</p>
 <h2 id="综合应用">综合应用</h2>
+<p>基于上一部分中层语义的「图像分析」技术，本部分将结合 <strong>图像分类</strong> 和 <strong>图像检索</strong> 技术展开最高层语义「图像理解」技术的学习。</p>
 <h3 id="图像分类">图像分类</h3>
 <h3 id="图像检索">图像检索</h3>
 
@@ -527,7 +570,7 @@ <h3 id="图像检索">图像检索</h3>
       
         <div class="license-meta-item license-meta-date">
           <div>更新于</div>
-          <div>2024年9月30日</div>
+          <div>2024年10月8日</div>
         </div>
       
       
diff --git a/GPA/5th-term/NeuralNetworkAndDeepLearning/index.html b/GPA/5th-term/NeuralNetworkAndDeepLearning/index.html
index 0ff6de77d..db9812c51 100644
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+++ b/GPA/5th-term/NeuralNetworkAndDeepLearning/index.html
@@ -28,7 +28,7 @@
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 <meta property="og:image" content="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409121508150.png">
 <meta property="article:published_time" content="2024-08-27T11:56:39.488Z">
-<meta property="article:modified_time" content="2024-09-29T12:09:54.910Z">
+<meta property="article:modified_time" content="2024-10-07T16:07:42.898Z">
 <meta property="article:author" content="Mr_Dwj">
 <meta property="article:tag" content="人工智能 机器学习 深度学习 算法 数据结构 计算机科学 编程">
 <meta name="twitter:card" content="summary_large_image">
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@@ -247,7 +247,7 @@
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         <i class="iconfont icon-chart"></i>
         
-          991 字
+          1k 字
         
       </span>
     
@@ -322,7 +322,7 @@ <h1 id="seo-header">NeuralNetworkAndDeepLearning</h1>
               <p id="updated-time" class="note note-info" style="">
                 
                   
-                    本文最后更新于 2024年9月29日 晚上
+                    本文最后更新于 2024年10月8日 凌晨
                   
                 
               </p>
@@ -410,15 +410,15 @@ <h3 id="绪论">绪论</h3>
 <h3 id="机器学习概述">机器学习概述</h3>
 <p>数据 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> 模型 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> 学习准则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> 优化算法</p>
 <h3 id="线性模型">线性模型</h3>
-<p>学习任务：回归、分类。</p>
+<p>学习任务：分类、回归。</p>
 <p>学习准则：</p>
 <ul>
-<li>经验风险最小化：通过 <strong>最小化训练集上的损失函数</strong> 来学习模型参数。</li>
+<li>经验风险最小化：通过 <strong>最小化训练集上的损失函数</strong> 来学习模型参数。例如：线性回归中的最小二乘损失。</li>
 <li>结构风险最小化：引入 <strong>正则化</strong> 来控制模型的复杂度，避免过拟合。</li>
-<li>最大似然估计：通过 <strong>最大化模型在给定数据上的似然函数</strong> 来估计模型参数。</li>
+<li>最大似然估计：通过 <strong>最大化模型在给定数据上的似然函数</strong> 来估计模型参数。例如二分类 logistic 和多分类 softmax 中的交叉熵损失。</li>
 <li>最大后验估计：结合了似然函数和 <strong>先验</strong> 分布，在贝叶斯框架下估计模型参数。</li>
 </ul>
-<p>优化算法：最小二乘法、梯度下降法。</p>
+<p>优化算法：最小二乘法、梯度下降法、拟牛顿法。</p>
 <h2 id="有监督学习">有监督学习</h2>
 <h3 id="基础神经网络模型">基础神经网络模型</h3>
 <h4 id="前馈神经网络">前馈神经网络</h4>
@@ -502,7 +502,7 @@ <h2 id="个人大作业">个人大作业</h2>
       
         <div class="license-meta-item license-meta-date">
           <div>更新于</div>
-          <div>2024年9月29日</div>
+          <div>2024年10月8日</div>
         </div>
       
       
diff --git a/GPA/5th-term/OperatingSystem/index.html b/GPA/5th-term/OperatingSystem/index.html
index 77c4ec81e..9cb6494bb 100644
--- a/GPA/5th-term/OperatingSystem/index.html
+++ b/GPA/5th-term/OperatingSystem/index.html
@@ -19,20 +19,21 @@
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-    <meta name="description" content="操作系统 学科地位：    主讲教师 学分配额 学科类别     段博佳 4+1 专业课    成绩组成：    理论课  实验课      平时 10% 小组作业 100%   期中 30%     期末 60%      教材情况：    课程名称 选用教材 版次 作者 出版社 ISBN 号     操作系统原理 《操作系统教程》 6 骆斌 葛季栋 费翔林 高等教育出版社 978-7-04-0">
+    <meta name="description" content="操作系统 前言 学科地位：    主讲教师 学分配额 学科类别     段博佳 4+1 专业课    成绩组成：    理论课  实验课      平时 10% 小组作业 100%   期中 30%     期末 60%      教材情况：    课程名称 选用教材 版次 作者 出版社 ISBN 号     操作系统原理 《操作系统教程》 6 骆斌 葛季栋 费翔林 高等教育出版社 978-7-0">
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-<meta property="og:description" content="操作系统 学科地位：    主讲教师 学分配额 学科类别     段博佳 4+1 专业课    成绩组成：    理论课  实验课      平时 10% 小组作业 100%   期中 30%     期末 60%      教材情况：    课程名称 选用教材 版次 作者 出版社 ISBN 号     操作系统原理 《操作系统教程》 6 骆斌 葛季栋 费翔林 高等教育出版社 978-7-04-0">
+<meta property="og:description" content="操作系统 前言 学科地位：    主讲教师 学分配额 学科类别     段博佳 4+1 专业课    成绩组成：    理论课  实验课      平时 10% 小组作业 100%   期中 30%     期末 60%      教材情况：    课程名称 选用教材 版次 作者 出版社 ISBN 号     操作系统原理 《操作系统教程》 6 骆斌 葛季栋 费翔林 高等教育出版社 978-7-0">
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+<meta property="og:image" content="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081443847.png">
 <meta property="article:published_time" content="2024-08-27T12:02:23.289Z">
-<meta property="article:modified_time" content="2024-09-29T07:38:34.766Z">
+<meta property="article:modified_time" content="2024-10-08T08:43:07.175Z">
 <meta property="article:author" content="Mr_Dwj">
 <meta property="article:tag" content="人工智能 机器学习 深度学习 算法 数据结构 计算机科学 编程">
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@@ -251,7 +252,7 @@
       <span class="post-meta mr-2">
         <i class="iconfont icon-chart"></i>
         
-          3.4k 字
+          3.6k 字
         
       </span>
     
@@ -262,7 +263,7 @@
         
         
         
-          21 分钟
+          23 分钟
         
       </span>
     
@@ -326,7 +327,7 @@ <h1 id="seo-header">OperatingSystem</h1>
               <p id="updated-time" class="note note-info" style="">
                 
                   
-                    本文最后更新于 2024年9月29日 下午
+                    本文最后更新于 2024年10月8日 下午
                   
                 
               </p>
@@ -335,6 +336,7 @@ <h1 id="seo-header">OperatingSystem</h1>
               <div class="markdown-body">
                 
                 <h2 id="操作系统">操作系统</h2>
+<h2 id="前言">前言</h2>
 <p>学科地位：</p>
 <table>
 <thead>
@@ -532,13 +534,17 @@ <h2 id="绪论">绪论</h2>
 <h2 id="并发">并发</h2>
 <h3 id="进程控制与管理">进程控制与管理</h3>
 <p>CPU 的核心关键是并发程序，为此每一个硬件设计与相应 OS 的运行逻辑，都和 <strong>进程管理、资源调度、并发</strong> 息息相关。</p>
+<h4 id="进程的定义">进程的定义</h4>
 <p><strong>进程和程序的区别是什么？动态性</strong>。程序是一个指令序列，当程序载入进程时就转换为了进程。</p>
+<h4 id="进程的状态">进程的状态</h4>
 <p>五态模型：</p>
 <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409241450175.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="五态模型 - 状态转换"></p>
 <p>七态模型：</p>
 <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409242052428.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="七态模型 - 状态转换"></p>
+<h4 id="进程的描述">进程的描述</h4>
 <p><strong>如何准确描述一个进程？通过进程映象</strong>。我们引入进程映象的概念。所谓进程映象就是某一时刻的进程状态，可以描述为以下 4 个部分：</p>
 <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409270845459.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="进程映象"></p>
+<h4 id="进程的组织">进程的组织</h4>
 <p><strong>操作系统如何跟踪每一个进程？通过 PCB</strong>。每一个进程都会有很多信息，OS 设计者将其统一存储在进程控制块 (Process Control Block, 简称 PCB) 中。每一个进程都有自己的 PCB，只要是跟进程管理相关的数据信息一定都存储于 PCB 中。PCB 中存储的内容主要有以下内容：</p>
 <ul>
 <li>
@@ -567,11 +573,15 @@ <h3 id="进程控制与管理">进程控制与管理</h3>
 </li>
 </ul>
 <p><strong>当有多个进程处于同一个状态时，操作系统如何访存不同的进程呢？通过等待队列存储 PCB 以及优先级调度算法实现</strong>。我们将状态相同的进程存放到同一个容器（如链表队列）中，操作系统拥有每一个容器的指针。容器中相同状态的不同进程可能会按照优先级的顺序链接。那么容器中存的是什么呢？从上文可以知道，每一个 PCB 都可以唯一确定一个进程，因此等待队列中只需要存储 PCB 即可。</p>
+<h4 id="进程的切换">进程的切换</h4>
 <p><strong>并发执行程序时，如何切换状态？如何继续运行原进程呢？通过中断/系统调用切换状态，通过 PCB 保存现场数据</strong>。PCB 中的寄存器可以在切换进程时保存必要的数据，确保进程在中断获得运行权后能够从刚才的状态继续运行。这个「触发中断、保存现场、处理中断、恢复现场、结束中断」的过程被称为进程上下文切换。如下图所示：</p>
 <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409270937298.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="进程上下文切换"></p>
-<p><strong>如果在执行中断任务时又中断了呢？这是不会发生的</strong>。很多操作是不允许被打断的，例如保存现场等和数据极其重要的操作。我们就需要最小操作单元，这被称作「原语」操作。</p>
+<h4 id="进程的最小执行单元">进程的最小执行单元</h4>
+<p><strong>如果在执行中断任务时又中断了呢？这是不会发生的</strong>。很多操作是不允许被打断的我们就需要最小操作单元，这被称作「原语」操作。例如更新进程信息（修改 PCB 中的信息）、更新进程状态（修改 PCB 所在的状态队列）、分配/回收资源。</p>
+<h4 id="线程">线程</h4>
 <p><strong>线程是什么</strong>？线程是从进程引申出来的概念与技术。进程是资源分配和调度的最小单位，而线程是 CPU 调度的最小单位。一个进程下可以有多个线程，每一个线程共享当前进程的所有资源，线程只会有执行指令最基本的资源（PC、寄存器等）。</p>
 <p><strong>为什么会有线程</strong>？当一个进程的任务很复杂以至于也需要并发时，线程就可以充当更细力度的并发单元。也就是说线程的出现是为了解决越来越复杂的进程多任务并发问题的。</p>
+<p><strong>多线程的几种模式是什么？分别有用户级线程、内核级线程和混合级线程</strong>。其中用户级线程 (User Level Thread) 的多线程逻辑顾名思义，就是由程序开发者定义，便于解决「逻辑并行」的问题。内核级线程 (Kernel Level Thread) 的多线程则由系统调用决定，便于解决「物理并行」的问题。</p>
 <h3 id="互斥">互斥</h3>
 <h3 id="同步">同步</h3>
 <h3 id="死锁">死锁</h3>
@@ -579,6 +589,15 @@ <h2 id="虚拟化">虚拟化</h2>
 <h2 id="内核">内核</h2>
 <h3 id="中断-系统调用-上下文切换">中断 | 系统调用 | 上下文切换</h3>
 <h3 id="处理器调度">处理器调度</h3>
+<p><strong>处理器调度的三级模式</strong>。分为高级调度、中级调度和低级调度。其中：</p>
+<ul>
+<li>高级调度是决定哪些进程从外存对换到内存中，也就是分配内存资源。</li>
+<li>中级调度是决定哪些进程从内存对换到外存中。</li>
+<li>低级调度是决定哪些内存中的进程可以获得合适的 CPU 计算资源。</li>
+</ul>
+<p><strong>处理器调度的评价指标</strong>。</p>
+<p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081443847.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="处理器调度的评价指标"></p>
+<p><strong>处理器调度的七种算法</strong>。</p>
 <h2 id="持久化">持久化</h2>
 
                 
@@ -644,7 +663,7 @@ <h2 id="持久化">持久化</h2>
       
         <div class="license-meta-item license-meta-date">
           <div>更新于</div>
-          <div>2024年9月29日</div>
+          <div>2024年10月8日</div>
         </div>
       
       
diff --git a/Operation/Nginx/nginx-basic/index.html b/Operation/Nginx/nginx-basic/index.html
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diff --git a/Operation/Nginx/solve-nginx-reload-error/index.html b/Operation/Nginx/solve-nginx-reload-error/index.html
index ec0afed25..eb94b7dc8 100644
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diff --git a/Operation/Shell/shell-basic/index.html b/Operation/Shell/shell-basic/index.html
index 0b432a3e6..e4bafecca 100644
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+<meta property="og:image" content="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081735300.png">
+<meta property="og:image" content="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081736525.png">
+<meta property="og:image" content="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081737564.png">
+<meta property="og:image" content="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081739971.png">
+<meta property="og:image" content="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081748528.png">
+<meta property="og:image" content="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081748818.png">
+<meta property="og:image" content="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081813738.png">
+<meta property="og:image" content="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081807870.png">
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+<meta property="og:image" content="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081948301.png">
+<meta property="og:image" content="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081949586.png">
+<meta property="og:image" content="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081951803.png">
 <meta property="article:published_time" content="2024-09-14T01:42:01.082Z">
-<meta property="article:modified_time" content="2024-10-02T09:55:24.296Z">
+<meta property="article:modified_time" content="2024-10-08T15:25:04.826Z">
 <meta property="article:author" content="Mr_Dwj">
 <meta property="article:tag" content="人工智能 机器学习 深度学习 算法 数据结构 计算机科学 编程">
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@@ -254,7 +266,7 @@
       <span class="post-meta mr-2">
         <i class="iconfont icon-chart"></i>
         
-          3.4k 字
+          4k 字
         
       </span>
     
@@ -265,7 +277,7 @@
         
         
         
-          22 分钟
+          26 分钟
         
       </span>
     
@@ -329,7 +341,7 @@ <h1 id="seo-header">shell-basic</h1>
               <p id="updated-time" class="note note-info" style="">
                 
                   
-                    本文最后更新于 2024年10月2日 下午
+                    本文最后更新于 2024年10月8日 晚上
                   
                 
               </p>
@@ -425,7 +437,7 @@ <h4 id="匹配-grep">匹配 grep *</h4>
 <ul>
 <li>option: <code>-n</code> 显示行号，<code>-i</code> 忽略大小写，<code>-r</code> 递归搜索，<code>-c</code> 打印匹配数量。</li>
 </ul>
-<h2 id="用户管理-权限管理（实验2）">用户管理 | 权限管理（实验2）</h2>
+<h2 id="用户与权限管理（实验2）">用户与权限管理（实验2）</h2>
 <p>在 Windows 中，我们对用户和权限的概念接触的并不多，因为很多东西都默认设置好了；但是在 Linux 中，很多文件的权限都需要自己配置和定义，因此「用户与权限管理」的操作方法十分重要。我们从现象入手逐个进行讲解。</p>
 <p>首先以 root 用户身份登录并进入 <code>/opt/OS/task2/</code> 目录，然后创建一个测试文件 <code>root_file.txt</code> 和一个测试文件夹 <code>root_folder</code>。使用 <code>ls -l</code> 命令列出当前目录下所有文件的详细信息：</p>
 <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409271003057.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="root 用户创建的文件和文件夹"></p>
@@ -567,18 +579,63 @@ <h4 id="默认权限-umask">默认权限 umask</h4>
 <p>可能是出于安全考虑，文件默认不允许拥有可执行权限，因此如果 <code>umask</code> 显示为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0022</mn></mrow><annotation encoding="application/x-tex">0022</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0022</span></span></span></span>，则创建的文件默认权限为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>644</mn></mrow><annotation encoding="application/x-tex">644</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">644</span></span></span></span>，即每一位都 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span> 以确保是偶数。</p>
 
     <div class="fold">
-      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-7af5d8f3" role="button" aria-expanded="false" aria-controls="collapse-7af5d8f3">
+      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-79236e41" role="button" aria-expanded="false" aria-controls="collapse-79236e41">
         <div class="fold-arrow">▶</div>练习
       </div>
-      <div class="fold-collapse collapse" id="collapse-7af5d8f3">
+      <div class="fold-collapse collapse" id="collapse-79236e41">
         <div class="fold-content">
-          <p>一、添加 4 个用户：alice、bob、john、mike</p><p>首先需要确保当前是 root 用户，使用 <code>su root</code> 切换到 root 用户。然后在创建用户时同时创建该用户对应的目录：</p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><code class="hljs bash">useradd -d /home/alice -m alice<br>useradd -d /home/bob -m bob<br>useradd -d /home/john -m john<br>useradd -d /home/mike -m mike<br></code></pre></td></tr></tbody></table></figure><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409272339074.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="添加 4 个用户"></p><p>二、为 alice 设置密码</p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">passwd alice<br></code></pre></td></tr></tbody></table></figure><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409272340323.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="为 alice 设置密码"></p><p>三、创建用户组 workgroup 并将 alice、bob、john 加入</p><ul><li><p>创建用户组：</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">groupadd workgroup<br></code></pre></td></tr></tbody></table></figure></li><li><p>添加到新组：</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><code class="hljs bash">usermod -a -G workgroup alice<br>usermod -a -G workgroup bob<br>usermod -a -G workgroup john<br></code></pre></td></tr></tbody></table></figure><ul><li><code>-a</code>：是 <code>--append</code> 的缩写，表示将用户添加到一个组，而不会移除她已有的其他组。这个选项必须与 <code>-G</code> 一起使用</li><li><code>-G</code>：指定要添加用户的附加组（即用户可以属于多个组），这里是 workgroup</li></ul></li><li><p>将 workgroup 作为各自的主组：</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><code class="hljs bash">usermod -g workgroup alice<br>usermod -g workgroup bob<br>usermod -g workgroup john<br></code></pre></td></tr></tbody></table></figure><ul><li><code>-g</code>：用于指定用户的主组（primary group）。主组是当用户创建文件或目录时默认分配的组</li></ul></li></ul><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409272341900.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="创建用户组 workgroup 并将 alice、bob、john 加入"></p><p>四、创建 <code>/home/work</code> 目录并将其属主改为 alice，属组改为 workgroup</p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-comment"># 创建目录</span><br><span class="hljs-built_in">mkdir</span> work<br><br><span class="hljs-comment"># 修改属主和属组</span><br><span class="hljs-built_in">chown</span> alice:workgroup work<br><br><span class="hljs-comment"># 或者</span><br><span class="hljs-built_in">chown</span> alice.workgroup work<br></code></pre></td></tr></tbody></table></figure><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409272343315.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="创建 /home/work 目录并将其属主改为 alice，属组改为 workgroup"></p><p>五、修改 work 目录的权限</p><p>使得属组内的用户对该目录具有所有权限，属组外的用户对该目录没有任何权限。</p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-comment"># 写法一</span><br><span class="hljs-built_in">chmod</span> ug+rwx,o-rwx work<br><br><span class="hljs-comment"># 写法二</span><br><span class="hljs-built_in">chmod</span> 770 work<br></code></pre></td></tr></tbody></table></figure><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409272345642.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="修改 work 目录的权限"></p><p>六、权限功能测试</p><p>以 bob 用户身份在 work 目录下创建 <code>bob.txt</code> 文件。可以看到符合默认创建文件的权限格式 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>644</mn></mrow><annotation encoding="application/x-tex">644</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">644</span></span></span></span>：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409272350694.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="以 bob 用户在 work 下创建 bob.txt 文件"></p><p>同组用户与不同组用户关于「目录/文件」的 <code>rw</code> 权限测试。</p><ul><li>关于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>770</mn></mrow><annotation encoding="application/x-tex">770</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">770</span></span></span></span> 目录。由于 work 目录被 bob 创建时权限设置为了 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>770</mn></mrow><annotation encoding="application/x-tex">770</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">770</span></span></span></span>，bob 用户与 john 用户属于同一个组 workgroup，因此 john 因为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo>=</mo><mn>7</mn></mrow><annotation encoding="application/x-tex">g=7</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">7</span></span></span></span> 可以进入 work 目录进行操作，而 bob 用户与 mike 用户不属于同一个组，因此 mike 因为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>o</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">o=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">o</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 无法进入 work 目录，更不用说查看或者修改 work 目录中的文件了。</li><li>关于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>644</mn></mrow><annotation encoding="application/x-tex">644</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">644</span></span></span></span> 文件。现在 john 由于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>770</mn></mrow><annotation encoding="application/x-tex">770</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">770</span></span></span></span> 中的第二个 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>7</mn></mrow><annotation encoding="application/x-tex">7</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">7</span></span></span></span> 进入了 work 目录。由文件默认的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>644</mn></mrow><annotation encoding="application/x-tex">644</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">644</span></span></span></span> 权限可以知道：john 因为第一个 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn></mrow><annotation encoding="application/x-tex">4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">4</span></span></span></span> 可以读文件，但是不可以写文件，因此如下图所示，可以执行 <code>cat</code> 查看文件内容，但是不可以执行 <code>echo</code> 编辑文件内容。至于 mike，可以看到无论起始是否在 work 目录，都没有权限 <code>cd</code> 到 work 目录或者 <code>ls</code> 查看 work 目录中的内容。</li></ul><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409280007514.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="权限测试"></p>
+          <p><strong>一、添加 4 个用户：alice、bob、john、mike</strong></p><p>首先需要确保当前是 root 用户，使用 <code>su root</code> 切换到 root 用户。然后在创建用户时同时创建该用户对应的目录：</p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><code class="hljs bash">useradd -d /home/alice -m alice<br>useradd -d /home/bob -m bob<br>useradd -d /home/john -m john<br>useradd -d /home/mike -m mike<br></code></pre></td></tr></tbody></table></figure><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409272339074.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="添加 4 个用户"></p><p><strong>二、为 alice 设置密码</strong></p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">passwd alice<br></code></pre></td></tr></tbody></table></figure><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409272340323.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="为 alice 设置密码"></p><p><strong>三、创建用户组 workgroup 并将 alice、bob、john 加入</strong></p><ul><li><p>创建用户组：</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">groupadd workgroup<br></code></pre></td></tr></tbody></table></figure></li><li><p>添加到新组：</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><code class="hljs bash">usermod -a -G workgroup alice<br>usermod -a -G workgroup bob<br>usermod -a -G workgroup john<br></code></pre></td></tr></tbody></table></figure><ul><li><code>-a</code>：是 <code>--append</code> 的缩写，表示将用户添加到一个组，而不会移除她已有的其他组。这个选项必须与 <code>-G</code> 一起使用</li><li><code>-G</code>：指定要添加用户的附加组（即用户可以属于多个组），这里是 workgroup</li></ul></li><li><p>将 workgroup 作为各自的主组：</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><code class="hljs bash">usermod -g workgroup alice<br>usermod -g workgroup bob<br>usermod -g workgroup john<br></code></pre></td></tr></tbody></table></figure><ul><li><code>-g</code>：用于指定用户的主组（primary group）。主组是当用户创建文件或目录时默认分配的组</li></ul></li></ul><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409272341900.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="创建用户组 workgroup 并将 alice、bob、john 加入"></p><p><strong>四、创建 <code>/home/work</code> 目录并将其属主改为 alice，属组改为 workgroup</strong></p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-comment"># 创建目录</span><br><span class="hljs-built_in">mkdir</span> work<br><br><span class="hljs-comment"># 修改属主和属组</span><br><span class="hljs-built_in">chown</span> alice:workgroup work<br><br><span class="hljs-comment"># 或者</span><br><span class="hljs-built_in">chown</span> alice.workgroup work<br></code></pre></td></tr></tbody></table></figure><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409272343315.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="创建 /home/work 目录并将其属主改为 alice，属组改为 workgroup"></p><p><strong>五、修改 work 目录的权限</strong></p><p>使得属组内的用户对该目录具有所有权限，属组外的用户对该目录没有任何权限。</p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-comment"># 写法一</span><br><span class="hljs-built_in">chmod</span> ug+rwx,o-rwx work<br><br><span class="hljs-comment"># 写法二</span><br><span class="hljs-built_in">chmod</span> 770 work<br></code></pre></td></tr></tbody></table></figure><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409272345642.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="修改 work 目录的权限"></p><p><strong>六、权限功能测试</strong></p><p>以 bob 用户身份在 work 目录下创建 <code>bob.txt</code> 文件。可以看到符合默认创建文件的权限格式 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>644</mn></mrow><annotation encoding="application/x-tex">644</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">644</span></span></span></span>：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409272350694.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="以 bob 用户在 work 下创建 bob.txt 文件"></p><p>同组用户与不同组用户关于「目录/文件」的 <code>rw</code> 权限测试。</p><ul><li>关于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>770</mn></mrow><annotation encoding="application/x-tex">770</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">770</span></span></span></span> 目录。由于 work 目录被 bob 创建时权限设置为了 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>770</mn></mrow><annotation encoding="application/x-tex">770</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">770</span></span></span></span>，bob 用户与 john 用户属于同一个组 workgroup，因此 john 因为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo>=</mo><mn>7</mn></mrow><annotation encoding="application/x-tex">g=7</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">7</span></span></span></span> 可以进入 work 目录进行操作，而 bob 用户与 mike 用户不属于同一个组，因此 mike 因为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>o</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">o=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">o</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 无法进入 work 目录，更不用说查看或者修改 work 目录中的文件了。</li><li>关于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>644</mn></mrow><annotation encoding="application/x-tex">644</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">644</span></span></span></span> 文件。现在 john 由于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>770</mn></mrow><annotation encoding="application/x-tex">770</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">770</span></span></span></span> 中的第二个 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>7</mn></mrow><annotation encoding="application/x-tex">7</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">7</span></span></span></span> 进入了 work 目录。由文件默认的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>644</mn></mrow><annotation encoding="application/x-tex">644</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">644</span></span></span></span> 权限可以知道：john 因为第一个 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn></mrow><annotation encoding="application/x-tex">4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">4</span></span></span></span> 可以读文件，但是不可以写文件，因此如下图所示，可以执行 <code>cat</code> 查看文件内容，但是不可以执行 <code>echo</code> 编辑文件内容。至于 mike，可以看到无论起始是否在 work 目录，都没有权限 <code>cd</code> 到 work 目录或者 <code>ls</code> 查看 work 目录中的内容。</li></ul><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409280007514.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="权限测试"></p>
+        </div>
+      </div>
+    </div>
+<h2 id="进程管理与调试（实验3）">进程管理与调试（实验3）</h2>
+<p>在熟悉了 bash shell 的基本命令以及 Linux 中用户与权限管理的基本概念后，我们就可以开始尝试管理 Linux 中的程序了。当然每一个程序在开始运行后都会成为一个或多个进程，因此接下来简单介绍一下 Linux 的进程管理。最后再通过调试一个 C 程序来熟悉 GNU 调试工具 gdb (GNU Debugger) 的使用。</p>
+<h3 id="进程管理">进程管理</h3>
+<p>查看进程状态 ps</p>
+<figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">ps<br></code></pre></td></tr></tbody></table></figure>
+<p>动态查看进程状态 top</p>
+<figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">top<br></code></pre></td></tr></tbody></table></figure>
+<p>杀死某个进程 kill</p>
+<figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">kill</span> -9 &lt;PID&gt;<br></code></pre></td></tr></tbody></table></figure>
+
+    <div class="fold">
+      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-d66a19be" role="button" aria-expanded="false" aria-controls="collapse-d66a19be">
+        <div class="fold-arrow">▶</div>练习
+      </div>
+      <div class="fold-collapse collapse" id="collapse-d66a19be">
+        <div class="fold-content">
+          <p><strong>一、编写一个 shell 程序 <code>badproc.sh</code> 使其不断循环</strong></p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-meta">#! /bin/bash</span><br><span class="hljs-keyword">while</span> <span class="hljs-built_in">echo</span> <span class="hljs-string">"I'm making files!"</span><br><span class="hljs-keyword">do</span><br>    <span class="hljs-built_in">mkdir</span> adir<br>    <span class="hljs-built_in">cd</span> adir<br>    <span class="hljs-built_in">touch</span> afile<br>    <span class="hljs-built_in">sleep</span> 10s<br><span class="hljs-keyword">done</span><br></code></pre></td></tr></tbody></table></figure><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081735300.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="编写 sh 文件"></p><p><strong>二、为 <code>badproc.sh</code> 增加可执行权限</strong></p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">chmod</span> u+x badproc.sh<br></code></pre></td></tr></tbody></table></figure><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081736525.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="增加可执行权限"></p><p><strong>三、在后台执行 <code>badproc.sh</code></strong></p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">./badproc.sh &amp;<br></code></pre></td></tr></tbody></table></figure><ul><li><code>&amp;</code> 表示后台执行</li></ul><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081737564.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="后台执行"></p><p><strong>四、利用 <code>ps</code> 命令查看其进程号</strong></p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">ps aux | grep badproc<br></code></pre></td></tr></tbody></table></figure><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081739971.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="查看进程号"></p><p><strong>五、利用 <code>kill</code> 命令杀死该进程</strong></p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">kill</span> -9 &lt;PID&gt;<br></code></pre></td></tr></tbody></table></figure><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081748528.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="杀死该进程"></p><p><strong>六、删除 <code>badproc.sh</code> 程序运行时创建的目录和文件</strong></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081748818.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="删除目录和文件"></p>
+        </div>
+      </div>
+    </div>
+<h3 id="gdb-调试">gdb 调试</h3>
+<p>基本命令参考 <a target="_blank" rel="noopener" href="https://www.cnblogs.com/HKUI/p/8955443.html">gdb调试的基本使用</a>。常用的如下：</p>
+<p>开始运行</p>
+<figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">r<br></code></pre></td></tr></tbody></table></figure>
+<ul>
+<li>r 即 run</li>
+</ul>
+<p>设置断点</p>
+<figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">break</span> &lt;line&gt;<br></code></pre></td></tr></tbody></table></figure>
+<ul>
+<li>line 即行号</li>
+</ul>
+<p>运行到下一个断点</p>
+<figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">c<br></code></pre></td></tr></tbody></table></figure>
+<ul>
+<li>c 即 continue</li>
+</ul>
+
+    <div class="fold">
+      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-91d325eb" role="button" aria-expanded="false" aria-controls="collapse-91d325eb">
+        <div class="fold-arrow">▶</div>练习
+      </div>
+      <div class="fold-collapse collapse" id="collapse-91d325eb">
+        <div class="fold-content">
+          <p><strong>一、创建 <code>fork.c</code> 文件</strong></p><figure class="highlight c"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br></pre></td><td class="code"><pre><code class="hljs c"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;stdio.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;stdlib.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;unistd.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;sys/types.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;sys/wait.h&gt;</span></span><br><br><span class="hljs-type">int</span> <span class="hljs-title function_">main</span><span class="hljs-params">()</span> {<br>    <span class="hljs-comment">/* fork another process */</span><br>    <span class="hljs-type">pid_t</span>  pid;<br>    pid = fork();<br><br>    <span class="hljs-keyword">if</span> (pid &lt; <span class="hljs-number">0</span>) {<br>        <span class="hljs-comment">/* error occurred */</span><br>        <span class="hljs-built_in">fprintf</span>(<span class="hljs-built_in">stderr</span>, <span class="hljs-string">"Fork Failed"</span>);<br>        <span class="hljs-built_in">exit</span>(<span class="hljs-number">-1</span>);<br>    } <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (pid == <span class="hljs-number">0</span>) {<br>        <span class="hljs-comment">/* child process */</span><br>        <span class="hljs-built_in">printf</span>(<span class="hljs-string">"This is child process, pid=%d\n"</span>, getpid());<br>        execlp(<span class="hljs-string">"/bin/ls"</span>, <span class="hljs-string">"ls"</span>, <span class="hljs-literal">NULL</span>);<br>        <span class="hljs-built_in">printf</span>(<span class="hljs-string">"Child process finished\n"</span>); <span class="hljs-comment">/*这句话不会被打印，除非execlp调用未成功*/</span><br>    } <span class="hljs-keyword">else</span> {<br>        <span class="hljs-comment">/* parent process */</span><br>        <span class="hljs-comment">/* parent will wait for the child to complete */</span><br>        <span class="hljs-built_in">printf</span>(<span class="hljs-string">"This is parent process, pid=%d\n"</span>, getpid());<br>        wait (<span class="hljs-literal">NULL</span>);<br>        <span class="hljs-built_in">printf</span> (<span class="hljs-string">"Child Complete\n"</span>);<br>        <span class="hljs-built_in">exit</span>(<span class="hljs-number">0</span>);<br>    }<br>}<br></code></pre></td></tr></tbody></table></figure><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081813738.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="创建文件"></p><p>这段程序首先通过调用 <code>fork()</code> 函数创建一个子进程，并通过 <code>pid</code> 信息来判断当前进程是父进程还是子进程。在并发的逻辑下，执行哪一个进程的逻辑是未知的。</p><p><strong>二、编译运行 <code>fork.c</code> 文件</strong></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081807870.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="编译运行"></p><p>从上述运行结果可以看出：并发时，首先执行父进程的逻辑，然后才执行子进程的逻辑。</p><p><strong>三、gdb 调试</strong></p><p>在 fork 创建子进程后追踪子进程：</p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><code class="hljs bash">gdb fork<br><span class="hljs-built_in">set</span> follow-fork-mode child<br>catch <span class="hljs-built_in">exec</span><br></code></pre></td></tr></tbody></table></figure><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081947476.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="追踪子进程"></p><p>运行到第一个断点时分别观察父进程 1510168 和子进程 1510171：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081948301.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="父进程 1510168"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081949586.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="子进程 1510171"></p><p>运行到第二个断点时观察子进程 1510171：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081951803.png" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="运行到第二个断点时观察子进程 1510171"></p><p>从上述子进程的追踪结果可以看出，在父进程结束之后，子进程成功执行了 <code>pid==0</code> 的逻辑并调用 <code>ls</code> 工具。</p>
         </div>
       </div>
     </div>
-<h2 id="进程管理-调试观察（实验3）">进程管理 | 调试观察（实验3）</h2>
-<h3 id="相关命令-2">相关命令</h3>
-<p>下面罗列一些进程相关的命令，打 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∗</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span> 表示重要命令。</p>
 <h2 id="Linux-编程（实验4）">Linux 编程（实验4）</h2>
 <h2 id="工具扩展">工具扩展</h2>
 <h3 id="tree">tree</h3>
@@ -608,6 +665,7 @@ <h3 id="wget">wget</h3>
 <h2 id="参考引用">参考引用</h2>
 <p><a target="_blank" rel="noopener" href="https://northword.cn/code/bash-for-windows/">在 Windows 中使用 Bash shell</a></p>
 <p><a target="_blank" rel="noopener" href="https://chatgpt.com/share/66f61785-7ed0-800c-bdf3-fbbfc84a56ae">Linux 用户权限信息</a></p>
+<p><a target="_blank" rel="noopener" href="https://www.cnblogs.com/HKUI/p/8955443.html">gdb调试的基本使用</a></p>
 
                 
               </div>
@@ -672,7 +730,7 @@ <h2 id="参考引用">参考引用</h2>
       
         <div class="license-meta-item license-meta-date">
           <div>更新于</div>
-          <div>2024年10月2日</div>
+          <div>2024年10月8日</div>
         </div>
       
       
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@@ -445,9 +445,9 @@ <h2 id="参考">参考</h2>
                   <article class="post-prev col-6">
                     
                     
-                      <a href="/RoadMap/road-map-guide/" title="road-map-guide">
+                      <a href="/BackEnd/Python/Flask/deploy-flask/" title="deploy-flask">
                         <i class="iconfont icon-arrowleft"></i>
-                        <span class="hidden-mobile">road-map-guide</span>
+                        <span class="hidden-mobile">deploy-flask</span>
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                       </a>
                     
@@ -455,8 +455,8 @@ <h2 id="参考">参考</h2>
                   <article class="post-next col-6">
                     
                     
-                      <a href="/GPA/4th-term/PyApply/" title="PyApply">
-                        <span class="hidden-mobile">PyApply</span>
+                      <a href="/RoadMap/road-map-guide/" title="road-map-guide">
+                        <span class="hidden-mobile">road-map-guide</span>
                         <span class="visible-mobile">下一篇</span>
                         <i class="iconfont icon-arrowright"></i>
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-                      <a href="/BackEnd/Python/Flask/deploy-flask/" title="deploy-flask">
+                      <a href="/Operation/operation-guide/" title="operation-guide">
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-                        <span class="hidden-mobile">deploy-flask</span>
+                        <span class="hidden-mobile">operation-guide</span>
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@@ -486,8 +486,8 @@ <h2 id="C">C++</h2>
                   <article class="post-next col-6">
                     
                     
-                      <a href="/Operation/operation-guide/" title="operation-guide">
-                        <span class="hidden-mobile">operation-guide</span>
+                      <a href="/GPA/4th-term/PyApply/" title="PyApply">
+                        <span class="hidden-mobile">PyApply</span>
                         <span class="visible-mobile">下一篇</span>
                         <i class="iconfont icon-arrowright"></i>
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diff --git a/about/index.html b/about/index.html
index 4eabdcb66..b87b11b0b 100644
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-<meta property="article:modified_time" content="2024-10-07T05:31:57.057Z">
+<meta property="article:published_time" content="2024-10-07T15:44:08.777Z">
+<meta property="article:modified_time" content="2024-10-07T15:44:08.777Z">
 <meta property="article:author" content="Mr_Dwj">
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 <h2 id="技术栈">技术栈</h2>
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+<img src="https://skillicons.dev/icons?i=flask,qt" srcset="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif" lazyload alt="技术栈">
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+    <a href="/Operation/operation-guide/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">road-map-guide</div>
+      <div class="list-group-item-title">operation-guide</div>
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-    <a href="/Operation/operation-guide/" class="list-group-item list-group-item-action">
+    <a href="/RoadMap/road-map-guide/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">operation-guide</div>
+      <div class="list-group-item-title">road-map-guide</div>
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+    <a href="/GPA/4th-term/OptMethod/" class="list-group-item list-group-item-action">
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-      <div class="list-group-item-title">ProbAndStat</div>
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@@ -248,9 +248,9 @@
       
       <p class="h5">2024</p>
     
-    <a href="/GPA/4th-term/OptMethod/" class="list-group-item list-group-item-action">
+    <a href="/GPA/4th-term/ProbAndStat/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">OptMethod</div>
+      <div class="list-group-item-title">ProbAndStat</div>
     </a>
   
     
@@ -266,21 +266,21 @@
     </a>
   
     
-    <a href="/GPA/3rd-term/DataStructure/" class="list-group-item list-group-item-action">
+    <a href="/GPA/3rd-term/DigitalLogicCircuit/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">DataStructure</div>
+      <div class="list-group-item-title">DigitalLogicCircuit</div>
     </a>
   
     
-    <a href="/GPA/3rd-term/DataStructureClassDesign/" class="list-group-item list-group-item-action">
+    <a href="/GPA/3rd-term/DataStructure/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">DataStructureClassDesign</div>
+      <div class="list-group-item-title">DataStructure</div>
     </a>
   
     
-    <a href="/GPA/3rd-term/DigitalLogicCircuit/" class="list-group-item list-group-item-action">
+    <a href="/GPA/3rd-term/DataStructureClassDesign/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">DigitalLogicCircuit</div>
+      <div class="list-group-item-title">DataStructureClassDesign</div>
     </a>
   
     
@@ -302,9 +302,9 @@
     </a>
   
     
-    <a href="/FrontEnd/Hexo/hexo-migrate/" class="list-group-item list-group-item-action">
+    <a href="/FrontEnd/Hexo/hexo-learning-record/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">hexo-migrate</div>
+      <div class="list-group-item-title">hexo-learning-record</div>
     </a>
   
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index 78472437d..aefa72d85 100644
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@@ -248,9 +248,9 @@
       
       <p class="h5">2024</p>
     
-    <a href="/FrontEnd/Hexo/hexo-learning-record/" class="list-group-item list-group-item-action">
+    <a href="/FrontEnd/Hexo/hexo-migrate/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">hexo-learning-record</div>
+      <div class="list-group-item-title">hexo-migrate</div>
     </a>
   
     
@@ -272,15 +272,15 @@
     </a>
   
     
-    <a href="/DevTools/DevCpp/devc-self-config/" class="list-group-item list-group-item-action">
+    <a href="/DevTools/CLion/solve-clion-decoding-error/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">self-config</div>
+      <div class="list-group-item-title">solve-clion-decoding-error</div>
     </a>
   
     
-    <a href="/DevTools/CLion/solve-clion-decoding-error/" class="list-group-item list-group-item-action">
+    <a href="/DevTools/DevCpp/devc-self-config/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">solve-clion-decoding-error</div>
+      <div class="list-group-item-title">self-config</div>
     </a>
   
     
@@ -296,15 +296,15 @@
     </a>
   
     
-    <a href="/BackEnd/back-end-guide/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/z_logic/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">back-end-guide</div>
+      <div class="list-group-item-title">z_logic</div>
     </a>
   
     
-    <a href="/Algorithm/z_logic/" class="list-group-item list-group-item-action">
+    <a href="/BackEnd/back-end-guide/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">z_logic</div>
+      <div class="list-group-item-title">back-end-guide</div>
     </a>
   
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diff --git a/archives/2024/03/page/4/index.html b/archives/2024/03/page/4/index.html
index 41779b756..182b8b3e9 100644
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@@ -248,39 +248,39 @@
       
       <p class="h5">2024</p>
     
-    <a href="/Algorithm/geometry/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/games/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">geometry</div>
+      <div class="list-group-item-title">games</div>
     </a>
   
     
-    <a href="/Algorithm/hashing/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/geometry/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">hashing</div>
+      <div class="list-group-item-title">geometry</div>
     </a>
   
     
-    <a href="/Algorithm/games/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/graphs/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">games</div>
+      <div class="list-group-item-title">graphs</div>
     </a>
   
     
-    <a href="/Algorithm/graphs/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/greedy/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">graphs</div>
+      <div class="list-group-item-title">greedy</div>
     </a>
   
     
-    <a href="/Algorithm/number-theory/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/hashing/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">number-theory</div>
+      <div class="list-group-item-title">hashing</div>
     </a>
   
     
-    <a href="/Algorithm/greedy/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/number-theory/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">greedy</div>
+      <div class="list-group-item-title">number-theory</div>
     </a>
   
     
@@ -302,9 +302,9 @@
     </a>
   
     
-    <a href="/Algorithm/binary-search/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/data-structure/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">binary-search</div>
+      <div class="list-group-item-title">data-structure</div>
     </a>
   
 </div>
diff --git a/archives/2024/03/page/5/index.html b/archives/2024/03/page/5/index.html
index 4524eec89..de2ad797d 100644
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@@ -248,9 +248,9 @@
       
       <p class="h5">2024</p>
     
-    <a href="/Algorithm/divide-and-conquer/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/binary-search/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">divide-and-conquer</div>
+      <div class="list-group-item-title">binary-search</div>
     </a>
   
     
@@ -260,9 +260,9 @@
     </a>
   
     
-    <a href="/Algorithm/data-structure/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/divide-and-conquer/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">data-structure</div>
+      <div class="list-group-item-title">divide-and-conquer</div>
     </a>
   
     
diff --git a/archives/2024/04/index.html b/archives/2024/04/index.html
index 190984a6c..54760ac25 100644
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diff --git a/archives/2024/05/index.html b/archives/2024/05/index.html
index c32ee7342..bf0e46fc2 100644
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-      <div class="list-group-item-title">road-map-guide</div>
+      <div class="list-group-item-title">operation-guide</div>
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-    <a href="/Operation/operation-guide/" class="list-group-item list-group-item-action">
+    <a href="/RoadMap/road-map-guide/" class="list-group-item list-group-item-action">
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-      <div class="list-group-item-title">operation-guide</div>
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       <time>03-21</time>
-      <div class="list-group-item-title">DataStructure</div>
+      <div class="list-group-item-title">DigitalLogicCircuit</div>
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-    <a href="/GPA/3rd-term/DataStructureClassDesign/" class="list-group-item list-group-item-action">
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-      <div class="list-group-item-title">DataStructureClassDesign</div>
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-    <a href="/GPA/3rd-term/DigitalLogicCircuit/" class="list-group-item list-group-item-action">
+    <a href="/GPA/3rd-term/DataStructureClassDesign/" class="list-group-item list-group-item-action">
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       <time>03-21</time>
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+    <a href="/FrontEnd/Hexo/hexo-migrate/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">hexo-learning-record</div>
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@@ -260,15 +260,15 @@
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-    <a href="/DevTools/DevCpp/devc-self-config/" class="list-group-item list-group-item-action">
+    <a href="/DevTools/CLion/solve-clion-decoding-error/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">self-config</div>
+      <div class="list-group-item-title">solve-clion-decoding-error</div>
     </a>
   
     
-    <a href="/DevTools/CLion/solve-clion-decoding-error/" class="list-group-item list-group-item-action">
+    <a href="/DevTools/DevCpp/devc-self-config/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">solve-clion-decoding-error</div>
+      <div class="list-group-item-title">self-config</div>
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@@ -284,27 +284,27 @@
     </a>
   
     
-    <a href="/BackEnd/back-end-guide/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/z_logic/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">back-end-guide</div>
+      <div class="list-group-item-title">z_logic</div>
     </a>
   
     
-    <a href="/Algorithm/z_logic/" class="list-group-item list-group-item-action">
+    <a href="/BackEnd/back-end-guide/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">z_logic</div>
+      <div class="list-group-item-title">back-end-guide</div>
     </a>
   
     
-    <a href="/Algorithm/geometry/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/games/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">geometry</div>
+      <div class="list-group-item-title">games</div>
     </a>
   
     
-    <a href="/Algorithm/hashing/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/geometry/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">hashing</div>
+      <div class="list-group-item-title">geometry</div>
     </a>
   
 </div>
diff --git a/archives/2024/page/8/index.html b/archives/2024/page/8/index.html
index a1f0b27c3..7281cbaae 100644
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@@ -248,27 +248,27 @@
       
       <p class="h5">2024</p>
     
-    <a href="/Algorithm/games/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/graphs/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">games</div>
+      <div class="list-group-item-title">graphs</div>
     </a>
   
     
-    <a href="/Algorithm/graphs/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/greedy/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">graphs</div>
+      <div class="list-group-item-title">greedy</div>
     </a>
   
     
-    <a href="/Algorithm/number-theory/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/hashing/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">number-theory</div>
+      <div class="list-group-item-title">hashing</div>
     </a>
   
     
-    <a href="/Algorithm/greedy/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/number-theory/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">greedy</div>
+      <div class="list-group-item-title">number-theory</div>
     </a>
   
     
@@ -290,15 +290,15 @@
     </a>
   
     
-    <a href="/Algorithm/binary-search/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/data-structure/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">binary-search</div>
+      <div class="list-group-item-title">data-structure</div>
     </a>
   
     
-    <a href="/Algorithm/divide-and-conquer/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/binary-search/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">divide-and-conquer</div>
+      <div class="list-group-item-title">binary-search</div>
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diff --git a/archives/2024/page/9/index.html b/archives/2024/page/9/index.html
index 6bc11caf3..4ddbe9868 100644
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+    <a href="/Algorithm/divide-and-conquer/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">data-structure</div>
+      <div class="list-group-item-title">divide-and-conquer</div>
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diff --git a/archives/index.html b/archives/index.html
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index 33df0b812..cccd767ef 100644
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-    <a href="/RoadMap/road-map-guide/" class="list-group-item list-group-item-action">
+    <a href="/Operation/operation-guide/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">road-map-guide</div>
+      <div class="list-group-item-title">operation-guide</div>
     </a>
   
     
-    <a href="/Operation/operation-guide/" class="list-group-item list-group-item-action">
+    <a href="/RoadMap/road-map-guide/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">operation-guide</div>
+      <div class="list-group-item-title">road-map-guide</div>
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@@ -278,9 +278,9 @@
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-    <a href="/GPA/4th-term/PyAlgo/" class="list-group-item list-group-item-action">
+    <a href="/GPA/4th-term/OptMethod/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">PyAlgo</div>
+      <div class="list-group-item-title">OptMethod</div>
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@@ -290,15 +290,15 @@
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-    <a href="/GPA/4th-term/ProbAndStat/" class="list-group-item list-group-item-action">
+    <a href="/GPA/4th-term/PyAlgo/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">ProbAndStat</div>
+      <div class="list-group-item-title">PyAlgo</div>
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-    <a href="/GPA/4th-term/OptMethod/" class="list-group-item list-group-item-action">
+    <a href="/GPA/4th-term/ProbAndStat/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">OptMethod</div>
+      <div class="list-group-item-title">ProbAndStat</div>
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diff --git a/archives/page/6/index.html b/archives/page/6/index.html
index 7a0baf90b..5083830d8 100644
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@@ -254,21 +254,21 @@
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-    <a href="/GPA/3rd-term/DataStructure/" class="list-group-item list-group-item-action">
+    <a href="/GPA/3rd-term/DigitalLogicCircuit/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">DataStructure</div>
+      <div class="list-group-item-title">DigitalLogicCircuit</div>
     </a>
   
     
-    <a href="/GPA/3rd-term/DataStructureClassDesign/" class="list-group-item list-group-item-action">
+    <a href="/GPA/3rd-term/DataStructure/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">DataStructureClassDesign</div>
+      <div class="list-group-item-title">DataStructure</div>
     </a>
   
     
-    <a href="/GPA/3rd-term/DigitalLogicCircuit/" class="list-group-item list-group-item-action">
+    <a href="/GPA/3rd-term/DataStructureClassDesign/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">DigitalLogicCircuit</div>
+      <div class="list-group-item-title">DataStructureClassDesign</div>
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@@ -290,15 +290,15 @@
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-    <a href="/FrontEnd/Hexo/hexo-migrate/" class="list-group-item list-group-item-action">
+    <a href="/FrontEnd/Hexo/hexo-learning-record/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">hexo-migrate</div>
+      <div class="list-group-item-title">hexo-learning-record</div>
     </a>
   
     
-    <a href="/FrontEnd/Hexo/hexo-learning-record/" class="list-group-item list-group-item-action">
+    <a href="/FrontEnd/Hexo/hexo-migrate/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">hexo-learning-record</div>
+      <div class="list-group-item-title">hexo-migrate</div>
     </a>
   
     
diff --git a/archives/page/7/index.html b/archives/page/7/index.html
index 41d1a67bf..2b41552b6 100644
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@@ -260,15 +260,15 @@
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-    <a href="/DevTools/DevCpp/devc-self-config/" class="list-group-item list-group-item-action">
+    <a href="/DevTools/CLion/solve-clion-decoding-error/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">self-config</div>
+      <div class="list-group-item-title">solve-clion-decoding-error</div>
     </a>
   
     
-    <a href="/DevTools/CLion/solve-clion-decoding-error/" class="list-group-item list-group-item-action">
+    <a href="/DevTools/DevCpp/devc-self-config/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">solve-clion-decoding-error</div>
+      <div class="list-group-item-title">self-config</div>
     </a>
   
     
@@ -284,27 +284,27 @@
     </a>
   
     
-    <a href="/BackEnd/back-end-guide/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/z_logic/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">back-end-guide</div>
+      <div class="list-group-item-title">z_logic</div>
     </a>
   
     
-    <a href="/Algorithm/z_logic/" class="list-group-item list-group-item-action">
+    <a href="/BackEnd/back-end-guide/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">z_logic</div>
+      <div class="list-group-item-title">back-end-guide</div>
     </a>
   
     
-    <a href="/Algorithm/geometry/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/games/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">geometry</div>
+      <div class="list-group-item-title">games</div>
     </a>
   
     
-    <a href="/Algorithm/hashing/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/geometry/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">hashing</div>
+      <div class="list-group-item-title">geometry</div>
     </a>
   
 </div>
diff --git a/archives/page/8/index.html b/archives/page/8/index.html
index f453cec93..f35b243fb 100644
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@@ -248,27 +248,27 @@
       
       <p class="h5">2024</p>
     
-    <a href="/Algorithm/games/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/graphs/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">games</div>
+      <div class="list-group-item-title">graphs</div>
     </a>
   
     
-    <a href="/Algorithm/graphs/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/greedy/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">graphs</div>
+      <div class="list-group-item-title">greedy</div>
     </a>
   
     
-    <a href="/Algorithm/number-theory/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/hashing/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">number-theory</div>
+      <div class="list-group-item-title">hashing</div>
     </a>
   
     
-    <a href="/Algorithm/greedy/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/number-theory/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">greedy</div>
+      <div class="list-group-item-title">number-theory</div>
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@@ -290,15 +290,15 @@
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-    <a href="/Algorithm/binary-search/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/data-structure/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">binary-search</div>
+      <div class="list-group-item-title">data-structure</div>
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-    <a href="/Algorithm/divide-and-conquer/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/binary-search/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">divide-and-conquer</div>
+      <div class="list-group-item-title">binary-search</div>
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diff --git a/archives/page/9/index.html b/archives/page/9/index.html
index 9dc54bd6b..a2b99c9e7 100644
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@@ -248,9 +248,9 @@
       
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-    <a href="/Algorithm/data-structure/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/divide-and-conquer/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">data-structure</div>
+      <div class="list-group-item-title">divide-and-conquer</div>
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-    <a href="/Algorithm/geometry/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/games/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">geometry</div>
+      <div class="list-group-item-title">games</div>
     </a>
   
     
-    <a href="/Algorithm/hashing/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/geometry/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">hashing</div>
+      <div class="list-group-item-title">geometry</div>
     </a>
   
     
-    <a href="/Algorithm/games/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/graphs/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">games</div>
+      <div class="list-group-item-title">graphs</div>
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-    <a href="/Algorithm/graphs/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/greedy/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">graphs</div>
+      <div class="list-group-item-title">greedy</div>
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-    <a href="/Algorithm/number-theory/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/hashing/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">number-theory</div>
+      <div class="list-group-item-title">hashing</div>
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-    <a href="/Algorithm/greedy/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/number-theory/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">greedy</div>
+      <div class="list-group-item-title">number-theory</div>
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@@ -302,9 +302,9 @@
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-    <a href="/Algorithm/binary-search/" class="list-group-item list-group-item-action">
+    <a href="/Algorithm/data-structure/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">binary-search</div>
+      <div class="list-group-item-title">data-structure</div>
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+    <a href="/Algorithm/binary-search/" class="list-group-item list-group-item-action">
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-      <div class="list-group-item-title">divide-and-conquer</div>
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       <time>03-21</time>
-      <div class="list-group-item-title">data-structure</div>
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-    <a href="/FrontEnd/Hexo/hexo-migrate/" class="list-group-item list-group-item-action">
+    <a href="/FrontEnd/Hexo/hexo-learning-record/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">hexo-migrate</div>
+      <div class="list-group-item-title">hexo-learning-record</div>
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-    <a href="/FrontEnd/Hexo/hexo-learning-record/" class="list-group-item list-group-item-action">
+    <a href="/FrontEnd/Hexo/hexo-migrate/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">hexo-learning-record</div>
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-    <a href="/FrontEnd/Hexo/hexo-migrate/" class="list-group-item list-group-item-action">
+    <a href="/FrontEnd/Hexo/hexo-learning-record/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">hexo-migrate</div>
+      <div class="list-group-item-title">hexo-learning-record</div>
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-    <a href="/FrontEnd/Hexo/hexo-learning-record/" class="list-group-item list-group-item-action">
+    <a href="/FrontEnd/Hexo/hexo-migrate/" class="list-group-item list-group-item-action">
       <time>03-21</time>
-      <div class="list-group-item-title">hexo-learning-record</div>
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-      <div class="list-group-item-title">PyAlgo</div>
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       <time>03-21</time>
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       <time>03-21</time>
-      <div class="list-group-item-title">DigitalLogicCircuit</div>
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diff --git a/local-search.xml b/local-search.xml
index d24c58394..8bd56df9f 100644
--- a/local-search.xml
+++ b/local-search.xml
@@ -29,7 +29,7 @@
     <link href="/Operation/Shell/shell-basic/"/>
     <url>/Operation/Shell/shell-basic/</url>
     
-    <content type="html"><![CDATA[<h2 id="Shell-基础">Shell 基础</h2><h2 id="前言">前言</h2><p>尝试在 Windows11 上使用 wget 通过一个 url 下载一个 js 静态文件。但是发现本地 bash 使用 wget 命令报错：command not found。于是借 wget 之名展开”古老“的 shell 之旅。</p><p><strong>Shell 是什么</strong>？我们用 GLM4 总结一下：</p><blockquote><p>Shell 的准确定义是：在操作系统中，Shell 是一个程序，它提供了一个用户界面，允许用户通过命令行输入指令来与操作系统交互。它充当用户与操作系统内核之间的中介，处理用户的输入，解释这些输入为操作系统可以理解和执行的操作，并返回执行结果。</p></blockquote><p>也就是说 shell 是一个命令解释器。可以将其理解为人机交互的中间层，通过解析字符串命令与操作系统交互，进而完成系统资源的调用与处理。可以将「解析命令并调用系统资源完成任务」这个过程简单地理解为：shell 解析输入 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> shell 调用对应的 C 函数向 OS 发起请求 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munderover><mo stretchy="true" minsize="3.0em">→</mo><mpadded width="+0.6em" lspace="0.3em"><mrow></mrow></mpadded><mpadded width="+0.6em" lspace="0.3em"><mtext>API</mtext></mpadded></munderover></mrow><annotation encoding="application/x-tex">\xrightarrow[]{\text{API}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2223em;vertical-align:-0.122em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1003em;"><span style="top:-3.322em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">API</span></span></span></span></span><span class="svg-align" style="top:-2.689em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.522em;min-width:1.469em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span><span style="top:-2.578em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.122em;"><span></span></span></span></span></span></span></span></span> OS 接受请求并执行相关底层操作 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> OS 返回执行结果给 C 函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> shell 接受 C 函数返回结果并显示提示信息。</p><p><strong>为什么用 Bash Shell</strong>？常见的命令解释器有：Bash (Bourne Again Shell)、Sh (Bourne Shell)、Zsh (Z Shell) 等。其中 Bash 是最流行的跨平台 Shell 之一。虽然它是 Linux 和 macOS 的默认 Shell，但也可以在 Windows 上通过 Windows Subsystem for Linux (WSL) 或第三方工具如 Git Bash 来运行。</p><p><strong>文章标题为什么不叫 Linux 基础</strong>？在我看来，Linux 基础的内容更多应当偏向于讲解  Linux 的理论知识，然而目前主流的讲解 Linux 基础的内容均偏向于指导大家如何使用 shell 与操作系统交互，那这与 python 等解释型编程语言有什么区别？python 是用 python 解释器与操作系统交互，相关的教程默认都叫「Python 基础」，shell 是用 shell 解释器与操作系统交互，那为什么不叫「Shell 基础」呢？</p><p>注：本文基于 Windows11 的 Git Bash 命令解释器配合 Ubuntu22.04 的 Bash 命令解释器展开。两个平台的 Bash 版本信息如下：</p><p>Windows11 <strong>Git Bash</strong> Version Info:</p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br></pre></td><td class="code"><pre><code class="hljs bash">$ bash --version<br>GNU bash, version 5.2.26(1)-release (x86_64-pc-msys)<br>Copyright (C) 2022 Free Software Foundation, Inc.<br>License GPLv3+: GNU GPL version 3 or later &lt;http://gnu.org/licenses/gpl.html&gt;<br><br>This is free software; you are free to change and redistribute it.<br>There is NO WARRANTY, to the extent permitted by law.<br></code></pre></td></tr></tbody></table></figure><p>Linux <strong>Bash</strong> Version Info:</p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br></pre></td><td class="code"><pre><code class="hljs bash">root@dwj2:~# bash --version<br>GNU bash，版本 5.1.16(1)-release (x86_64-pc-linux-gnu)<br>Copyright (C) 2020 Free Software Foundation, Inc.<br>许可证 GPLv3+：GNU GPL 许可证第三版或者更新版本 &lt;http://gnu.org/licenses/gpl.html&gt;<br><br>本软件是自由软件，您可以自由地更改和重新发布。<br>在法律许可的情况下特此明示，本软件不提供任何担保。<br></code></pre></td></tr></tbody></table></figure><h2 id="Shell-基本命令（实验1）">Shell 基本命令（实验1）</h2><p>在开始之前，我们可以使用 <code>echo $SHELL</code> 命令查看当前的命令解释器是什么。</p><h3 id="常用命令">常用命令</h3><h4 id="改变目录-cd">改变目录 cd</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">cd</span> ../<br></code></pre></td></tr></tbody></table></figure><p><code>../</code> 表示上一级，<code>./</code> 表示当前一级（也可以不写），<code>/</code> 表示从根目录开始。</p><h4 id="打印目录内容-ls">打印目录内容 ls</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">ls</span><br></code></pre></td></tr></tbody></table></figure><ul><li>添加 <code>-l</code> 参数打印详细信息。</li></ul><h4 id="打印当前路径-pwd">打印当前路径 pwd</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">pwd</span><br></code></pre></td></tr></tbody></table></figure><h4 id="打印当前用户名">打印当前用户名</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">whoami</span><br></code></pre></td></tr></tbody></table></figure><h4 id="创建文件夹-mkdir">创建文件夹 mkdir</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">mkdir</span> &lt;FolderName&gt;<br></code></pre></td></tr></tbody></table></figure><h4 id="创建文件-touch">创建文件 touch</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">touch</span> &lt;FileName&gt;<br></code></pre></td></tr></tbody></table></figure><h4 id="复制-cp">复制 cp</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">cp</span> [option] &lt;<span class="hljs-built_in">source</span>&gt; &lt;target&gt;<br></code></pre></td></tr></tbody></table></figure><ul><li>option 中：<code>-r</code> 表示递归复制，<code>-i</code> 用来当出现重名文件时进行提示。</li><li>source 表示被拷贝的资源。</li><li>target 表示拷贝后的资源名称或者路径。</li></ul><h4 id="移动-mv">移动 mv</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">mv</span> [option] &lt;<span class="hljs-built_in">source</span>&gt; &lt;target&gt;<br></code></pre></td></tr></tbody></table></figure><ul><li>option 中：<code>-i</code> 用来当出现重名文件时进行提示。</li><li>source 表示被移动的资源。</li><li>target 表示移动后的资源名称或者路径（可以以此进行重命名）。</li></ul><h4 id="删除-rm">删除 rm</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">rm</span> [option] &lt;<span class="hljs-built_in">source</span>&gt;<br></code></pre></td></tr></tbody></table></figure><ul><li>option 中：<code>-i</code> 需要一个一个确认，<code>-f</code> 表示强制删除，<code>-r</code> 表示递归删除。</li></ul><h4 id="打印-echo">打印 echo</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">echo</span> <span class="hljs-string">"hello"</span><br></code></pre></td></tr></tbody></table></figure><p>将 echo 后面紧跟着的内容打印到命令行解释器中。可以用来查看环境变量的具体值。</p><p>也可以配合输出重定向符号 <code>&gt;</code> 将信息打印到文件中实现创建新文件的作用。例如 <code>echo "hello" &gt; file.txt</code> 用于创建 file.txt 文件并且在其中写下 <code>hello</code> 信息。</p><h4 id="查看-cat">查看 cat</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">cat</span> [option] &lt;<span class="hljs-built_in">source</span>&gt;<br></code></pre></td></tr></tbody></table></figure><ul><li>option: -n 或 --number 表示将 source 中的内容打印出来的同时带上行号。</li><li>也可以配合输出重定位符号 <code>&gt;</code> 将信息输出到文件中。例如 <code>cat -n a.txt &gt; b.txt</code> 表示将 a.txt 文件中的内容带上行号输出到 b.txt 文件中。</li></ul><h4 id="分页查看-more">分页查看 more</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">more &lt;<span class="hljs-built_in">source</span>&gt;<br></code></pre></td></tr></tbody></table></figure><p>与 cat 类似，只不过可以分页展示。按空格键下一页，b 键上一页，q 键退出。</p><p>可以配合管道符 <code>|</code> 与别的命令组合从而实现分页展示，例如 <code>tree | more</code> 表示分页打印文件目录。</p><h4 id="输出重定向">输出重定向</h4><p>标准输出（stdout）默认是显示器。</p><p><code>&gt;</code> 表示创建或覆盖，<code>&gt;&gt;</code> 表示追加。</p><h3 id="重要命令">重要命令</h3><h4 id="查找-find">查找 find *</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">find &lt;path&gt; &lt;expression&gt;<br></code></pre></td></tr></tbody></table></figure><ul><li>-maxdepth LEVELS, -mindepth LEVELS: 最大、最小搜索深度</li></ul><h4 id="匹配-grep">匹配 grep *</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">grep [option] &lt;pattern&gt; &lt;<span class="hljs-built_in">source</span>&gt;<br></code></pre></td></tr></tbody></table></figure><p>使用正则表达式在指定文件中进行模式匹配。</p><ul><li>option: <code>-n</code> 显示行号，<code>-i</code> 忽略大小写，<code>-r</code> 递归搜索，<code>-c</code> 打印匹配数量。</li></ul><h2 id="用户管理-权限管理（实验2）">用户管理 | 权限管理（实验2）</h2><p>在 Windows 中，我们对用户和权限的概念接触的并不多，因为很多东西都默认设置好了；但是在 Linux 中，很多文件的权限都需要自己配置和定义，因此「用户与权限管理」的操作方法十分重要。我们从现象入手逐个进行讲解。</p><p>首先以 root 用户身份登录并进入 <code>/opt/OS/task2/</code> 目录，然后创建一个测试文件 <code>root_file.txt</code> 和一个测试文件夹 <code>root_folder</code>。使用 <code>ls -l</code> 命令列出当前目录下所有文件的详细信息：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409271003057.png" alt="root 用户创建的文件和文件夹"></p><p>可以看到一共有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>6</mn></mrow><annotation encoding="application/x-tex">6</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">6</span></span></span></span> 列信息，从左到右依次为：用户访问权限、与文件链接的个数、文件属主、文件属组、文件大小、最后修改日期、文件/目录名。<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mo>−</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">2-5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">5</span></span></span></span> 列的信息都很显然，第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 列信息一共有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10</mn></mrow><annotation encoding="application/x-tex">10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span></span></span></span> 个字符，其中第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 个字符表示当前文件的类型，共有如下几种：</p><table><thead><tr><th style="text-align:center">文件类型</th><th style="text-align:center">符号</th></tr></thead><tbody><tr><td style="text-align:center">普通文件</td><td style="text-align:center"><code>-</code></td></tr><tr><td style="text-align:center">目录</td><td style="text-align:center"><code>d</code></td></tr><tr><td style="text-align:center">字符设备文件</td><td style="text-align:center"><code>c</code></td></tr><tr><td style="text-align:center">块设备文件</td><td style="text-align:center"><code>b</code></td></tr><tr><td style="text-align:center">符号链接</td><td style="text-align:center"><code>l</code></td></tr><tr><td style="text-align:center">本地域套接口</td><td style="text-align:center"><code>s</code></td></tr><tr><td style="text-align:center">有名管道</td><td style="text-align:center"><code>p</code></td></tr></tbody></table><p>第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mo>−</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">2-10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span></span></span></span> 共 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>9</mn></mrow><annotation encoding="application/x-tex">9</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">9</span></span></span></span> 个字符分 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn></mrow><annotation encoding="application/x-tex">3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span></span></span></span> 个一组，分别表示「属主 user 权限、属组 group 权限、其他用户 other 权限」，下面分别介绍「用户和组」以及「权限管理」两个概念。</p><h3 id="关于用户和组">关于用户和组</h3><p>首先我们需要明确用户和组这两个概念的定义：</p><ul><li><strong>文件用户（user/u）</strong>：文件的拥有者。</li><li><strong>所属组（group/g）</strong>：与该文件关联的用户组，组内成员享有特定的权限。</li><li><strong>其他用户（others/o）</strong>：系统中不属于拥有者或组的其他用户。</li></ul><p>举个例子。对于当前用户 <code>now_user</code> 以及当前用户所在的组 <code>now_group</code>，同组用户 <code>adj_user</code> 和其他用户 <code>other_user</code> 可以形象的理解为以下的集合关系：</p><pre><code class=" mermaid">graph TB    subgraph ag[another_group]    other_user1    other_user2    end    subgraph ng[now_group]    now_user    adj_user    end</code></pre><h3 id="关于权限管理">关于权限管理</h3><p>一共有 3 种权限，如下表所示：</p><table><thead><tr><th style="text-align:center"></th><th style="text-align:center">文件</th><th style="text-align:center">目录</th></tr></thead><tbody><tr><td style="text-align:center"><code>r</code></td><td style="text-align:center">可查看文件</td><td style="text-align:center">能列出目录下内容</td></tr><tr><td style="text-align:center"><code>w</code></td><td style="text-align:center">可修改文件</td><td style="text-align:center">能在目录中创建、删除和重命名文件</td></tr><tr><td style="text-align:center"><code>x</code></td><td style="text-align:center">可执行文件</td><td style="text-align:center">能进入目录</td></tr></tbody></table><p>那么我们平时看到的关于权限还有数字的配置，是怎么回事呢？其实是对上述字符配置的八进制数字化。读 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span> 对应 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn></mrow><annotation encoding="application/x-tex">4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">4</span></span></span></span>，写 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span></span></span></span> 对应 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span>，可执行 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 对应 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，例如如果一个文件对于所有用户都拥有可读、可写、可执行权限，那么就是 <code>rwxrwxrwx</code>，对应到数字就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>777</mn></mrow><annotation encoding="application/x-tex">777</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">777</span></span></span></span>。</p><h3 id="相关命令">相关命令</h3><p>下面罗列一些用户与权限管理相关的命令，打 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∗</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span> 表示重要命令。</p><h4 id="提升权限-sudo">提升权限 sudo</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">sudo</span> ...<br></code></pre></td></tr></tbody></table></figure><p>使用 sudo 前缀可以使得当前用户的权限提升到 root 权限，如果是 root 用户则无需添加。当然使用 sudo 的前提是将当前用户添加到</p><h4 id="查看当前用户-whoami">查看当前用户 whoami</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">whoami</span><br></code></pre></td></tr></tbody></table></figure><p>打印当前用户名。</p><h4 id="创建用户-useradd">创建用户 useradd</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">useradd &lt;username&gt;<br></code></pre></td></tr></tbody></table></figure><p>添加用户。</p><h4 id="删除用户-userdel">删除用户 userdel</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">userdel &lt;username&gt;<br></code></pre></td></tr></tbody></table></figure><p>删除用户。</p><ul><li><code>-r</code> 表示同时删除数据信息。</li></ul><h4 id="修改用户信息-usermod">修改用户信息 usermod</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">usermod<br></code></pre></td></tr></tbody></table></figure><p>使用 <code>-h</code> 参数查看所有用法。</p><h4 id="修改用户密码-passwd">修改用户密码 passwd</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">passwd &lt;username&gt;<br></code></pre></td></tr></tbody></table></figure><h4 id="切换用户-su">切换用户 su</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">su &lt;username&gt;<br></code></pre></td></tr></tbody></table></figure><p>添加 <code>-</code> 参数则直接进入 <code>/home/&lt;username&gt;/</code> 目录（如果有的话）。</p><h4 id="查看当前用户所属组-groups">查看当前用户所属组 groups</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">groups</span><br></code></pre></td></tr></tbody></table></figure><h4 id="创建用户组-groupadd">创建用户组 groupadd</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">groupadd<br></code></pre></td></tr></tbody></table></figure><h4 id="删除用户组-groupdel">删除用户组 groupdel</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">groupdel<br></code></pre></td></tr></tbody></table></figure><h4 id="改变属主-chown">改变属主 chown *</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">chown</span> &lt;user&gt; &lt;filename&gt;<br></code></pre></td></tr></tbody></table></figure><p>将指定文件 filename 更改属主为 user。</p><h4 id="改变属组-chgrp">改变属组 chgrp *</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">chgrp</span> &lt;group&gt; &lt;filename&gt;<br></code></pre></td></tr></tbody></table></figure><p>将指定文件 filename 更改属组为 group。</p><h4 id="改变权限-chmod">改变权限 chmod *</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">chmod</span> &lt;option&gt; &lt;filename&gt;<br></code></pre></td></tr></tbody></table></figure><p>将文件 filename 更改所有用户对应的权限。举个例子就知道了：让 <a href="http://demo.py">demo.py</a> 文件只能让所有者拥有可读、可写和可执行权限，其余任何用户都只有可读和可写权限。</p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-comment"># 写法一</span><br><span class="hljs-built_in">chmod</span> u=rwx,go=rw demo.py<br><br><span class="hljs-comment"># 写法二</span><br><span class="hljs-built_in">chmod</span> 766 demo.py<br></code></pre></td></tr></tbody></table></figure><p>至于为什么数字表示法会用 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">4,2,1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span></span></span></span>，是因为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">4,2,1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span></span></span></span> 刚好对应了二进制的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>001</mn><mo separator="true">,</mo><mn>010</mn><mo separator="true">,</mo><mn>100</mn></mrow><annotation encoding="application/x-tex">001, 010, 100</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">001</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">010</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">100</span></span></span></span>，三者的组合可以完美的表示出 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mn>7</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0,7]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">7</span><span class="mclose">]</span></span></span></span> 范围内的任何一个数。</p><h4 id="默认权限-umask">默认权限 umask</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">umask</span><br></code></pre></td></tr></tbody></table></figure><ul><li><code>-S</code> 显示字符型默认权限</li></ul><p>直接使用 <code>umask</code> 会显示 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn></mrow><annotation encoding="application/x-tex">4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">4</span></span></span></span> 位八进制数，第一位是当前用户的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mi>i</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">uid</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">u</span><span class="mord mathnormal">i</span><span class="mord mathnormal">d</span></span></span></span>，后三位分别表示当前用户创建文件时的默认权限的补，例如 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0022</mn></mrow><annotation encoding="application/x-tex">0022</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0022</span></span></span></span> 表示当前用户 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mi>i</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">uid</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">u</span><span class="mord mathnormal">i</span><span class="mord mathnormal">d</span></span></span></span> 为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，创建的文件/目录默认权限为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>777</mn><mo>−</mo><mn>022</mn><mo>=</mo><mn>755</mn></mrow><annotation encoding="application/x-tex">777-022=755</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">777</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">022</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">755</span></span></span></span>。</p><p>可能是出于安全考虑，文件默认不允许拥有可执行权限，因此如果 <code>umask</code> 显示为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0022</mn></mrow><annotation encoding="application/x-tex">0022</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0022</span></span></span></span>，则创建的文件默认权限为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>644</mn></mrow><annotation encoding="application/x-tex">644</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">644</span></span></span></span>，即每一位都 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span> 以确保是偶数。</p>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-7af5d8f3" role="button" aria-expanded="false" aria-controls="collapse-7af5d8f3">        <div class="fold-arrow">▶</div>练习      </div>      <div class="fold-collapse collapse" id="collapse-7af5d8f3">        <div class="fold-content">          <p>一、添加 4 个用户：alice、bob、john、mike</p><p>首先需要确保当前是 root 用户，使用 <code>su root</code> 切换到 root 用户。然后在创建用户时同时创建该用户对应的目录：</p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><code class="hljs bash">useradd -d /home/alice -m alice<br>useradd -d /home/bob -m bob<br>useradd -d /home/john -m john<br>useradd -d /home/mike -m mike<br></code></pre></td></tr></tbody></table></figure><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409272339074.png" alt="添加 4 个用户"></p><p>二、为 alice 设置密码</p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">passwd alice<br></code></pre></td></tr></tbody></table></figure><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409272340323.png" alt="为 alice 设置密码"></p><p>三、创建用户组 workgroup 并将 alice、bob、john 加入</p><ul><li><p>创建用户组：</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">groupadd workgroup<br></code></pre></td></tr></tbody></table></figure></li><li><p>添加到新组：</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><code class="hljs bash">usermod -a -G workgroup alice<br>usermod -a -G workgroup bob<br>usermod -a -G workgroup john<br></code></pre></td></tr></tbody></table></figure><ul><li><code>-a</code>：是 <code>--append</code> 的缩写，表示将用户添加到一个组，而不会移除她已有的其他组。这个选项必须与 <code>-G</code> 一起使用</li><li><code>-G</code>：指定要添加用户的附加组（即用户可以属于多个组），这里是 workgroup</li></ul></li><li><p>将 workgroup 作为各自的主组：</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><code class="hljs bash">usermod -g workgroup alice<br>usermod -g workgroup bob<br>usermod -g workgroup john<br></code></pre></td></tr></tbody></table></figure><ul><li><code>-g</code>：用于指定用户的主组（primary group）。主组是当用户创建文件或目录时默认分配的组</li></ul></li></ul><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409272341900.png" alt="创建用户组 workgroup 并将 alice、bob、john 加入"></p><p>四、创建 <code>/home/work</code> 目录并将其属主改为 alice，属组改为 workgroup</p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-comment"># 创建目录</span><br><span class="hljs-built_in">mkdir</span> work<br><br><span class="hljs-comment"># 修改属主和属组</span><br><span class="hljs-built_in">chown</span> alice:workgroup work<br><br><span class="hljs-comment"># 或者</span><br><span class="hljs-built_in">chown</span> alice.workgroup work<br></code></pre></td></tr></tbody></table></figure><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409272343315.png" alt="创建 /home/work 目录并将其属主改为 alice，属组改为 workgroup"></p><p>五、修改 work 目录的权限</p><p>使得属组内的用户对该目录具有所有权限，属组外的用户对该目录没有任何权限。</p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-comment"># 写法一</span><br><span class="hljs-built_in">chmod</span> ug+rwx,o-rwx work<br><br><span class="hljs-comment"># 写法二</span><br><span class="hljs-built_in">chmod</span> 770 work<br></code></pre></td></tr></tbody></table></figure><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409272345642.png" alt="修改 work 目录的权限"></p><p>六、权限功能测试</p><p>以 bob 用户身份在 work 目录下创建 <code>bob.txt</code> 文件。可以看到符合默认创建文件的权限格式 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>644</mn></mrow><annotation encoding="application/x-tex">644</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">644</span></span></span></span>：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409272350694.png" alt="以 bob 用户在 work 下创建 bob.txt 文件"></p><p>同组用户与不同组用户关于「目录/文件」的 <code>rw</code> 权限测试。</p><ul><li>关于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>770</mn></mrow><annotation encoding="application/x-tex">770</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">770</span></span></span></span> 目录。由于 work 目录被 bob 创建时权限设置为了 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>770</mn></mrow><annotation encoding="application/x-tex">770</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">770</span></span></span></span>，bob 用户与 john 用户属于同一个组 workgroup，因此 john 因为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo>=</mo><mn>7</mn></mrow><annotation encoding="application/x-tex">g=7</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">7</span></span></span></span> 可以进入 work 目录进行操作，而 bob 用户与 mike 用户不属于同一个组，因此 mike 因为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>o</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">o=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">o</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 无法进入 work 目录，更不用说查看或者修改 work 目录中的文件了。</li><li>关于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>644</mn></mrow><annotation encoding="application/x-tex">644</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">644</span></span></span></span> 文件。现在 john 由于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>770</mn></mrow><annotation encoding="application/x-tex">770</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">770</span></span></span></span> 中的第二个 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>7</mn></mrow><annotation encoding="application/x-tex">7</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">7</span></span></span></span> 进入了 work 目录。由文件默认的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>644</mn></mrow><annotation encoding="application/x-tex">644</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">644</span></span></span></span> 权限可以知道：john 因为第一个 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn></mrow><annotation encoding="application/x-tex">4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">4</span></span></span></span> 可以读文件，但是不可以写文件，因此如下图所示，可以执行 <code>cat</code> 查看文件内容，但是不可以执行 <code>echo</code> 编辑文件内容。至于 mike，可以看到无论起始是否在 work 目录，都没有权限 <code>cd</code> 到 work 目录或者 <code>ls</code> 查看 work 目录中的内容。</li></ul><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409280007514.png" alt="权限测试"></p>        </div>      </div>    </div><h2 id="进程管理-调试观察（实验3）">进程管理 | 调试观察（实验3）</h2><h3 id="相关命令-2">相关命令</h3><p>下面罗列一些进程相关的命令，打 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∗</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span> 表示重要命令。</p><h2 id="Linux-编程（实验4）">Linux 编程（实验4）</h2><h2 id="工具扩展">工具扩展</h2><h3 id="tree">tree</h3><p>目录可视化工具。</p><p>下载地址：<a href="hhttps://gnuwin32.sourceforge.net/packages/tree.htm">Tree for Windows (sourceforge.net)open in new window</a>，下载 Binaries 的 Zip 文件。解压完成后，将 bin 目录下的 tree.exe 复制到 Git Bash 安装路径下的 usr/bin 文件夹下，即可使用 tree 命令。</p><p>基本命令格式：<code>tree [-option] [dir]</code></p><ul><li>显示中文：<code>-N</code>。如果中文名是中文，不加 <code>-N</code> 有些电脑上是乱码的。</li><li>选择展示的层级：<code>-L [n]</code>。</li><li>只显示文件夹：<code>-d</code>。</li><li>区分文件夹、普通文件、可执行文件：<code>-FC</code>。<code>C</code> 是加上颜色。</li><li>起别名：<code>alias tree='tree -FCN'</code>。</li><li>输出目录结构到文件： <code>tree -L 2 -I '*.js|node_modules|*.md|*.json|*.css|*.ht' &gt; tree.txt</code>。</li></ul><h3 id="wget">wget</h3><p>url 下载工具。</p><p>下载地址：<a href="https://eternallybored.org/misc/wget/">Windows binaries of GNU Wget</a>，选择合适的版本和架构包进行下载。将 wget.exe 移动到 Git Bash 安装路径下的 usr/bin 文件夹下，即可使用 wget 命令。</p><p>基本命令格式：<code>wget [url]</code></p><ul><li>指定文件名：<code>-O</code>。</li><li>指定目录：<code>-P</code>。</li><li>下载多个文件：<code>wget -i [url.txt]</code>。</li><li>断点续传：<code>wget -c -t [n] [url]</code>。<code>n</code> 代表尝试的次数，<code>0</code> 代表一直尝试。</li><li>后台执行：<code>wget -b [url]</code>。执行该命令的回显信息都会自动存储在 <code>wget.log</code> 文件中。</li><li>下载一个网站的所有图片、视频和 pdf 文件：<code>wget -r -A.pdf url</code>。</li></ul><h2 id="参考引用">参考引用</h2><p><a href="https://northword.cn/code/bash-for-windows/">在 Windows 中使用 Bash shell</a></p><p><a href="https://chatgpt.com/share/66f61785-7ed0-800c-bdf3-fbbfc84a56ae">Linux 用户权限信息</a></p>]]></content>
+    <content type="html"><![CDATA[<h2 id="Shell-基础">Shell 基础</h2><h2 id="前言">前言</h2><p>尝试在 Windows11 上使用 wget 通过一个 url 下载一个 js 静态文件。但是发现本地 bash 使用 wget 命令报错：command not found。于是借 wget 之名展开”古老“的 shell 之旅。</p><p><strong>Shell 是什么</strong>？我们用 GLM4 总结一下：</p><blockquote><p>Shell 的准确定义是：在操作系统中，Shell 是一个程序，它提供了一个用户界面，允许用户通过命令行输入指令来与操作系统交互。它充当用户与操作系统内核之间的中介，处理用户的输入，解释这些输入为操作系统可以理解和执行的操作，并返回执行结果。</p></blockquote><p>也就是说 shell 是一个命令解释器。可以将其理解为人机交互的中间层，通过解析字符串命令与操作系统交互，进而完成系统资源的调用与处理。可以将「解析命令并调用系统资源完成任务」这个过程简单地理解为：shell 解析输入 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> shell 调用对应的 C 函数向 OS 发起请求 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munderover><mo stretchy="true" minsize="3.0em">→</mo><mpadded width="+0.6em" lspace="0.3em"><mrow></mrow></mpadded><mpadded width="+0.6em" lspace="0.3em"><mtext>API</mtext></mpadded></munderover></mrow><annotation encoding="application/x-tex">\xrightarrow[]{\text{API}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2223em;vertical-align:-0.122em;"></span><span class="mrel x-arrow"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1003em;"><span style="top:-3.322em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">API</span></span></span></span></span><span class="svg-align" style="top:-2.689em;"><span class="pstrut" style="height:2.7em;"></span><span class="hide-tail" style="height:0.522em;min-width:1.469em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="0.522em" viewBox="0 0 400000 522" preserveAspectRatio="xMaxYMin slice"><path d="M0 241v40h399891c-47.3 35.3-84 78-110 128-16.7 32-27.7 63.7-33 95 0 1.3-.2 2.7-.5 4-.3 1.3-.5 2.3-.5 3 0 7.3 6.7 11 20 11 8 0 13.2-.8 15.5-2.5 2.3-1.7 4.2-5.5 5.5-11.5 2-13.3 5.7-27 11-41 14.7-44.7 39-84.5 73-119.5s73.7-60.2 119-75.5c6-2 9-5.7 9-11s-3-9-9-11c-45.3-15.3-85-40.5-119-75.5s-58.3-74.8-73-119.5c-4.7-14-8.3-27.3-11-40-1.3-6.7-3.2-10.8-5.5-12.5-2.3-1.7-7.5-2.5-15.5-2.5-14 0-21 3.7-21 11 0 2 2 10.3 6 25 20.7 83.3 67 151.7 139 205zm0 0v40h399900v-40z"></path></svg></span></span><span style="top:-2.578em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight x-arrow-pad"><span class="mord mtight"></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.122em;"><span></span></span></span></span></span></span></span></span> OS 接受请求并执行相关底层操作 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> OS 返回执行结果给 C 函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> shell 接受 C 函数返回结果并显示提示信息。</p><p><strong>为什么用 Bash Shell</strong>？常见的命令解释器有：Bash (Bourne Again Shell)、Sh (Bourne Shell)、Zsh (Z Shell) 等。其中 Bash 是最流行的跨平台 Shell 之一。虽然它是 Linux 和 macOS 的默认 Shell，但也可以在 Windows 上通过 Windows Subsystem for Linux (WSL) 或第三方工具如 Git Bash 来运行。</p><p><strong>文章标题为什么不叫 Linux 基础</strong>？在我看来，Linux 基础的内容更多应当偏向于讲解  Linux 的理论知识，然而目前主流的讲解 Linux 基础的内容均偏向于指导大家如何使用 shell 与操作系统交互，那这与 python 等解释型编程语言有什么区别？python 是用 python 解释器与操作系统交互，相关的教程默认都叫「Python 基础」，shell 是用 shell 解释器与操作系统交互，那为什么不叫「Shell 基础」呢？</p><p>注：本文基于 Windows11 的 Git Bash 命令解释器配合 Ubuntu22.04 的 Bash 命令解释器展开。两个平台的 Bash 版本信息如下：</p><p>Windows11 <strong>Git Bash</strong> Version Info:</p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br></pre></td><td class="code"><pre><code class="hljs bash">$ bash --version<br>GNU bash, version 5.2.26(1)-release (x86_64-pc-msys)<br>Copyright (C) 2022 Free Software Foundation, Inc.<br>License GPLv3+: GNU GPL version 3 or later &lt;http://gnu.org/licenses/gpl.html&gt;<br><br>This is free software; you are free to change and redistribute it.<br>There is NO WARRANTY, to the extent permitted by law.<br></code></pre></td></tr></tbody></table></figure><p>Linux <strong>Bash</strong> Version Info:</p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br></pre></td><td class="code"><pre><code class="hljs bash">root@dwj2:~# bash --version<br>GNU bash，版本 5.1.16(1)-release (x86_64-pc-linux-gnu)<br>Copyright (C) 2020 Free Software Foundation, Inc.<br>许可证 GPLv3+：GNU GPL 许可证第三版或者更新版本 &lt;http://gnu.org/licenses/gpl.html&gt;<br><br>本软件是自由软件，您可以自由地更改和重新发布。<br>在法律许可的情况下特此明示，本软件不提供任何担保。<br></code></pre></td></tr></tbody></table></figure><h2 id="Shell-基本命令（实验1）">Shell 基本命令（实验1）</h2><p>在开始之前，我们可以使用 <code>echo $SHELL</code> 命令查看当前的命令解释器是什么。</p><h3 id="常用命令">常用命令</h3><h4 id="改变目录-cd">改变目录 cd</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">cd</span> ../<br></code></pre></td></tr></tbody></table></figure><p><code>../</code> 表示上一级，<code>./</code> 表示当前一级（也可以不写），<code>/</code> 表示从根目录开始。</p><h4 id="打印目录内容-ls">打印目录内容 ls</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">ls</span><br></code></pre></td></tr></tbody></table></figure><ul><li>添加 <code>-l</code> 参数打印详细信息。</li></ul><h4 id="打印当前路径-pwd">打印当前路径 pwd</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">pwd</span><br></code></pre></td></tr></tbody></table></figure><h4 id="打印当前用户名">打印当前用户名</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">whoami</span><br></code></pre></td></tr></tbody></table></figure><h4 id="创建文件夹-mkdir">创建文件夹 mkdir</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">mkdir</span> &lt;FolderName&gt;<br></code></pre></td></tr></tbody></table></figure><h4 id="创建文件-touch">创建文件 touch</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">touch</span> &lt;FileName&gt;<br></code></pre></td></tr></tbody></table></figure><h4 id="复制-cp">复制 cp</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">cp</span> [option] &lt;<span class="hljs-built_in">source</span>&gt; &lt;target&gt;<br></code></pre></td></tr></tbody></table></figure><ul><li>option 中：<code>-r</code> 表示递归复制，<code>-i</code> 用来当出现重名文件时进行提示。</li><li>source 表示被拷贝的资源。</li><li>target 表示拷贝后的资源名称或者路径。</li></ul><h4 id="移动-mv">移动 mv</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">mv</span> [option] &lt;<span class="hljs-built_in">source</span>&gt; &lt;target&gt;<br></code></pre></td></tr></tbody></table></figure><ul><li>option 中：<code>-i</code> 用来当出现重名文件时进行提示。</li><li>source 表示被移动的资源。</li><li>target 表示移动后的资源名称或者路径（可以以此进行重命名）。</li></ul><h4 id="删除-rm">删除 rm</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">rm</span> [option] &lt;<span class="hljs-built_in">source</span>&gt;<br></code></pre></td></tr></tbody></table></figure><ul><li>option 中：<code>-i</code> 需要一个一个确认，<code>-f</code> 表示强制删除，<code>-r</code> 表示递归删除。</li></ul><h4 id="打印-echo">打印 echo</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">echo</span> <span class="hljs-string">"hello"</span><br></code></pre></td></tr></tbody></table></figure><p>将 echo 后面紧跟着的内容打印到命令行解释器中。可以用来查看环境变量的具体值。</p><p>也可以配合输出重定向符号 <code>&gt;</code> 将信息打印到文件中实现创建新文件的作用。例如 <code>echo "hello" &gt; file.txt</code> 用于创建 file.txt 文件并且在其中写下 <code>hello</code> 信息。</p><h4 id="查看-cat">查看 cat</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">cat</span> [option] &lt;<span class="hljs-built_in">source</span>&gt;<br></code></pre></td></tr></tbody></table></figure><ul><li>option: -n 或 --number 表示将 source 中的内容打印出来的同时带上行号。</li><li>也可以配合输出重定位符号 <code>&gt;</code> 将信息输出到文件中。例如 <code>cat -n a.txt &gt; b.txt</code> 表示将 a.txt 文件中的内容带上行号输出到 b.txt 文件中。</li></ul><h4 id="分页查看-more">分页查看 more</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">more &lt;<span class="hljs-built_in">source</span>&gt;<br></code></pre></td></tr></tbody></table></figure><p>与 cat 类似，只不过可以分页展示。按空格键下一页，b 键上一页，q 键退出。</p><p>可以配合管道符 <code>|</code> 与别的命令组合从而实现分页展示，例如 <code>tree | more</code> 表示分页打印文件目录。</p><h4 id="输出重定向">输出重定向</h4><p>标准输出（stdout）默认是显示器。</p><p><code>&gt;</code> 表示创建或覆盖，<code>&gt;&gt;</code> 表示追加。</p><h3 id="重要命令">重要命令</h3><h4 id="查找-find">查找 find *</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">find &lt;path&gt; &lt;expression&gt;<br></code></pre></td></tr></tbody></table></figure><ul><li>-maxdepth LEVELS, -mindepth LEVELS: 最大、最小搜索深度</li></ul><h4 id="匹配-grep">匹配 grep *</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">grep [option] &lt;pattern&gt; &lt;<span class="hljs-built_in">source</span>&gt;<br></code></pre></td></tr></tbody></table></figure><p>使用正则表达式在指定文件中进行模式匹配。</p><ul><li>option: <code>-n</code> 显示行号，<code>-i</code> 忽略大小写，<code>-r</code> 递归搜索，<code>-c</code> 打印匹配数量。</li></ul><h2 id="用户与权限管理（实验2）">用户与权限管理（实验2）</h2><p>在 Windows 中，我们对用户和权限的概念接触的并不多，因为很多东西都默认设置好了；但是在 Linux 中，很多文件的权限都需要自己配置和定义，因此「用户与权限管理」的操作方法十分重要。我们从现象入手逐个进行讲解。</p><p>首先以 root 用户身份登录并进入 <code>/opt/OS/task2/</code> 目录，然后创建一个测试文件 <code>root_file.txt</code> 和一个测试文件夹 <code>root_folder</code>。使用 <code>ls -l</code> 命令列出当前目录下所有文件的详细信息：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409271003057.png" alt="root 用户创建的文件和文件夹"></p><p>可以看到一共有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>6</mn></mrow><annotation encoding="application/x-tex">6</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">6</span></span></span></span> 列信息，从左到右依次为：用户访问权限、与文件链接的个数、文件属主、文件属组、文件大小、最后修改日期、文件/目录名。<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mo>−</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">2-5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">5</span></span></span></span> 列的信息都很显然，第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 列信息一共有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10</mn></mrow><annotation encoding="application/x-tex">10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span></span></span></span> 个字符，其中第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 个字符表示当前文件的类型，共有如下几种：</p><table><thead><tr><th style="text-align:center">文件类型</th><th style="text-align:center">符号</th></tr></thead><tbody><tr><td style="text-align:center">普通文件</td><td style="text-align:center"><code>-</code></td></tr><tr><td style="text-align:center">目录</td><td style="text-align:center"><code>d</code></td></tr><tr><td style="text-align:center">字符设备文件</td><td style="text-align:center"><code>c</code></td></tr><tr><td style="text-align:center">块设备文件</td><td style="text-align:center"><code>b</code></td></tr><tr><td style="text-align:center">符号链接</td><td style="text-align:center"><code>l</code></td></tr><tr><td style="text-align:center">本地域套接口</td><td style="text-align:center"><code>s</code></td></tr><tr><td style="text-align:center">有名管道</td><td style="text-align:center"><code>p</code></td></tr></tbody></table><p>第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mo>−</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">2-10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span></span></span></span> 共 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>9</mn></mrow><annotation encoding="application/x-tex">9</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">9</span></span></span></span> 个字符分 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn></mrow><annotation encoding="application/x-tex">3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span></span></span></span> 个一组，分别表示「属主 user 权限、属组 group 权限、其他用户 other 权限」，下面分别介绍「用户和组」以及「权限管理」两个概念。</p><h3 id="关于用户和组">关于用户和组</h3><p>首先我们需要明确用户和组这两个概念的定义：</p><ul><li><strong>文件用户（user/u）</strong>：文件的拥有者。</li><li><strong>所属组（group/g）</strong>：与该文件关联的用户组，组内成员享有特定的权限。</li><li><strong>其他用户（others/o）</strong>：系统中不属于拥有者或组的其他用户。</li></ul><p>举个例子。对于当前用户 <code>now_user</code> 以及当前用户所在的组 <code>now_group</code>，同组用户 <code>adj_user</code> 和其他用户 <code>other_user</code> 可以形象的理解为以下的集合关系：</p><pre><code class=" mermaid">graph TB    subgraph ag[another_group]    other_user1    other_user2    end    subgraph ng[now_group]    now_user    adj_user    end</code></pre><h3 id="关于权限管理">关于权限管理</h3><p>一共有 3 种权限，如下表所示：</p><table><thead><tr><th style="text-align:center"></th><th style="text-align:center">文件</th><th style="text-align:center">目录</th></tr></thead><tbody><tr><td style="text-align:center"><code>r</code></td><td style="text-align:center">可查看文件</td><td style="text-align:center">能列出目录下内容</td></tr><tr><td style="text-align:center"><code>w</code></td><td style="text-align:center">可修改文件</td><td style="text-align:center">能在目录中创建、删除和重命名文件</td></tr><tr><td style="text-align:center"><code>x</code></td><td style="text-align:center">可执行文件</td><td style="text-align:center">能进入目录</td></tr></tbody></table><p>那么我们平时看到的关于权限还有数字的配置，是怎么回事呢？其实是对上述字符配置的八进制数字化。读 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span> 对应 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn></mrow><annotation encoding="application/x-tex">4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">4</span></span></span></span>，写 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span></span></span></span> 对应 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span>，可执行 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 对应 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，例如如果一个文件对于所有用户都拥有可读、可写、可执行权限，那么就是 <code>rwxrwxrwx</code>，对应到数字就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>777</mn></mrow><annotation encoding="application/x-tex">777</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">777</span></span></span></span>。</p><h3 id="相关命令">相关命令</h3><p>下面罗列一些用户与权限管理相关的命令，打 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∗</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4653em;"></span><span class="mord">∗</span></span></span></span> 表示重要命令。</p><h4 id="提升权限-sudo">提升权限 sudo</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">sudo</span> ...<br></code></pre></td></tr></tbody></table></figure><p>使用 sudo 前缀可以使得当前用户的权限提升到 root 权限，如果是 root 用户则无需添加。当然使用 sudo 的前提是将当前用户添加到</p><h4 id="查看当前用户-whoami">查看当前用户 whoami</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">whoami</span><br></code></pre></td></tr></tbody></table></figure><p>打印当前用户名。</p><h4 id="创建用户-useradd">创建用户 useradd</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">useradd &lt;username&gt;<br></code></pre></td></tr></tbody></table></figure><p>添加用户。</p><h4 id="删除用户-userdel">删除用户 userdel</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">userdel &lt;username&gt;<br></code></pre></td></tr></tbody></table></figure><p>删除用户。</p><ul><li><code>-r</code> 表示同时删除数据信息。</li></ul><h4 id="修改用户信息-usermod">修改用户信息 usermod</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">usermod<br></code></pre></td></tr></tbody></table></figure><p>使用 <code>-h</code> 参数查看所有用法。</p><h4 id="修改用户密码-passwd">修改用户密码 passwd</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">passwd &lt;username&gt;<br></code></pre></td></tr></tbody></table></figure><h4 id="切换用户-su">切换用户 su</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">su &lt;username&gt;<br></code></pre></td></tr></tbody></table></figure><p>添加 <code>-</code> 参数则直接进入 <code>/home/&lt;username&gt;/</code> 目录（如果有的话）。</p><h4 id="查看当前用户所属组-groups">查看当前用户所属组 groups</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">groups</span><br></code></pre></td></tr></tbody></table></figure><h4 id="创建用户组-groupadd">创建用户组 groupadd</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">groupadd<br></code></pre></td></tr></tbody></table></figure><h4 id="删除用户组-groupdel">删除用户组 groupdel</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">groupdel<br></code></pre></td></tr></tbody></table></figure><h4 id="改变属主-chown">改变属主 chown *</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">chown</span> &lt;user&gt; &lt;filename&gt;<br></code></pre></td></tr></tbody></table></figure><p>将指定文件 filename 更改属主为 user。</p><h4 id="改变属组-chgrp">改变属组 chgrp *</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">chgrp</span> &lt;group&gt; &lt;filename&gt;<br></code></pre></td></tr></tbody></table></figure><p>将指定文件 filename 更改属组为 group。</p><h4 id="改变权限-chmod">改变权限 chmod *</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">chmod</span> &lt;option&gt; &lt;filename&gt;<br></code></pre></td></tr></tbody></table></figure><p>将文件 filename 更改所有用户对应的权限。举个例子就知道了：让 <a href="http://demo.py">demo.py</a> 文件只能让所有者拥有可读、可写和可执行权限，其余任何用户都只有可读和可写权限。</p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-comment"># 写法一</span><br><span class="hljs-built_in">chmod</span> u=rwx,go=rw demo.py<br><br><span class="hljs-comment"># 写法二</span><br><span class="hljs-built_in">chmod</span> 766 demo.py<br></code></pre></td></tr></tbody></table></figure><p>至于为什么数字表示法会用 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">4,2,1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span></span></span></span>，是因为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">4,2,1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span></span></span></span> 刚好对应了二进制的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>001</mn><mo separator="true">,</mo><mn>010</mn><mo separator="true">,</mo><mn>100</mn></mrow><annotation encoding="application/x-tex">001, 010, 100</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">001</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">010</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">100</span></span></span></span>，三者的组合可以完美的表示出 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mn>7</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0,7]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">7</span><span class="mclose">]</span></span></span></span> 范围内的任何一个数。</p><h4 id="默认权限-umask">默认权限 umask</h4><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">umask</span><br></code></pre></td></tr></tbody></table></figure><ul><li><code>-S</code> 显示字符型默认权限</li></ul><p>直接使用 <code>umask</code> 会显示 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn></mrow><annotation encoding="application/x-tex">4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">4</span></span></span></span> 位八进制数，第一位是当前用户的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mi>i</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">uid</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">u</span><span class="mord mathnormal">i</span><span class="mord mathnormal">d</span></span></span></span>，后三位分别表示当前用户创建文件时的默认权限的补，例如 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0022</mn></mrow><annotation encoding="application/x-tex">0022</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0022</span></span></span></span> 表示当前用户 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mi>i</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">uid</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">u</span><span class="mord mathnormal">i</span><span class="mord mathnormal">d</span></span></span></span> 为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，创建的文件/目录默认权限为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>777</mn><mo>−</mo><mn>022</mn><mo>=</mo><mn>755</mn></mrow><annotation encoding="application/x-tex">777-022=755</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">777</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">022</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">755</span></span></span></span>。</p><p>可能是出于安全考虑，文件默认不允许拥有可执行权限，因此如果 <code>umask</code> 显示为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0022</mn></mrow><annotation encoding="application/x-tex">0022</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0022</span></span></span></span>，则创建的文件默认权限为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>644</mn></mrow><annotation encoding="application/x-tex">644</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">644</span></span></span></span>，即每一位都 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span> 以确保是偶数。</p>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-79236e41" role="button" aria-expanded="false" aria-controls="collapse-79236e41">        <div class="fold-arrow">▶</div>练习      </div>      <div class="fold-collapse collapse" id="collapse-79236e41">        <div class="fold-content">          <p><strong>一、添加 4 个用户：alice、bob、john、mike</strong></p><p>首先需要确保当前是 root 用户，使用 <code>su root</code> 切换到 root 用户。然后在创建用户时同时创建该用户对应的目录：</p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><code class="hljs bash">useradd -d /home/alice -m alice<br>useradd -d /home/bob -m bob<br>useradd -d /home/john -m john<br>useradd -d /home/mike -m mike<br></code></pre></td></tr></tbody></table></figure><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409272339074.png" alt="添加 4 个用户"></p><p><strong>二、为 alice 设置密码</strong></p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">passwd alice<br></code></pre></td></tr></tbody></table></figure><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409272340323.png" alt="为 alice 设置密码"></p><p><strong>三、创建用户组 workgroup 并将 alice、bob、john 加入</strong></p><ul><li><p>创建用户组：</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">groupadd workgroup<br></code></pre></td></tr></tbody></table></figure></li><li><p>添加到新组：</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><code class="hljs bash">usermod -a -G workgroup alice<br>usermod -a -G workgroup bob<br>usermod -a -G workgroup john<br></code></pre></td></tr></tbody></table></figure><ul><li><code>-a</code>：是 <code>--append</code> 的缩写，表示将用户添加到一个组，而不会移除她已有的其他组。这个选项必须与 <code>-G</code> 一起使用</li><li><code>-G</code>：指定要添加用户的附加组（即用户可以属于多个组），这里是 workgroup</li></ul></li><li><p>将 workgroup 作为各自的主组：</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><code class="hljs bash">usermod -g workgroup alice<br>usermod -g workgroup bob<br>usermod -g workgroup john<br></code></pre></td></tr></tbody></table></figure><ul><li><code>-g</code>：用于指定用户的主组（primary group）。主组是当用户创建文件或目录时默认分配的组</li></ul></li></ul><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409272341900.png" alt="创建用户组 workgroup 并将 alice、bob、john 加入"></p><p><strong>四、创建 <code>/home/work</code> 目录并将其属主改为 alice，属组改为 workgroup</strong></p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-comment"># 创建目录</span><br><span class="hljs-built_in">mkdir</span> work<br><br><span class="hljs-comment"># 修改属主和属组</span><br><span class="hljs-built_in">chown</span> alice:workgroup work<br><br><span class="hljs-comment"># 或者</span><br><span class="hljs-built_in">chown</span> alice.workgroup work<br></code></pre></td></tr></tbody></table></figure><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409272343315.png" alt="创建 /home/work 目录并将其属主改为 alice，属组改为 workgroup"></p><p><strong>五、修改 work 目录的权限</strong></p><p>使得属组内的用户对该目录具有所有权限，属组外的用户对该目录没有任何权限。</p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-comment"># 写法一</span><br><span class="hljs-built_in">chmod</span> ug+rwx,o-rwx work<br><br><span class="hljs-comment"># 写法二</span><br><span class="hljs-built_in">chmod</span> 770 work<br></code></pre></td></tr></tbody></table></figure><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409272345642.png" alt="修改 work 目录的权限"></p><p><strong>六、权限功能测试</strong></p><p>以 bob 用户身份在 work 目录下创建 <code>bob.txt</code> 文件。可以看到符合默认创建文件的权限格式 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>644</mn></mrow><annotation encoding="application/x-tex">644</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">644</span></span></span></span>：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409272350694.png" alt="以 bob 用户在 work 下创建 bob.txt 文件"></p><p>同组用户与不同组用户关于「目录/文件」的 <code>rw</code> 权限测试。</p><ul><li>关于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>770</mn></mrow><annotation encoding="application/x-tex">770</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">770</span></span></span></span> 目录。由于 work 目录被 bob 创建时权限设置为了 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>770</mn></mrow><annotation encoding="application/x-tex">770</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">770</span></span></span></span>，bob 用户与 john 用户属于同一个组 workgroup，因此 john 因为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo>=</mo><mn>7</mn></mrow><annotation encoding="application/x-tex">g=7</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">7</span></span></span></span> 可以进入 work 目录进行操作，而 bob 用户与 mike 用户不属于同一个组，因此 mike 因为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>o</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">o=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">o</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 无法进入 work 目录，更不用说查看或者修改 work 目录中的文件了。</li><li>关于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>644</mn></mrow><annotation encoding="application/x-tex">644</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">644</span></span></span></span> 文件。现在 john 由于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>770</mn></mrow><annotation encoding="application/x-tex">770</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">770</span></span></span></span> 中的第二个 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>7</mn></mrow><annotation encoding="application/x-tex">7</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">7</span></span></span></span> 进入了 work 目录。由文件默认的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>644</mn></mrow><annotation encoding="application/x-tex">644</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">644</span></span></span></span> 权限可以知道：john 因为第一个 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>4</mn></mrow><annotation encoding="application/x-tex">4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">4</span></span></span></span> 可以读文件，但是不可以写文件，因此如下图所示，可以执行 <code>cat</code> 查看文件内容，但是不可以执行 <code>echo</code> 编辑文件内容。至于 mike，可以看到无论起始是否在 work 目录，都没有权限 <code>cd</code> 到 work 目录或者 <code>ls</code> 查看 work 目录中的内容。</li></ul><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409280007514.png" alt="权限测试"></p>        </div>      </div>    </div><h2 id="进程管理与调试（实验3）">进程管理与调试（实验3）</h2><p>在熟悉了 bash shell 的基本命令以及 Linux 中用户与权限管理的基本概念后，我们就可以开始尝试管理 Linux 中的程序了。当然每一个程序在开始运行后都会成为一个或多个进程，因此接下来简单介绍一下 Linux 的进程管理。最后再通过调试一个 C 程序来熟悉 GNU 调试工具 gdb (GNU Debugger) 的使用。</p><h3 id="进程管理">进程管理</h3><p>查看进程状态 ps</p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">ps<br></code></pre></td></tr></tbody></table></figure><p>动态查看进程状态 top</p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">top<br></code></pre></td></tr></tbody></table></figure><p>杀死某个进程 kill</p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">kill</span> -9 &lt;PID&gt;<br></code></pre></td></tr></tbody></table></figure>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-d66a19be" role="button" aria-expanded="false" aria-controls="collapse-d66a19be">        <div class="fold-arrow">▶</div>练习      </div>      <div class="fold-collapse collapse" id="collapse-d66a19be">        <div class="fold-content">          <p><strong>一、编写一个 shell 程序 <code>badproc.sh</code> 使其不断循环</strong></p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-meta">#! /bin/bash</span><br><span class="hljs-keyword">while</span> <span class="hljs-built_in">echo</span> <span class="hljs-string">"I'm making files!"</span><br><span class="hljs-keyword">do</span><br>    <span class="hljs-built_in">mkdir</span> adir<br>    <span class="hljs-built_in">cd</span> adir<br>    <span class="hljs-built_in">touch</span> afile<br>    <span class="hljs-built_in">sleep</span> 10s<br><span class="hljs-keyword">done</span><br></code></pre></td></tr></tbody></table></figure><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081735300.png" alt="编写 sh 文件"></p><p><strong>二、为 <code>badproc.sh</code> 增加可执行权限</strong></p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">chmod</span> u+x badproc.sh<br></code></pre></td></tr></tbody></table></figure><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081736525.png" alt="增加可执行权限"></p><p><strong>三、在后台执行 <code>badproc.sh</code></strong></p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">./badproc.sh &amp;<br></code></pre></td></tr></tbody></table></figure><ul><li><code>&amp;</code> 表示后台执行</li></ul><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081737564.png" alt="后台执行"></p><p><strong>四、利用 <code>ps</code> 命令查看其进程号</strong></p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">ps aux | grep badproc<br></code></pre></td></tr></tbody></table></figure><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081739971.png" alt="查看进程号"></p><p><strong>五、利用 <code>kill</code> 命令杀死该进程</strong></p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">kill</span> -9 &lt;PID&gt;<br></code></pre></td></tr></tbody></table></figure><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081748528.png" alt="杀死该进程"></p><p><strong>六、删除 <code>badproc.sh</code> 程序运行时创建的目录和文件</strong></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081748818.png" alt="删除目录和文件"></p>        </div>      </div>    </div><h3 id="gdb-调试">gdb 调试</h3><p>基本命令参考 <a href="https://www.cnblogs.com/HKUI/p/8955443.html">gdb调试的基本使用</a>。常用的如下：</p><p>开始运行</p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">r<br></code></pre></td></tr></tbody></table></figure><ul><li>r 即 run</li></ul><p>设置断点</p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">break</span> &lt;line&gt;<br></code></pre></td></tr></tbody></table></figure><ul><li>line 即行号</li></ul><p>运行到下一个断点</p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">c<br></code></pre></td></tr></tbody></table></figure><ul><li>c 即 continue</li></ul>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-91d325eb" role="button" aria-expanded="false" aria-controls="collapse-91d325eb">        <div class="fold-arrow">▶</div>练习      </div>      <div class="fold-collapse collapse" id="collapse-91d325eb">        <div class="fold-content">          <p><strong>一、创建 <code>fork.c</code> 文件</strong></p><figure class="highlight c"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br></pre></td><td class="code"><pre><code class="hljs c"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;stdio.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;stdlib.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;unistd.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;sys/types.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;sys/wait.h&gt;</span></span><br><br><span class="hljs-type">int</span> <span class="hljs-title function_">main</span><span class="hljs-params">()</span> {<br>    <span class="hljs-comment">/* fork another process */</span><br>    <span class="hljs-type">pid_t</span>  pid;<br>    pid = fork();<br><br>    <span class="hljs-keyword">if</span> (pid &lt; <span class="hljs-number">0</span>) {<br>        <span class="hljs-comment">/* error occurred */</span><br>        <span class="hljs-built_in">fprintf</span>(<span class="hljs-built_in">stderr</span>, <span class="hljs-string">"Fork Failed"</span>);<br>        <span class="hljs-built_in">exit</span>(<span class="hljs-number">-1</span>);<br>    } <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (pid == <span class="hljs-number">0</span>) {<br>        <span class="hljs-comment">/* child process */</span><br>        <span class="hljs-built_in">printf</span>(<span class="hljs-string">"This is child process, pid=%d\n"</span>, getpid());<br>        execlp(<span class="hljs-string">"/bin/ls"</span>, <span class="hljs-string">"ls"</span>, <span class="hljs-literal">NULL</span>);<br>        <span class="hljs-built_in">printf</span>(<span class="hljs-string">"Child process finished\n"</span>); <span class="hljs-comment">/*这句话不会被打印，除非execlp调用未成功*/</span><br>    } <span class="hljs-keyword">else</span> {<br>        <span class="hljs-comment">/* parent process */</span><br>        <span class="hljs-comment">/* parent will wait for the child to complete */</span><br>        <span class="hljs-built_in">printf</span>(<span class="hljs-string">"This is parent process, pid=%d\n"</span>, getpid());<br>        wait (<span class="hljs-literal">NULL</span>);<br>        <span class="hljs-built_in">printf</span> (<span class="hljs-string">"Child Complete\n"</span>);<br>        <span class="hljs-built_in">exit</span>(<span class="hljs-number">0</span>);<br>    }<br>}<br></code></pre></td></tr></tbody></table></figure><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081813738.png" alt="创建文件"></p><p>这段程序首先通过调用 <code>fork()</code> 函数创建一个子进程，并通过 <code>pid</code> 信息来判断当前进程是父进程还是子进程。在并发的逻辑下，执行哪一个进程的逻辑是未知的。</p><p><strong>二、编译运行 <code>fork.c</code> 文件</strong></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081807870.png" alt="编译运行"></p><p>从上述运行结果可以看出：并发时，首先执行父进程的逻辑，然后才执行子进程的逻辑。</p><p><strong>三、gdb 调试</strong></p><p>在 fork 创建子进程后追踪子进程：</p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><code class="hljs bash">gdb fork<br><span class="hljs-built_in">set</span> follow-fork-mode child<br>catch <span class="hljs-built_in">exec</span><br></code></pre></td></tr></tbody></table></figure><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081947476.png" alt="追踪子进程"></p><p>运行到第一个断点时分别观察父进程 1510168 和子进程 1510171：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081948301.png" alt="父进程 1510168"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081949586.png" alt="子进程 1510171"></p><p>运行到第二个断点时观察子进程 1510171：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081951803.png" alt="运行到第二个断点时观察子进程 1510171"></p><p>从上述子进程的追踪结果可以看出，在父进程结束之后，子进程成功执行了 <code>pid==0</code> 的逻辑并调用 <code>ls</code> 工具。</p>        </div>      </div>    </div><h2 id="Linux-编程（实验4）">Linux 编程（实验4）</h2><h2 id="工具扩展">工具扩展</h2><h3 id="tree">tree</h3><p>目录可视化工具。</p><p>下载地址：<a href="hhttps://gnuwin32.sourceforge.net/packages/tree.htm">Tree for Windows (sourceforge.net)open in new window</a>，下载 Binaries 的 Zip 文件。解压完成后，将 bin 目录下的 tree.exe 复制到 Git Bash 安装路径下的 usr/bin 文件夹下，即可使用 tree 命令。</p><p>基本命令格式：<code>tree [-option] [dir]</code></p><ul><li>显示中文：<code>-N</code>。如果中文名是中文，不加 <code>-N</code> 有些电脑上是乱码的。</li><li>选择展示的层级：<code>-L [n]</code>。</li><li>只显示文件夹：<code>-d</code>。</li><li>区分文件夹、普通文件、可执行文件：<code>-FC</code>。<code>C</code> 是加上颜色。</li><li>起别名：<code>alias tree='tree -FCN'</code>。</li><li>输出目录结构到文件： <code>tree -L 2 -I '*.js|node_modules|*.md|*.json|*.css|*.ht' &gt; tree.txt</code>。</li></ul><h3 id="wget">wget</h3><p>url 下载工具。</p><p>下载地址：<a href="https://eternallybored.org/misc/wget/">Windows binaries of GNU Wget</a>，选择合适的版本和架构包进行下载。将 wget.exe 移动到 Git Bash 安装路径下的 usr/bin 文件夹下，即可使用 wget 命令。</p><p>基本命令格式：<code>wget [url]</code></p><ul><li>指定文件名：<code>-O</code>。</li><li>指定目录：<code>-P</code>。</li><li>下载多个文件：<code>wget -i [url.txt]</code>。</li><li>断点续传：<code>wget -c -t [n] [url]</code>。<code>n</code> 代表尝试的次数，<code>0</code> 代表一直尝试。</li><li>后台执行：<code>wget -b [url]</code>。执行该命令的回显信息都会自动存储在 <code>wget.log</code> 文件中。</li><li>下载一个网站的所有图片、视频和 pdf 文件：<code>wget -r -A.pdf url</code>。</li></ul><h2 id="参考引用">参考引用</h2><p><a href="https://northword.cn/code/bash-for-windows/">在 Windows 中使用 Bash shell</a></p><p><a href="https://chatgpt.com/share/66f61785-7ed0-800c-bdf3-fbbfc84a56ae">Linux 用户权限信息</a></p><p><a href="https://www.cnblogs.com/HKUI/p/8955443.html">gdb调试的基本使用</a></p>]]></content>
     
     
     <categories>
@@ -451,7 +451,7 @@
     <link href="/GPA/5th-term/DigitalImageProcessing/"/>
     <url>/GPA/5th-term/DigitalImageProcessing/</url>
     
-    <content type="html"><![CDATA[<h2 id="数字图像处理">数字图像处理</h2><h2 id="前言">前言</h2><p>学科地位：</p><table><thead><tr><th style="text-align:center">主讲教师</th><th style="text-align:center">学分配额</th><th style="text-align:center">学科类别</th></tr></thead><tbody><tr><td style="text-align:center">李峻</td><td style="text-align:center">3</td><td style="text-align:center">专业课</td></tr></tbody></table><p>成绩组成：</p><table><thead><tr><th style="text-align:center">作业</th><th style="text-align:center">实验</th><th style="text-align:center">期中</th><th style="text-align:center">期末</th></tr></thead><tbody><tr><td style="text-align:center">10%</td><td style="text-align:center">10%</td><td style="text-align:center">30%</td><td style="text-align:center">50%</td></tr></tbody></table><p>教材情况：</p><table><thead><tr><th style="text-align:center">课程名称</th><th style="text-align:center">选用教材</th><th style="text-align:center">版次</th><th style="text-align:center">作者</th><th style="text-align:center">出版社</th><th style="text-align:center">ISBN 号</th></tr></thead><tbody><tr><td style="text-align:center">数字图像处理</td><td style="text-align:center">《数字图像处理 使用 MATLAB 分析与实现》</td><td style="text-align:center">2</td><td style="text-align:center">蔡利梅 王利娟</td><td style="text-align:center">清华大学出版社</td><td style="text-align:center">9787302654100</td></tr><tr><td style="text-align:center">-</td><td style="text-align:center">《智能图像处理 Python 和 OpenCV 实现》</td><td style="text-align:center">1</td><td style="text-align:center">赵云龙 葛广英</td><td style="text-align:center">机械工业出版社</td><td style="text-align:center">9787111694038</td></tr></tbody></table><p>为什么要学这门课？</p><blockquote><p>说实在的我也不知道这门课的意义到底有多大，感觉与目前为止主流的炼丹或者 sys 的关系不大，更像是某个分支，比如平面图形学。但是转念一想，图像其实也是一种数据，也可以用来炼丹；同样的，二维图像的各种处理技术也对应了很多的数学原理和算法策略。这么一想好像也就和其他的课程串起来了。</p></blockquote><p>会收获什么？</p><blockquote><p><strong>先从编程说起</strong>。终于有机会可以正式的任务驱动式的学习 MATLAB 编程和 OpenCV (Python) 编程了。</p><p><strong>当然，没有理论何谈编程</strong>？图像处理的数学理论和算法策略，也会是一个全新且不小的收获，且学且更新。目前看来与炼丹并不是一回事，与传统的数据结构也没有很大的交集。更多的是一种矩阵论、线性代数论。</p></blockquote><h2 id="图像获取">图像获取</h2><p>TODO</p><h2 id="图像处理">图像处理</h2><p>图像是数据存储和表达的一种方式。然而模拟图像无法进行更深入的研究，于是乎，数字图像应运而生。数字图像处理技术也就有了更多的研究意义与应用价值。</p><p><strong>像素是什么</strong>？一张模拟图像被离散化后的 <strong>最小单元</strong>。</p><p><strong>分辨率是什么</strong>？一张模拟图像被离散化后的 <strong>像素数量</strong>。例如，如果一张模拟图像被划分为由 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2560</mn><mo>×</mo><mn>1920</mn></mrow><annotation encoding="application/x-tex">2560\times1920</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2560</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1920</span></span></span></span> 个像素点组成的数字图像，那么这张数字图像的分辨率就是像素点的数量，约 500 万。</p><p><strong>在单一变量原则的情况下，分辨率越高，数字图像就越清晰吗</strong>？显然不是。由分辨率的定义，一张图像的分辨率表示图像中像素点的个数。如果 A, B 两张图的分辨率一致，但是 A 图的面积远超过 B，那么显然的 B 比 A 更清晰。因为 A 的一像素对应的信息量太多了，不够精细，也就导致 A 更模糊。因此我们引出了图像清晰度的度量理念。</p><p><strong>如何定义图像的清晰度</strong>？从上面的分析可知，图像的清晰度不是由像素的数量决定的，而是由 <a href="https://zhuanlan.zhihu.com/p/146713168">像素的密度</a> 决定的。于是引出了以下两种像素密度度量方式：</p><ul><li><p>PPI 用于描述显示设备（如显示器、手机屏幕等）的分辨率（注：下面的屏幕尺寸是指屏幕对角线的长度）：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>PPI</mtext><mo>=</mo><mfrac><msqrt><mrow><mo stretchy="false">(</mo><mtext>水平像素数</mtext><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><mtext>垂直像素数</mtext><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></msqrt><mtext>屏幕尺寸（英寸）</mtext></mfrac></mrow><annotation encoding="application/x-tex">\text{PPI} = \frac{\sqrt{(\text{水平像素数})^2 + (\text{垂直像素数})^2}}{\text{屏幕尺寸（英寸）}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord">PPI</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.316em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.63em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">屏幕尺寸（英寸）</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.695em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.935em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mopen">(</span><span class="mord text"><span class="mord cjk_fallback">水平像素数</span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord text"><span class="mord cjk_fallback">垂直像素数</span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-2.895em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.28em" viewBox="0 0 400000 1296" preserveAspectRatio="xMinYMin slice"><path d="M263,681c0.7,0,18,39.7,52,119c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120c340,-704.7,510.7,-1060.3,512,-1067l0 -0c4.7,-7.3,11,-11,19,-11H40000v40H1012.3s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60zM1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305em;"><span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p></li><li><p>DPI 用于描述打印设备（如打印机）的分辨率：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>DPI</mtext><mo>=</mo><mfrac><mtext>打印的点数</mtext><mtext>每英寸的距离</mtext></mfrac></mrow><annotation encoding="application/x-tex">\text{DPI} = \frac{\text{打印的点数}}{\text{每英寸的距离}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord">DPI</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0463em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">每英寸的距离</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">打印的点数</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p></li></ul><h3 id="图像运算">图像运算</h3><p>首先需要掌握一些最基本的概念：齐次坐标表示、变换矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span>、图像插值策略。</p><h4 id="几何变换">几何变换</h4><p>图像的几何变换基本步骤我们称之为 <strong>后向映射法</strong>。逻辑很简单，根据原图像的四个顶点计算出新图像的尺寸后，逐个填充新图像的像素值即可。那么问题来了：如何计算新图像的尺寸？如何填充新图像的像素值？</p><ul><li>平移</li><li>镜像</li><li>旋转</li><li>缩放</li><li>错切</li></ul><h4 id="代数运算">代数运算</h4><h4 id="领域及模版运算">领域及模版运算</h4><h3 id="图像正交变换">图像正交变换</h3><p>本章我们重点学习傅里叶变换。</p><h3 id="图像增强">图像增强</h3><h3 id="图像平滑">图像平滑</h3><h3 id="图像锐化">图像锐化</h3><h3 id="图像复原">图像复原</h3><h3 id="图像的数学形态学处理">图像的数学形态学处理</h3><h2 id="图像编码">图像编码</h2><h2 id="图像分析">图像分析</h2><h3 id="图像分割">图像分割</h3><h3 id="图像描述与分析">图像描述与分析</h3><h2 id="综合应用">综合应用</h2><h3 id="图像分类">图像分类</h3><h3 id="图像检索">图像检索</h3>]]></content>
+    <content type="html"><![CDATA[<h2 id="数字图像处理">数字图像处理</h2><h2 id="前言">前言</h2><p>学科地位：</p><table><thead><tr><th style="text-align:center">主讲教师</th><th style="text-align:center">学分配额</th><th style="text-align:center">学科类别</th></tr></thead><tbody><tr><td style="text-align:center">李峻</td><td style="text-align:center">3</td><td style="text-align:center">专业课</td></tr></tbody></table><p>成绩组成：</p><table><thead><tr><th style="text-align:center">作业</th><th style="text-align:center">实验</th><th style="text-align:center">期中</th><th style="text-align:center">期末</th></tr></thead><tbody><tr><td style="text-align:center">10%</td><td style="text-align:center">10%</td><td style="text-align:center">30%</td><td style="text-align:center">50%</td></tr></tbody></table><p>教材情况：</p><table><thead><tr><th style="text-align:center">课程名称</th><th style="text-align:center">选用教材</th><th style="text-align:center">版次</th><th style="text-align:center">作者</th><th style="text-align:center">出版社</th><th style="text-align:center">ISBN 号</th></tr></thead><tbody><tr><td style="text-align:center">数字图像处理</td><td style="text-align:center">《数字图像处理 使用 MATLAB 分析与实现》</td><td style="text-align:center">2</td><td style="text-align:center">蔡利梅 王利娟</td><td style="text-align:center">清华大学出版社</td><td style="text-align:center">9787302654100</td></tr><tr><td style="text-align:center">-</td><td style="text-align:center">《智能图像处理 Python 和 OpenCV 实现》</td><td style="text-align:center">1</td><td style="text-align:center">赵云龙 葛广英</td><td style="text-align:center">机械工业出版社</td><td style="text-align:center">9787111694038</td></tr></tbody></table><p>为什么要学这门课？</p><blockquote><p>图像是数据的一种，图像的应用场景极为广泛，因此如何获取、处理、存储数据也就显得十分重要。在本课程中我们将主要学习图像的处理技术。当然，如果非要说数字图像处理和别的课程的关联，愚以为既然图像是数据，那么就可以用到数据算法那一套上，也就与别的课程有了关联。</p></blockquote><p>会收获什么？</p><blockquote><p>图像简单来说就是一个矩阵。如何在这种数据结构上进行理论研究与算法应用显得十分有学习意义。本课程也将使用 MATLAB 和 OpenCV-Python 进行编码实现。</p></blockquote><h2 id="图像获取">图像获取</h2><p><strong>如何从现实世界中的模拟图像转化为计算机世界的数字图像</strong>？由于现实世界中的模拟图像是连续的，而计算机无法处理连续数据，因此我们只能对模拟图像进行离散化采样。进而引出了下面的一些概念。</p><p><strong>像素是什么</strong>？一张模拟图像被离散化后的 <strong>最小单元</strong>。</p><p><strong>分辨率是什么</strong>？一张模拟图像被离散化后的 <strong>像素数量</strong>。例如，如果一张模拟图像被划分为由 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2560</mn><mo>×</mo><mn>1920</mn></mrow><annotation encoding="application/x-tex">2560\times1920</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2560</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1920</span></span></span></span> 个像素点组成的数字图像，那么这张数字图像的分辨率就是像素点的数量，约 500 万。</p><p><strong>在单一变量原则的情况下，分辨率越高，数字图像就越清晰吗</strong>？显然不是。由分辨率的定义，一张图像的分辨率表示图像中像素点的个数。如果 A, B 两张图的分辨率一致，但是 A 图的面积远超过 B，那么显然的 B 比 A 更清晰。因为 A 的一像素对应的信息量太多了，不够精细，也就导致 A 更模糊。因此我们引出了图像清晰度的度量理念。</p><p><strong>如何定义图像的清晰度</strong>？从上面的分析可知，图像的清晰度不是由像素的数量决定的，而是由 <a href="https://zhuanlan.zhihu.com/p/146713168">像素的密度</a> 决定的。于是引出了以下两种像素密度度量方式：</p><ul><li><p>PPI 用于描述显示设备（如显示器、手机屏幕等）的分辨率（注：下面的屏幕尺寸是指屏幕对角线的长度）：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>PPI</mtext><mo>=</mo><mfrac><msqrt><mrow><mo stretchy="false">(</mo><mtext>水平像素数</mtext><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><mtext>垂直像素数</mtext><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></msqrt><mtext>屏幕尺寸（英寸）</mtext></mfrac></mrow><annotation encoding="application/x-tex">\text{PPI} = \frac{\sqrt{(\text{水平像素数})^2 + (\text{垂直像素数})^2}}{\text{屏幕尺寸（英寸）}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord">PPI</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.316em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.63em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">屏幕尺寸（英寸）</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.695em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.935em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mopen">(</span><span class="mord text"><span class="mord cjk_fallback">水平像素数</span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord text"><span class="mord cjk_fallback">垂直像素数</span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-2.895em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.28em" viewBox="0 0 400000 1296" preserveAspectRatio="xMinYMin slice"><path d="M263,681c0.7,0,18,39.7,52,119c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120c340,-704.7,510.7,-1060.3,512,-1067l0 -0c4.7,-7.3,11,-11,19,-11H40000v40H1012.3s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60zM1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305em;"><span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p></li><li><p>DPI 用于描述打印设备（如打印机）的分辨率：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>DPI</mtext><mo>=</mo><mfrac><mtext>打印的点数</mtext><mtext>每英寸的距离</mtext></mfrac></mrow><annotation encoding="application/x-tex">\text{DPI} = \frac{\text{打印的点数}}{\text{每英寸的距离}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord">DPI</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0463em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">每英寸的距离</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">打印的点数</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p></li></ul><p><strong>离散化采样出来的数字图像是如何量化表示模拟图像的</strong>？介绍三种，分别是二值图像、灰度图像和彩色图像。其中二值图像中，每一个像素点就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 或 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，灰度图像中，每一个像素点取值在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mn>255</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0,255]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">255</span><span class="mclose">]</span></span></span></span> 之间，彩色图像中每一个像素点是一个三通道向量，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi><mo separator="true">,</mo><mi>G</mi><mo separator="true">,</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">R,G,B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">G</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> 的取值均在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mn>255</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0,255]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">255</span><span class="mclose">]</span></span></span></span> 之间。</p><h2 id="图像编码">图像编码 *</h2><p>这一部分本课程不作要求，但愚以为这是十分重要的一个部分。</p><p>承接上一部分的「图像获取」技术。现在我们有了图像数据，就需要对其进行传输和存储，但是图像的数据量极大，如果不进行压缩编码，必将在传输和存储的过程中造成极大的消耗，因此图像的压缩编码是很重要的。目前主要有两种压缩编码策略，分别是 <strong>无损压缩编码</strong> 和 <strong>有损压缩编码</strong>。</p><h2 id="图像处理">图像处理</h2><p>有了图像数据，我们就可以进行最基本、最底层语义的图像处理，进而才能有后续更高语义上的 <strong>图像分析</strong> 和 <strong>图像理解</strong>。这一部分我们将从图像的各个处理策略入手，分别进行学习。</p><h3 id="1-图像基本运算">1. 图像基本运算</h3><p>首先需要掌握一些最基本的概念：齐次坐标表示与变换矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span></span></span>、图像插值策略、图像映射策略。</p><p><strong>关于齐次坐标表示和变换矩阵</strong>。就是将图像中的点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><msup><mo stretchy="false">)</mo><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">(x,y)^T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0913em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span></span></span> 用 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">(x,y,1)^T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0913em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span></span></span> 来表示。之所以要这样表示，从 <a href="https://blog.csdn.net/wangmj_hdu/article/details/119143771">这篇博客</a> 可以看出是为了「统一图像变换的运算形式」，但应该没这么简单，可能还有很多别的意义。至于所谓的变换矩阵，其实就是利用矩阵乘法对图像的像素点转换进行格式上的统一运算，所有的变换操作都可以统一为对像素点进行矩阵乘法，例如：对于当前的一个像素点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">p=(x,y,1)^T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0913em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span></span></span> 和一个变换矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>k</mi><mi>x</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>k</mi><mi>y</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><mi>y</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>p</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>q</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>s</mi></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \begin{bmatrix} k_x &amp; b &amp; \Delta x \\ c &amp; k_y &amp;\Delta y \\ p &amp; q &amp; s \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.6em;vertical-align:-1.55em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M403 1759 V84 H666 V0 H319 V1759 v0 v1759 h347 v-84H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Δ</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M347 1759 V0 H0 V84 H263 V1759 v0 v1759 H0 v84 H347zM347 1759 V0 H263 V1759 v0 v1759 h84z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span></span></span></span>。新像素点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>p</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mo stretchy="false">(</mo><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo separator="true">,</mo><msup><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo separator="true">,</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">p'=(x',y',1)^T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9463em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0913em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span></span></span> 可以通过矩阵乘法 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>p</mi><mrow><mn>3</mn><mo>×</mo><mn>1</mn></mrow><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo>=</mo><msub><mi>T</mi><mrow><mn>3</mn><mo>×</mo><mn>3</mn></mrow></msub><mo>×</mo><msub><mi>p</mi><mrow><mn>3</mn><mo>×</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">p'_{3\times 1}=T_{3\times 3} \times p_{3\times 1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0583em;vertical-align:-0.3064em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-2.4519em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span><span class="mbin mtight">×</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3064em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span><span class="mbin mtight">×</span><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span><span class="mbin mtight">×</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span> 得到，即：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msup><mi>x</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msup><mi>y</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>k</mi><mi>x</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>k</mi><mi>y</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">Δ</mi><mi>y</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>p</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>q</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>s</mi></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>x</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>y</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} =\begin{bmatrix} k_x &amp; b &amp; \Delta x \\ c &amp; k_y &amp;\Delta y \\ p &amp; q &amp; s \end{bmatrix}\begin{bmatrix} x \\ y\\ 1 \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.6em;vertical-align:-1.55em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M403 1759 V84 H666 V0 H319 V1759 v0 v1759 h347 v-84H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M347 1759 V0 H0 V84 H263 V1759 v0 v1759 H0 v84 H347zM347 1759 V0 H263 V1759 v0 v1759 h84z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.6em;vertical-align:-1.55em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M403 1759 V84 H666 V0 H319 V1759 v0 v1759 h347 v-84H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Δ</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M347 1759 V0 H0 V84 H263 V1759 v0 v1759 H0 v84 H347zM347 1759 V0 H263 V1759 v0 v1759 h84z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M403 1759 V84 H666 V0 H319 V1759 v0 v1759 h347 v-84H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M347 1759 V0 H0 V84 H263 V1759 v0 v1759 H0 v84 H347zM347 1759 V0 H263 V1759 v0 v1759 h84z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p><strong>关于图像插值策略</strong>。常用的有三种，分别是最邻近差值、双线性插值、双三次插值。其中最邻近插值很显然，双三次插值最精准但是计算量很大，因此最常用的还是双线性插值。</p><p><strong>关于图像映射策略</strong>。有前向映射法和后向映射法，常用的是后向映射法，因此讲讲后向映射法的流程。逻辑很简单，根据原图像的「四个顶点」与「几何变换逻辑」确定新图像的位置后，基于几何变换的 <strong>逆变换</strong> 逐个填充新图像的像素值即可。填充策略也很显然，当我们对新图像的像素点使用逆变换寻找在原图像的对应点时，会发生以下两种情况：</p><ol><li>新图像在原图像的对应点超过了原图像的范围。说明这个像素点是由于几何变换而凭空出现的，直接赋背景色即可。</li><li>新图像在原图像的对应点没超过原图像的范围。说明这个像素点一定是有实际意义的，此时又有两种情况：<ol><li>对应点存在的。说明找到了「最佳配对」，这是最完美的对应情况，直接赋原图的像素值即可；</li><li>对应点不存在。说明没有最佳配对，此时我们只能退而求其次给这个像素点生成一个配对的像素值，怎么生成呢？就可以用到上面介绍的插值策略了。</li></ol></li></ol><h4 id="1-1-几何变换">1.1 几何变换</h4><p>有了上述的基础知识之后，就可以对图像进行下面 5 种常见的几何变换了。</p><p><strong>平移</strong></p><p><strong>镜像</strong></p><p><strong>旋转</strong></p><p><strong>缩放</strong></p><p><strong>错切</strong></p>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-1d81f2f5" role="button" aria-expanded="false" aria-controls="collapse-1d81f2f5">        <div class="fold-arrow">▶</div>实战演练      </div>      <div class="fold-collapse collapse" id="collapse-1d81f2f5">        <div class="fold-content">          <p>一、放大+双线性插值</p><p>图像为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>4</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>7</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>5</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>8</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>3</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>6</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>9</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mstyle></mrow><annotation encoding="application/x-tex">f=\displaystyle \begin{bmatrix} 1 &amp; 4 &amp; 7 \\ 2 &amp; 5 &amp; 8 \\ 3 &amp; 6 &amp; 9 \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.6em;vertical-align:-1.55em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M403 1759 V84 H666 V0 H319 V1759 v0 v1759 h347 v-84H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">5</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">7</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">8</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">9</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M347 1759 V0 H0 V84 H263 V1759 v0 v1759 H0 v84 H347zM347 1759 V0 H263 V1759 v0 v1759 h84z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span></span></span></span>，放大比例为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mi>x</mi></msub><mo>=</mo><mn>2.3</mn><mo separator="true">,</mo><msub><mi>k</mi><mi>y</mi></msub><mo>=</mo><mn>1.6</mn></mrow><annotation encoding="application/x-tex">k_x=2.3,k_y=1.6</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord">2.3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1.6</span></span></span></span>。</p><p>MATLAB 代码：</p><figure class="highlight matlab"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br></pre></td><td class="code"><pre><code class="hljs matlab">f = [<span class="hljs-number">1</span> <span class="hljs-number">4</span> <span class="hljs-number">7</span>; <span class="hljs-number">2</span> <span class="hljs-number">5</span> <span class="hljs-number">8</span>; <span class="hljs-number">3</span> <span class="hljs-number">6</span> <span class="hljs-number">9</span>];<br>k_x = <span class="hljs-number">2.3</span>;<br>k_y = <span class="hljs-number">1.6</span>;<br><br><span class="hljs-comment">% 计算新图像的尺寸</span><br>[original_height, original_width] = <span class="hljs-built_in">size</span>(f);<br>new_height = <span class="hljs-built_in">round</span>(original_height * k_y);<br>new_width = <span class="hljs-built_in">round</span>(original_width * k_x);<br><br><span class="hljs-comment">% 后向映射法填充所有像素</span><br>g = <span class="hljs-built_in">zeros</span>(new_height, new_width);<br><span class="hljs-keyword">for</span> <span class="hljs-built_in">i</span> = <span class="hljs-number">1</span>:new_height<br>    <span class="hljs-keyword">for</span> <span class="hljs-built_in">j</span> = <span class="hljs-number">1</span>:new_width<br>        <span class="hljs-comment">% 计算原图像中的坐标</span><br>        orig_x = (<span class="hljs-built_in">j</span> - <span class="hljs-number">0.5</span>) / k_x + <span class="hljs-number">0.5</span>;<br>        orig_y = (<span class="hljs-built_in">i</span> - <span class="hljs-number">0.5</span>) / k_y + <span class="hljs-number">0.5</span>;<br><br>        <span class="hljs-comment">% 找到原图像中的邻近像素</span><br>        x1 = <span class="hljs-built_in">floor</span>(orig_x);<br>        x2 = <span class="hljs-built_in">ceil</span>(orig_x);<br>        y1 = <span class="hljs-built_in">floor</span>(orig_y);<br>        y2 = <span class="hljs-built_in">ceil</span>(orig_y);<br><br>        <span class="hljs-comment">% 限制边界</span><br>        x1 = <span class="hljs-built_in">max</span>(<span class="hljs-built_in">min</span>(x1, original_width), <span class="hljs-number">1</span>);<br>        x2 = <span class="hljs-built_in">max</span>(<span class="hljs-built_in">min</span>(x2, original_width), <span class="hljs-number">1</span>);<br>        y1 = <span class="hljs-built_in">max</span>(<span class="hljs-built_in">min</span>(y1, original_height), <span class="hljs-number">1</span>);<br>        y2 = <span class="hljs-built_in">max</span>(<span class="hljs-built_in">min</span>(y2, original_height), <span class="hljs-number">1</span>);<br><br>        <span class="hljs-comment">% 双线性插值</span><br>        Q11 = f(y1, x1);<br>        Q12 = f(y1, x2);<br>        Q21 = f(y2, x1);<br>        Q22 = f(y2, x2);<br>        <br>        <span class="hljs-comment">% 插值计算</span><br>        g(<span class="hljs-built_in">i</span>, <span class="hljs-built_in">j</span>) = (Q11 * (x2 - orig_x) * (y2 - orig_y) + ...<br>                   Q12 * (orig_x - x1) * (y2 - orig_y) + ...<br>                   Q21 * (x2 - orig_x) * (orig_y - y1) + ...<br>                   Q22 * (orig_x - x1) * (orig_y - y1));<br>    <span class="hljs-keyword">end</span><br><span class="hljs-keyword">end</span><br><br><span class="hljs-comment">% 显示结果</span><br><span class="hljs-built_in">disp</span>(g)<br></code></pre></td></tr></tbody></table></figure><p>运行结果：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410082035267.png" alt="运行结果"></p><p>OpenCV-Python 代码：</p><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">import</span> numpy <span class="hljs-keyword">as</span> np<br><br>f = np.array([[<span class="hljs-number">1</span>, <span class="hljs-number">4</span>, <span class="hljs-number">7</span>], [<span class="hljs-number">2</span>, <span class="hljs-number">5</span>, <span class="hljs-number">8</span>], [<span class="hljs-number">3</span>, <span class="hljs-number">6</span>, <span class="hljs-number">9</span>]], dtype=np.float32)<br>k_x = <span class="hljs-number">2.3</span><br>k_y = <span class="hljs-number">1.6</span><br><br><span class="hljs-comment"># 计算新图像的尺寸</span><br>original_height, original_width = f.shape<br>new_height = <span class="hljs-built_in">round</span>(original_height * k_y)<br>new_width = <span class="hljs-built_in">round</span>(original_width * k_x)<br><br><span class="hljs-comment"># 后向映射法填充所有像素</span><br>g = np.zeros((new_height, new_width), dtype=np.float32)<br><span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(new_height):<br>    <span class="hljs-keyword">for</span> j <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(new_width):<br>        <span class="hljs-comment"># 计算原图像中的坐标</span><br>        orig_x = (j + <span class="hljs-number">0.5</span>) / k_x - <span class="hljs-number">0.5</span><br>        orig_y = (i + <span class="hljs-number">0.5</span>) / k_y - <span class="hljs-number">0.5</span><br><br>        <span class="hljs-comment"># 找到原图像中的邻近像素</span><br>        x1 = <span class="hljs-built_in">int</span>(np.floor(orig_x))<br>        x2 = <span class="hljs-built_in">int</span>(np.ceil(orig_x))<br>        y1 = <span class="hljs-built_in">int</span>(np.floor(orig_y))<br>        y2 = <span class="hljs-built_in">int</span>(np.ceil(orig_y))<br><br>        <span class="hljs-comment"># 限制边界</span><br>        x1 = <span class="hljs-built_in">max</span>(<span class="hljs-built_in">min</span>(x1, original_width - <span class="hljs-number">1</span>), <span class="hljs-number">0</span>)<br>        x2 = <span class="hljs-built_in">max</span>(<span class="hljs-built_in">min</span>(x2, original_width - <span class="hljs-number">1</span>), <span class="hljs-number">0</span>)<br>        y1 = <span class="hljs-built_in">max</span>(<span class="hljs-built_in">min</span>(y1, original_height - <span class="hljs-number">1</span>), <span class="hljs-number">0</span>)<br>        y2 = <span class="hljs-built_in">max</span>(<span class="hljs-built_in">min</span>(y2, original_height - <span class="hljs-number">1</span>), <span class="hljs-number">0</span>)<br><br>        <span class="hljs-comment"># 双线性插值</span><br>        Q11 = f[y1, x1]<br>        Q12 = f[y1, x2]<br>        Q21 = f[y2, x1]<br>        Q22 = f[y2, x2]<br><br>        <span class="hljs-comment"># 插值计算</span><br>        g[i, j] = (Q11 * (x2 - orig_x) * (y2 - orig_y) +<br>                   Q12 * (orig_x - x1) * (y2 - orig_y) +<br>                   Q21 * (x2 - orig_x) * (orig_y - y1) +<br>                   Q22 * (orig_x - x1) * (orig_y - y1))<br><br><span class="hljs-comment"># 显示结果</span><br><span class="hljs-keyword">for</span> line <span class="hljs-keyword">in</span> g:<br>    <span class="hljs-keyword">for</span> d <span class="hljs-keyword">in</span> line:<br>        <span class="hljs-built_in">print</span>(<span class="hljs-string">f'<span class="hljs-subst">{d:<span class="hljs-number">.4</span>f}</span>'</span>, end=<span class="hljs-string">' '</span>)<br>    <span class="hljs-built_in">print</span>()<br></code></pre></td></tr></tbody></table></figure><p>运行结果：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410082044689.png" alt="运行结果"></p><p>二、旋转+双线性插值</p><p>图像为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>4</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>7</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>5</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>8</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>3</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>6</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>9</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mstyle></mrow><annotation encoding="application/x-tex">f=\displaystyle \begin{bmatrix} 1 &amp; 4 &amp; 7 \\ 2 &amp; 5 &amp; 8 \\ 3 &amp; 6 &amp; 9 \end{bmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.6em;vertical-align:-1.55em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M403 1759 V84 H666 V0 H319 V1759 v0 v1759 h347 v-84H403z M403 1759 V0 H319 V1759 v0 v1759 h84z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">5</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">7</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">8</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">9</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.667em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="3.600em" viewBox="0 0 667 3600"><path d="M347 1759 V0 H0 V84 H263 V1759 v0 v1759 H0 v84 H347zM347 1759 V0 H263 V1759 v0 v1759 h84z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span></span></span></span>，旋转角度为逆时针 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>60</mn></mrow><annotation encoding="application/x-tex">60</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">60</span></span></span></span> 度。</p><p>MATLAB 代码：</p><figure class="highlight matlab"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br></pre></td><td class="code"><pre><code class="hljs matlab">f = [<span class="hljs-number">1</span> <span class="hljs-number">4</span> <span class="hljs-number">7</span>; <span class="hljs-number">2</span> <span class="hljs-number">5</span> <span class="hljs-number">8</span>; <span class="hljs-number">3</span> <span class="hljs-number">6</span> <span class="hljs-number">9</span>];<br><br><span class="hljs-comment">% 计算新图像尺寸</span><br>theta = <span class="hljs-number">60</span>; <span class="hljs-comment">% 角度</span><br>theta_rad = deg2rad(theta); <span class="hljs-comment">% 转换为弧度</span><br>[original_height, original_width] = <span class="hljs-built_in">size</span>(f);<br>new_height = <span class="hljs-built_in">round</span>(<span class="hljs-built_in">abs</span>(original_height * <span class="hljs-built_in">cos</span>(theta_rad)) + <span class="hljs-built_in">abs</span>(original_width * <span class="hljs-built_in">sin</span>(theta_rad)));<br>new_width = <span class="hljs-built_in">round</span>(<span class="hljs-built_in">abs</span>(original_height * <span class="hljs-built_in">sin</span>(theta_rad)) + <span class="hljs-built_in">abs</span>(original_width * <span class="hljs-built_in">cos</span>(theta_rad)));<br><br><span class="hljs-comment">% 后向映射法</span><br>g = <span class="hljs-built_in">zeros</span>(new_height, new_width);<br>center_x = original_width / <span class="hljs-number">2</span>;<br>center_y = original_height / <span class="hljs-number">2</span>;<br><span class="hljs-keyword">for</span> <span class="hljs-built_in">i</span> = <span class="hljs-number">1</span>:new_height<br>    <span class="hljs-keyword">for</span> <span class="hljs-built_in">j</span> = <span class="hljs-number">1</span>:new_width<br>        <span class="hljs-comment">% 计算原图像中的坐标</span><br>        orig_x = (<span class="hljs-built_in">j</span> - new_width / <span class="hljs-number">2</span>) * <span class="hljs-built_in">cos</span>(theta_rad) + (<span class="hljs-built_in">i</span> - new_height / <span class="hljs-number">2</span>) * <span class="hljs-built_in">sin</span>(theta_rad) + center_x;<br>        orig_y = -(<span class="hljs-built_in">j</span> - new_width / <span class="hljs-number">2</span>) * <span class="hljs-built_in">sin</span>(theta_rad) + (<span class="hljs-built_in">i</span> - new_height / <span class="hljs-number">2</span>) * <span class="hljs-built_in">cos</span>(theta_rad) + center_y;<br><br>        <span class="hljs-comment">% 找到原图像中的邻近像素</span><br>        x1 = <span class="hljs-built_in">floor</span>(orig_x);<br>        x2 = <span class="hljs-built_in">ceil</span>(orig_x);<br>        y1 = <span class="hljs-built_in">floor</span>(orig_y);<br>        y2 = <span class="hljs-built_in">ceil</span>(orig_y);<br><br>        <span class="hljs-comment">% 限制边界</span><br>        x1 = <span class="hljs-built_in">max</span>(<span class="hljs-built_in">min</span>(x1, original_width), <span class="hljs-number">1</span>);<br>        x2 = <span class="hljs-built_in">max</span>(<span class="hljs-built_in">min</span>(x2, original_width), <span class="hljs-number">1</span>);<br>        y1 = <span class="hljs-built_in">max</span>(<span class="hljs-built_in">min</span>(y1, original_height), <span class="hljs-number">1</span>);<br>        y2 = <span class="hljs-built_in">max</span>(<span class="hljs-built_in">min</span>(y2, original_height), <span class="hljs-number">1</span>);<br><br>        <span class="hljs-comment">% 双线性插值</span><br>        Q11 = f(y1, x1);<br>        Q12 = f(y1, x2);<br>        Q21 = f(y2, x1);<br>        Q22 = f(y2, x2);<br>        <br>        <span class="hljs-comment">% 插值计算</span><br>        g(<span class="hljs-built_in">i</span>, <span class="hljs-built_in">j</span>) = (Q11 * (x2 - orig_x) * (y2 - orig_y) + ...<br>                   Q12 * (orig_x - x1) * (y2 - orig_y) + ...<br>                   Q21 * (x2 - orig_x) * (orig_y - y1) + ...<br>                   Q22 * (orig_x - x1) * (orig_y - y1));<br>    <span class="hljs-keyword">end</span><br><span class="hljs-keyword">end</span><br><br><span class="hljs-comment">% 显示结果</span><br><span class="hljs-built_in">disp</span>(g)<br></code></pre></td></tr></tbody></table></figure><p>运行结果：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410082043712.png" alt="运行结果"></p><p>OpenCV-Python 代码：</p><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">import</span> numpy <span class="hljs-keyword">as</span> np<br><br>f = np.array([[<span class="hljs-number">1</span>, <span class="hljs-number">4</span>, <span class="hljs-number">7</span>], [<span class="hljs-number">2</span>, <span class="hljs-number">5</span>, <span class="hljs-number">8</span>], [<span class="hljs-number">3</span>, <span class="hljs-number">6</span>, <span class="hljs-number">9</span>]], dtype=np.float32)<br><br><span class="hljs-comment"># 计算新图像尺寸</span><br>theta = <span class="hljs-number">60</span>  <span class="hljs-comment"># 角度</span><br>theta_rad = np.radians(theta)  <span class="hljs-comment"># 转换为弧度</span><br>original_height, original_width = f.shape<br>new_height = <span class="hljs-built_in">int</span>(<span class="hljs-built_in">round</span>(<span class="hljs-built_in">abs</span>(original_height * np.cos(theta_rad)) + <span class="hljs-built_in">abs</span>(original_width * np.sin(theta_rad))))<br>new_width = <span class="hljs-built_in">int</span>(<span class="hljs-built_in">round</span>(<span class="hljs-built_in">abs</span>(original_height * np.sin(theta_rad)) + <span class="hljs-built_in">abs</span>(original_width * np.cos(theta_rad))))<br><br><span class="hljs-comment"># 后向映射法</span><br>g = np.zeros((new_height, new_width), dtype=np.float32)<br>center_x = original_width / <span class="hljs-number">2</span><br>center_y = original_height / <span class="hljs-number">2</span><br><span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(new_height):<br>    <span class="hljs-keyword">for</span> j <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(new_width):<br>        <span class="hljs-comment"># 计算原图像中的坐标</span><br>        orig_x = (j - new_width / <span class="hljs-number">2</span>) * np.cos(theta_rad) + (i - new_height / <span class="hljs-number">2</span>) * np.sin(theta_rad) + center_x<br>        orig_y = -(j - new_width / <span class="hljs-number">2</span>) * np.sin(theta_rad) + (i - new_height / <span class="hljs-number">2</span>) * np.cos(theta_rad) + center_y<br><br>        <span class="hljs-comment"># 找到原图像中的邻近像素</span><br>        x1 = <span class="hljs-built_in">int</span>(np.floor(orig_x))<br>        x2 = <span class="hljs-built_in">int</span>(np.ceil(orig_x))<br>        y1 = <span class="hljs-built_in">int</span>(np.floor(orig_y))<br>        y2 = <span class="hljs-built_in">int</span>(np.ceil(orig_y))<br><br>        <span class="hljs-comment"># 限制边界</span><br>        x1 = <span class="hljs-built_in">max</span>(<span class="hljs-built_in">min</span>(x1, original_width - <span class="hljs-number">1</span>), <span class="hljs-number">0</span>)<br>        x2 = <span class="hljs-built_in">max</span>(<span class="hljs-built_in">min</span>(x2, original_width - <span class="hljs-number">1</span>), <span class="hljs-number">0</span>)<br>        y1 = <span class="hljs-built_in">max</span>(<span class="hljs-built_in">min</span>(y1, original_height - <span class="hljs-number">1</span>), <span class="hljs-number">0</span>)<br>        y2 = <span class="hljs-built_in">max</span>(<span class="hljs-built_in">min</span>(y2, original_height - <span class="hljs-number">1</span>), <span class="hljs-number">0</span>)<br><br>        <span class="hljs-comment"># 双线性插值</span><br>        Q11 = f[y1, x1]<br>        Q12 = f[y1, x2]<br>        Q21 = f[y2, x1]<br>        Q22 = f[y2, x2]<br><br>        <span class="hljs-comment"># 插值计算</span><br>        g[i, j] = (Q11 * (x2 - orig_x) * (y2 - orig_y) +<br>                   Q12 * (orig_x - x1) * (y2 - orig_y) +<br>                   Q21 * (x2 - orig_x) * (orig_y - y1) +<br>                   Q22 * (orig_x - x1) * (orig_y - y1))<br><br><span class="hljs-comment"># 显示结果</span><br><span class="hljs-keyword">for</span> line <span class="hljs-keyword">in</span> g:<br>    <span class="hljs-keyword">for</span> d <span class="hljs-keyword">in</span> line:<br>        <span class="hljs-built_in">print</span>(<span class="hljs-string">f'<span class="hljs-subst">{d:<span class="hljs-number">.4</span>f}</span>'</span>, end=<span class="hljs-string">' '</span>)<br>    <span class="hljs-built_in">print</span>()<br></code></pre></td></tr></tbody></table></figure><p>运行结果：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410082043419.png" alt="运行结果"></p><p>三、旋转+镜像+错切+缩小+双线性插值</p><p>选择一张图片，顺时针旋转 20 度，水平镜像，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mi>x</mi></msub><mo>=</mo><mn>0.3</mn><mo separator="true">,</mo><msub><mi>k</mi><mi>y</mi></msub><mo>=</mo><mn>0.5</mn></mrow><annotation encoding="application/x-tex">k_x=0.3,k_y=0.5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord">0.3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.5</span></span></span></span> 做错切变换，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mi>x</mi></msub><mo>=</mo><mn>0.6</mn><mo separator="true">,</mo><msub><mi>k</mi><mi>y</mi></msub><mo>=</mo><mn>0.6</mn></mrow><annotation encoding="application/x-tex">k_x=0.6,k_y=0.6</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord">0.6</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.6</span></span></span></span> 缩小图片。以下图为例：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410082054211.jpg" alt="bird"></p><p>MATLAB 代码：</p><figure class="highlight matlab"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br></pre></td><td class="code"><pre><code class="hljs matlab"><span class="hljs-comment">% 读取图片</span><br>img = imread(<span class="hljs-string">'bird.jpg'</span>);<br>[h, w, ~] = <span class="hljs-built_in">size</span>(img);<br><br><span class="hljs-built_in">figure</span>;<br>subplot(<span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">1</span>), imshow(img), title(<span class="hljs-string">'原始图像'</span>);<br><br><span class="hljs-comment">% 1. 顺时针旋转 20 度</span><br>rotated_img = imrotate(img, <span class="hljs-number">-20</span>, <span class="hljs-string">'bilinear'</span>, <span class="hljs-string">'crop'</span>);<br>subplot(<span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">2</span>), imshow(rotated_img), title(<span class="hljs-string">'顺时针旋转 20 度后'</span>);<br><br><span class="hljs-comment">% 2. 水平镜像</span><br>mirrored_img = flip(rotated_img, <span class="hljs-number">2</span>);<br>subplot(<span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">3</span>), imshow(mirrored_img), title(<span class="hljs-string">'水平镜像后'</span>);<br><br><span class="hljs-comment">% 3. 错切变换</span><br>tform = affine2d([<span class="hljs-number">1</span> <span class="hljs-number">0.3</span> <span class="hljs-number">0</span>; <span class="hljs-number">0.5</span> <span class="hljs-number">1</span> <span class="hljs-number">0</span>; <span class="hljs-number">0</span> <span class="hljs-number">0</span> <span class="hljs-number">1</span>]); <br>sheared_img = imwarp(mirrored_img, tform, <span class="hljs-string">'Interp'</span>, <span class="hljs-string">'bilinear'</span>);<br>subplot(<span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">4</span>), imshow(sheared_img), title(<span class="hljs-string">'错切变换后'</span>);<br><br><span class="hljs-comment">% 4. 缩放并用白色背景填充</span><br>scale_x = <span class="hljs-number">0.6</span>;<br>scale_y = <span class="hljs-number">0.6</span>;<br>scaled_img = imresize(sheared_img, [<span class="hljs-built_in">round</span>(h*scale_y), <span class="hljs-built_in">round</span>(w*scale_x)], <span class="hljs-string">'bilinear'</span>);<br>background = uint8(<span class="hljs-number">255</span> * <span class="hljs-built_in">ones</span>(h, w, <span class="hljs-number">3</span>));<br>x_offset = <span class="hljs-built_in">round</span>((w - <span class="hljs-built_in">size</span>(scaled_img, <span class="hljs-number">2</span>)) / <span class="hljs-number">2</span>);<br>y_offset = <span class="hljs-built_in">round</span>((h - <span class="hljs-built_in">size</span>(scaled_img, <span class="hljs-number">1</span>)) / <span class="hljs-number">2</span>);<br>background(y_offset:(y_offset + <span class="hljs-built_in">size</span>(scaled_img, <span class="hljs-number">1</span>) - <span class="hljs-number">1</span>), x_offset:(x_offset + <span class="hljs-built_in">size</span>(scaled_img, <span class="hljs-number">2</span>) - <span class="hljs-number">1</span>), :) = scaled_img;<br><br>subplot(<span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">5</span>), imshow(background), title(<span class="hljs-string">'缩小后'</span>);<br></code></pre></td></tr></tbody></table></figure><p>运行结果：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410082105689.jpg" alt="运行结果"></p><p>OpenCV-Python 代码：</p><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">import</span> cv2<br><span class="hljs-keyword">import</span> numpy <span class="hljs-keyword">as</span> np<br><span class="hljs-keyword">import</span> matplotlib.pyplot <span class="hljs-keyword">as</span> plt<br><br>plt.rcParams[<span class="hljs-string">'font.sans-serif'</span>] = [<span class="hljs-string">'SimHei'</span>]<br>plt.rcParams[<span class="hljs-string">'axes.unicode_minus'</span>] = <span class="hljs-literal">False</span><br><br>img = cv2.imread(<span class="hljs-string">'E:/python/OpenCV/Homework_1/bird.png'</span>)<br>img = cv2.cvtColor(img, cv2.COLOR_BGR2RGB)<br>(h, w) = img.shape[:<span class="hljs-number">2</span>]<br>plt.figure(figsize=(<span class="hljs-number">12</span>, <span class="hljs-number">8</span>))<br><br><span class="hljs-comment"># 显示原始图像</span><br>plt.subplot(<span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">1</span>)<br>plt.imshow(img)<br>plt.title(<span class="hljs-string">'原始图像'</span>)<br><br><span class="hljs-comment"># 1. 顺时针旋转 20 度</span><br>center = (w // <span class="hljs-number">2</span>, h // <span class="hljs-number">2</span>)<br>M = cv2.getRotationMatrix2D(center, -<span class="hljs-number">20</span>, <span class="hljs-number">1.0</span>)<br>rotated_img = cv2.warpAffine(img, M, (w, h), flags=cv2.INTER_LINEAR)<br>plt.subplot(<span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">2</span>)<br>plt.imshow(rotated_img)<br>plt.title(<span class="hljs-string">'旋转 20 度后'</span>)<br><br><span class="hljs-comment"># 2. 水平镜像</span><br>mirrored_img = cv2.flip(rotated_img, <span class="hljs-number">1</span>)<br>plt.subplot(<span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">3</span>)<br>plt.imshow(mirrored_img)<br>plt.title(<span class="hljs-string">'水平镜像后'</span>)<br><br><span class="hljs-comment"># 3. 错切变换</span><br>kx, ky = <span class="hljs-number">0.3</span>, <span class="hljs-number">0.5</span><br>M_shear = np.array([[<span class="hljs-number">1</span>, kx, <span class="hljs-number">0</span>], [ky, <span class="hljs-number">1</span>, <span class="hljs-number">0</span>]], dtype=np.float32)<br><span class="hljs-comment"># sheared_img = cv2.warpAffine(mirrored_img, M_shear, (w, h), flags=cv2.INTER_LINEAR)</span><br>sheared_img = cv2.warpAffine(mirrored_img, M_shear, (<span class="hljs-built_in">int</span>(w + h * kx), <span class="hljs-built_in">int</span>(h + w * ky)), flags=cv2.INTER_LINEAR)<br>plt.subplot(<span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">4</span>)<br>plt.imshow(sheared_img)<br>plt.title(<span class="hljs-string">'错切变换后'</span>)<br><br><span class="hljs-comment"># 4. 缩放并添加白色背景</span><br>kx, ky = <span class="hljs-number">0.6</span>, <span class="hljs-number">0.6</span><br>scaled_img = cv2.resize(sheared_img, (<span class="hljs-number">0</span>, <span class="hljs-number">0</span>), fx=kx, fy=ky, interpolation=cv2.INTER_LINEAR)<br>background = np.ones_like(sheared_img, dtype=np.uint8) * <span class="hljs-number">255</span><br>x_offset = (background.shape[<span class="hljs-number">1</span>] - scaled_img.shape[<span class="hljs-number">1</span>]) // <span class="hljs-number">2</span><br>y_offset = (background.shape[<span class="hljs-number">0</span>] - scaled_img.shape[<span class="hljs-number">0</span>]) // <span class="hljs-number">2</span><br>background[y_offset:y_offset + scaled_img.shape[<span class="hljs-number">0</span>], x_offset:x_offset + scaled_img.shape[<span class="hljs-number">1</span>]] = scaled_img<br>plt.subplot(<span class="hljs-number">2</span>, <span class="hljs-number">3</span>, <span class="hljs-number">5</span>)<br>plt.imshow(background)<br>plt.title(<span class="hljs-string">'缩小后'</span>)<br><br><span class="hljs-comment"># 显示所有子图</span><br>plt.tight_layout()<br>plt.show()<br></code></pre></td></tr></tbody></table></figure><p>运行结果：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410082115300.png" alt="运行结果"></p>        </div>      </div>    </div><h4 id="1-2-代数运算">1.2 代数运算</h4><h4 id="1-3-模版运算">1.3 模版运算</h4><h3 id="2-图像正交变换">2. 图像正交变换</h3><h3 id="3-图像增强">3. 图像增强</h3><h3 id="4-图像平滑">4. 图像平滑</h3><h3 id="5-图像锐化">5. 图像锐化</h3><h3 id="6-图像复原">6. 图像复原</h3><h3 id="7-图像的数学形态学处理">7. 图像的数学形态学处理</h3><h3 id="8-图像分割">8. 图像分割</h3><h2 id="图像分析">图像分析</h2><p>基于上一部分最底层语义的「图像处理」技术，本部分将展开更高一层语义「图像分析」技术的学习。</p><h2 id="综合应用">综合应用</h2><p>基于上一部分中层语义的「图像分析」技术，本部分将结合 <strong>图像分类</strong> 和 <strong>图像检索</strong> 技术展开最高层语义「图像理解」技术的学习。</p><h3 id="图像分类">图像分类</h3><h3 id="图像检索">图像检索</h3>]]></content>
     
     
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@@ -472,7 +472,7 @@
     <link href="/GPA/5th-term/ComputerOrganization/"/>
     <url>/GPA/5th-term/ComputerOrganization/</url>
     
-    <content type="html"><![CDATA[<h2 id="计算机组成原理">计算机组成原理</h2><h2 id="前言">前言</h2><p>学科地位：</p><table><thead><tr><th style="text-align:center">主讲教师</th><th style="text-align:center">学分配额</th><th style="text-align:center">学科类别</th></tr></thead><tbody><tr><td style="text-align:center">闫文珠</td><td style="text-align:center">3.5</td><td style="text-align:center">自发课</td></tr></tbody></table><p>成绩组成：</p><table><thead><tr><th style="text-align:center">作业+实验</th><th style="text-align:center">期末</th></tr></thead><tbody><tr><td style="text-align:center">50%</td><td style="text-align:center">50%</td></tr></tbody></table><p>教材情况：</p><table><thead><tr><th style="text-align:center">课程名称</th><th style="text-align:center">选用教材</th><th style="text-align:center">版次</th><th style="text-align:center">作者</th><th style="text-align:center">出版社</th><th style="text-align:center">ISBN 号</th></tr></thead><tbody><tr><td style="text-align:center">计算机组成原理</td><td style="text-align:center">《计算机组成与系统结构》</td><td style="text-align:center">3</td><td style="text-align:center">袁春风</td><td style="text-align:center">清华大学出版社</td><td style="text-align:center">978-7-302-59988-3</td></tr></tbody></table><p>最基本的知识普及：</p>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-d6ca2bab" role="button" aria-expanded="false" aria-controls="collapse-d6ca2bab">        <div class="fold-arrow">▶</div>最基本的知识普及      </div>      <div class="fold-collapse collapse" id="collapse-d6ca2bab">        <div class="fold-content">          <ul><li><a href="https://www.bilibili.com/video/BV1xQ4y1b7JS/">【硬核科普】从零开始认识主板</a></li><li><a href="https://www.bilibili.com/video/BV1xE421j7Uv/">【硬核科普】从零开始认识显卡</a></li></ul><p><strong>主板（Motherboard）</strong></p><ul><li><strong>作用</strong>：主板是所有硬件的连接平台，它连接并协调 CPU、内存、存储设备、显卡、网络设备和其他外部接口。主板上还包含芯片组，负责管理数据传输和设备间的通信。</li><li><strong>与其他部分的关系</strong>：主板是各个硬件组件的核心枢纽，确保数据能够在各个组件之间高效传输和通信。</li></ul><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409102038540.png" alt="机箱视角中的主板"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409102039187.png" alt="主板的 6 大区域"></p><p><strong>输入/输出设备（I/O Devices）</strong></p><ul><li><strong>作用</strong>：输入设备（如键盘、鼠标、触摸屏）用于用户与计算机的交互。输出设备（如显示器、打印机）用于显示计算机的操作结果。网络设备（如网卡、Wi-Fi 模块）则允许电脑连接到网络。</li><li><strong>与其他部分的关系</strong>：输入设备通过主板将用户的操作传递给 CPU 进行处理，输出设备则从 CPU 或显卡接收处理后的数据进行展示。</li></ul><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409102039284.png" alt="I/O 区域"></p><p><strong>中央处理器（CPU）</strong></p><ul><li><strong>作用</strong>：CPU 是电脑的大脑，负责执行计算任务和指令。它从内存中读取指令，进行运算，然后将结果写回内存。现代 CPU 具有多个核心，可以并行处理多项任务，提高计算效率。</li><li><strong>与其他部分的关系</strong>：CPU 与内存和存储设备密切配合。它从内存中获取数据和指令，然后进行处理，最终将结果写回内存或存储设备。</li></ul><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409102040689.png" alt="CPU 区域"></p><p><strong>内存（RAM）</strong></p><ul><li><strong>作用</strong>：内存是计算机中的临时存储器，用于存储当前运行的程序和数据。它的访问速度极快，但数据在断电后会丢失。内存用于支持 CPU 的高速操作。</li><li><strong>与其他部分的关系</strong>：内存是 CPU 执行任务时的工作区域，存储从存储设备（如硬盘）中读取的数据和指令。内存的数据会在需要时被写回到存储设备中。</li></ul><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409102040651.png" alt="内存区域"></p><p><strong>显卡（GPU）</strong></p><ul><li><strong>作用</strong>：显卡负责处理图形计算任务，特别是在图像和视频渲染、游戏以及某些并行计算任务中（如深度学习）。高性能显卡能够加速这些计算任务。</li><li><strong>与其他部分的关系</strong>：显卡通过主板与 CPU 和内存通信，从内存中读取图形数据，并进行处理，然后将渲染结果输出到显示器。</li></ul><p><strong>存储设备（Storage Devices）</strong></p><ul><li><strong>作用</strong>：存储设备如机械硬盘 hard disk (HDD)、固态硬盘 solid state disk (SSD)，用于长期保存数据和程序，即使在电脑断电后数据也不会丢失。SSD 的速度比传统硬盘更快，因此越来越常用。</li><li><strong>与其他部分的关系</strong>：存储设备保存操作系统、应用程序和用户数据。程序执行时，数据会从存储设备加载到内存中供 CPU 使用。</li></ul><p><strong>网卡（Network Interface Card, NIC）</strong></p><ul><li><strong>作用</strong>：网卡是计算机与网络之间的桥梁，负责处理计算机与网络间的数据通信。它将计算机的数据转换为网络信号，以太网网卡通过有线或无线方式连接局域网（LAN）或互联网。</li><li><strong>与其他部分的关系</strong>：网卡通过主板与南桥芯片组或 PCIe 总线连接，接收来自 CPU 的数据，经过处理后通过网络接口发送到外部网络。同时，它也接收来自网络的数据，传递给 CPU 进行处理，确保计算机能够与其他网络设备进行通信。</li></ul><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409102040086.png" alt="扩展区域"></p><p><strong>南桥芯片组（Southbridge）</strong></p><ul><li><strong>作用</strong>：南桥芯片组是主板上的一个重要芯片，为了避免过多的模块与 CPU 直连导致主板排线困难以及布局问题，产生了南桥芯片组用于负责管理较慢的外围设备的连接和数据传输。包括 USB 接口、SATA 接口、网络接口、音频设备和其他 I/O 设备。南桥还处理 BIOS、系统时钟、硬盘和光驱的控制。</li><li><strong>与其他部分的关系</strong>：南桥通过主板与 CPU 和北桥芯片组（或直接与 CPU）连接，接收来自 CPU 的指令，并将其转化为外设可以理解的信号。它还通过 I/O 总线连接各种外设，确保数据在外围设备与系统之间顺畅传输。</li></ul><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409102040366.png" alt="南桥芯片组区域"></p><p><strong>BIOS（Basic Input/Output System）</strong></p><ul><li><strong>作用</strong>：BIOS 是主板的固件程序，它存储在主板的 BIOS 芯片里。负责在计算机启动时进行硬件初始化，指导所有的硬件运行，并启动操作系统。</li><li><strong>与其他部分的关系</strong>：BIOS 在开机时检查并初始化硬件，然后引导操作系统，从而开始系统的工作。</li></ul><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409102040350.png" alt="BIOS 芯片"></p>        </div>      </div>    </div><p>为什么要学这门课？</p><blockquote><p>上上学期学习了《数字逻辑电路》，云里雾里；上学期学习了《计算机系统基础》，继续云里雾里。本学期开始学习《计算机组成原理》？如果用一句话来概括我对数字逻辑电路的理解，大概可以这样说：原来计算机是一个由“门电路”、“导线”和“时钟脉冲”组成的机器。如果用一句话来概括我对计算机系统基础的理解，大概可以这样说：原来计算机所有的活动都是由翻译过来的 01 序列驱动的。</p><p>现在我们让 AI 模仿上面我对课程理解的概括逻辑，用一句话来概括计算机组成原理，ta 是这样回答的：<strong>原来计算机的高效运作是通过硬件架构的精妙设计，将复杂的指令和数据流转化为有序的电子信号运动来实现的</strong>。</p></blockquote><p>会收获什么？</p><blockquote><p>对计算机整体有一个宏观的把握，并且能够知道计算机各个模块之间是如何关联与调度的，以及每一个模块内部的工作原理。</p><p>最后能够从逻辑上实现一个单核 CPU。</p></blockquote><p>注：</p><blockquote><p><strong>本书 ISA 采用 MIPS 指令系统</strong>。之前学习的 <a href="https://blog.dwj601.cn/GPA/4th-term/SysBasic/#3-2-IA-32-%E6%8C%87%E4%BB%A4%E7%B3%BB%E7%BB%9F%E6%A6%82%E8%BF%B0">《计算机系统基础》</a> 中 ISA 采用的是 IA-32 指令系统，也就是大名鼎鼎的 x86-64 指令系统的前身。</p><p>为了形式化地描述 CPU 的运行逻辑，需要使用寄存器传送级 (register transfer level, 简称 RTL) 语言。<strong>本书 RTL 语言有以下规定</strong>：R[r] 表示通用寄存器 r 的内容，M[addr] 表示存储单元 addr 的内容；M[R[r]] 表示寄存器 r 的内容所指存储单元的内容；PC 表示 PC 的内容，M[PC] 表示 PC 所指存储单元的内容；SEXT[imm] 表示对 imm 进行符号扩展，ZEXT[imm] 表示对 imm 进行零扩展；传送方向用 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>←</mo></mrow><annotation encoding="application/x-tex">\leftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">←</span></span></span></span> 表示，即传送源在右，传送目的在左。</p></blockquote><h2 id="一、计算机系统概述">一、计算机系统概述</h2><p>注：本章与《计算机系统基础》的第 1 章重复，详见 <a href="https://blog.dwj601.cn/GPA/4th-term/SysBasic/#%E7%AC%AC1%E7%AB%A0-%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%B3%BB%E7%BB%9F%E6%A6%82%E8%BF%B0">https://blog.dwj601.cn/GPA/4th-term/SysBasic/#第1章-计算机系统概述</a></p><p>主要掌握 <strong>冯诺依曼状态机</strong> 和 <strong>计算机性能度量</strong> 两个知识点。</p><h2 id="二、数据的机器级表示">二、数据的机器级表示</h2><p>注：本章与《计算机系统基础》的第 2 章重复，详见 <a href="https://blog.dwj601.cn/GPA/4th-term/SysBasic/#%E7%AC%AC2%E7%AB%A0-%E6%95%B0%E6%8D%AE%E7%9A%84%E6%9C%BA%E5%99%A8%E7%BA%A7%E8%A1%A8%E7%A4%BA%E4%B8%8E%E5%A4%84%E7%90%86">https://blog.dwj601.cn/GPA/4th-term/SysBasic/#第2章-数据的机器级表示与处理</a></p><p>主要掌握 <strong>数值/非数值数据的表示</strong>、<strong>数据宽度</strong>、<strong>存储对齐</strong> 和 <strong>纠/检错</strong> 四个知识点。</p><p>数据的纠错/检错见 7.4 节。</p><h2 id="三、运算方法和运算部件">三、运算方法和运算部件</h2><p>本章我们讲讲 CPU 中的 <strong>算数逻辑单元（Arithmetic and Logic Unit，简称 ALU）</strong>。ALU 在计算机中的地位大致如下图所示：</p><pre><code class=" mermaid">graph RL    实现功能    执行程序    机器指令    subgraph ALU    direction RL    算数运算逻辑    subgraph 运算部件    direction LR    加法器    移位器    end    end        运算部件 --&gt;|硬件支持| 算数运算逻辑    ALU --&gt; 机器指令    机器指令 --&gt; 执行程序    执行程序 --&gt; 实现功能</code></pre><p>也就有了这样的学习路线：涉及运算的机器指令（MIPS 指令系统） <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> 运算部件支持（逻辑电路图） <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> 算数运算逻辑（算法设计）</p><h3 id="3-1-涉及运算的机器指令">3.1 涉及运算的机器指令</h3><p><strong>都有哪些涉及运算的机器指令</strong>？机器指令的种类有很多，我们在计算机系统基础中已经学习到了比如：数据传输指令、控制指令、跳转指令等等。由于本章讲解的是算数运算，因此我们主要学习涉及到运算的机器指令。我们将所有涉及到运算的机器指令抽丝剥茧，凝练出了以下共 11 条指令：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409252041301.png" alt="11 条涉及运算指令的 RTL 描述"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409252041551.png" alt="11 条涉及运算指令的 RTL 描述 - 续"></p><p>不难发现，上述 11 种除了最后一个跳转指令，其余的 10 条运算指令只有 3 大类：加减法、按位或、小于置 1。</p><h3 id="3-2-运算部件支持">3.2 运算部件支持</h3><p><strong>有了上述的程序运算需求，如何设计出对应的硬件来支持呢？CPU 中的 ALU 单元就是负责运算的部件</strong>。ALU 内部的逻辑结构如下图所示，其输入有两个运算数 A 和 B 以及一个控制输入 ALUctr：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409250813229.png" alt="ALU 实现"></p><p>ALUctr 有 4 个控制信号，下面借书中原文分别解释 4 个控制信号对应的逻辑：</p><blockquote><p>SUBctr 用来控制 ALU 执行加法还是减法运算。当 SUBctr=1 时，做减法；当 SUBctr=0 时，做加法。</p><p>OPctr 用来控制选择哪种运算的结果作为 Result 输出。因为所实现的 11 条指令中只可能有加/减、按位或、小于置 1 这 3 大类运算，所以 OPctr 有两位。</p><p>OVctr 用来控制是否要进行溢出判断。当 OVctr=1 时，进行溢出判断，此时，若结果发生溢出，则溢出标志 Overflow 为 1，当 OVctr=0 时，无须溢出判断，此时，即使结果发生溢出，溢出标志 Overflow 也不为 1。</p><p>SIGctr 信号控制 ALU 是执行「带符号整数比较小于置 1」还是执行「无符号数比较小于置 1」的功能。当 SIGctr=0，执行「无符号数比较小于置 1」的功能；当 SIGctr=1 时，执行「带符号整数比较小于置 1」的功能。</p></blockquote><p>完美实现了所需的 3 大类运算逻辑。</p><p><strong>ALU 是如何控制以精准实现上述 11 条指令的？本质上就是通过 ALUctr 来实现运算控制的</strong>。除了最后一个跳转，难道 10 个指令就一定需要 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⌈</mo><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo><mo stretchy="false">⌉</mo><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">\lceil \log_2(10) \rceil =4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⌈</span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.207em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">10</span><span class="mclose">)⌉</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">4</span></span></span></span> 位进行信号选择控制吗？并不是，其中一些虽然功能不同，但是运算逻辑相同：</p><ul><li>指令 addiu、lw、sw 和 beq 转移目标地址计算的 ALU 控制信号取值一样，都是进行加法运算并不判溢出，记为 addu 操作.</li><li>指令 subu 和 beq 判 0 操作的 ALU 控制信号可看成一样，都做减法运算并不判溢出，记为 subu 操作。</li></ul><p>下表详细解释了每一条指令的运算类型与 ALUctr 取值之间的关系：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409261336957.png" alt="运算类型与 ALUctr 取值之间的关系"></p><p>因此，这 11 条指令可以归纳为 7 种操作：addu、add、or、subu、sub、sltu、slt，也就是说 ALUctr 只需要 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⌈</mo><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo><mo stretchy="false">⌉</mo><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">\lceil \log_2(7) \rceil =3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⌈</span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.207em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">7</span><span class="mclose">)⌉</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span></span></span></span> 个选择控制位！如下表所示，列出了 ALUctr 的选择控制逻辑：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409261337035.png" alt="ALUctr 的选择控制逻辑"></p><p><strong>为什么可以这样控制</strong>？</p><p>加减运算控制（SUBctr）：很显然，加法（SUBctr=0），减法（SUBctr=1）。</p><p>输出内容控制（OPctr）：很显然， 3 大类运算就对应 3 个取值。</p><p>溢出判断控制（OVctr）：很显然，有符号运算需要溢出判断，其他运算都不需要。</p><p>小于置一控制（SIGctr）：这个逻辑比较有意思。为什么无符号数比大小时 <code>Less = Carry ^ SUBctr</code>；有符号数比大小时 <code>Less = Sign ^ Overflow</code> 呢？首先我们知道，比大小的本质是使用加法器做减法运算，那么 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>−</mo><mi>B</mi><mo>=</mo><mi>A</mi><mo>+</mo><msub><mi>B</mi><mtext>补</mtext></msub><mo>=</mo><mi>A</mi><mo>+</mo><mo stretchy="false">(</mo><mo>∼</mo><mi>B</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A-B=A+B_{\text{补}}=A+(\sim B +1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">补</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span>。</p><ul><li>对于无符号数的比大小逻辑。由于是减法，SUBctr 一定是 1，因此此时的输出其实可以进一步归纳为 <code>Less = Carry ^ 1 = !Carry</code>。也就是说我们只需要分析加法运算的进位结果即可，而这就是之前学过的无符号整数加减运算逻辑了。如果 A 严格小于 B，则 B 的补码与 A 相加后不会产生进位，此时 <code>!Carry = !0 = 1</code> 表示 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>&lt;</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A&lt;B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>；如果 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>≥</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A\ge B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>，则 B 的补码与 A 相加后就会超过无符号整数表示的范围产生进位，此时 <code>!Carry = !1 = 0</code> 表示 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>≥</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \ge B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>。很巧妙的逻辑。</li><li>对于有符号数的比大小逻辑。<ul><li>如果运算没有溢出，即 <code>Overflow=0</code>，此时 A 与 B 的正负一定相同。A 与 B 的比大小结果可以直接根据加法器运算结果的符号位来确定。如果运算结果是负的，即 <code>Sign=1</code>，那么显然 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>&lt;</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A&lt;B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>；反之如果运算结果是正的，即 <code>Sign=0</code>，那么显然 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>≥</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A\ge B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>。</li><li>如果运算发生溢出，即 <code>Overflow=1</code>，此时 A 与 B 的正负一定不同。但我们不知道谁正谁负，根据有符号整数加减运算的溢出符号判定逻辑可知：<ul><li>若 A 为正数 B 为负数。溢出发生时运算结果一定是负数（正溢出），即 <code>Sign=1</code>，此时 <code>Less = Sign ^ OverFlow = 0</code>，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>≥</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \ge B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>。</li><li>若 A 为负数 B 为正数，溢出发生时运算结果一定是正数（负溢出），即 <code>Sign=0</code>，此时 <code>Less = Sign ^ OverFlow = 1</code>，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>&lt;</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A&lt; B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>。</li></ul></li></ul></li></ul><h3 id="3-3-算数运算逻辑">3.3 算数运算逻辑</h3><p>程序的运算需求有了（算数运算），能进行基本运算的硬件也设计出来了（ALU）。如何利用已有的运算部件来巧妙地设计算法以高效地实现我们常见的数学运算呢？让我们一探究竟！</p><h4 id="整数加减运算">整数加减运算</h4><p>不重复造轮子，见上学期记的笔记：<a href="https://blog.dwj601.cn/GPA/4th-term/SysBasic/#2-7-4-%E6%95%B4%E6%95%B0%E5%8A%A0%E5%87%8F%E8%BF%90%E7%AE%97">https://blog.dwj601.cn/GPA/4th-term/SysBasic/#2-7-4-整数加减运算</a></p><h4 id="原码乘法运算">原码乘法运算</h4><p><strong>怎么算的？按照手算乘法的逻辑进行运算</strong>。但是在计算机内部进行了一定的改进，有三点：</p><ol><li>并不是全部算完每一步的乘法结果后再一次性累加，而是使用一个"局部变量"保存前缀和（部分积）；</li><li>由于每一位的乘法结果都是在最高位进行的，因此我们不是对当前的乘法运算结果左移一位，而是将前面计算出的前缀和右移一位；</li><li>由于单步乘法运算时只有 0 和 1，显然若当前为 0 则对答案的贡献也为 0，因此当乘法位为 0 时只需要前缀和右移一位即可，而不需要执行相加操作。</li></ol>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-3fe975e9" role="button" aria-expanded="false" aria-controls="collapse-3fe975e9">        <div class="fold-arrow">▶</div>推导过程      </div>      <div class="fold-collapse collapse" id="collapse-3fe975e9">        <div class="fold-content">          <p>其实算法过程很简单，就是模拟了乘法运算的过程，这里就不罗列了。只不过其中有一些关于运算部件的巧妙利用。比如将每次部分积右移后多出来的一位存放到 Y 中，反正 Y 右移后的最后一位已经没用并且舍弃掉了，前面空出来的一位正好就用来存储部分积的最后一位。</p><p>我们将二进制位从低到高的下标从 1 开始计数，进位位记作 C，部分积记作 P，乘数位记作 Y，则有这样的递推式：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>P</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><mi>X</mi><msub><mi>y</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>≫</mo><mn>1</mn><mo separator="true">,</mo><mspace width="1em"></mspace><msub><mi>P</mi><mn>0</mn></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">P_i = (P_{i-1}+Xy_i) \gg 1,\quad P_0=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≫</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p><p>模拟过程如下：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409291051084.png" alt="算例"></p>        </div>      </div>    </div><p><strong>有哪些应用场景？浮点数尾数运算</strong>。在浮点数的尾数运算中，对数值位直接使用原码乘法即可，符号位就是两个乘数的符号位相异或的结果，例如：设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>x</mi><msub><mo stretchy="false">]</mo><mtext>原</mtext></msub><mo>=</mo><mn>0.1110</mn></mrow><annotation encoding="application/x-tex">[x]_\text{原}=0.1110</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal">x</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">原</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.1110</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>y</mi><msub><mo stretchy="false">]</mo><mtext>原</mtext></msub><mo>=</mo><mn>1.1101</mn></mrow><annotation encoding="application/x-tex">[y]_\text{原}=1.1101</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">原</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1.1101</span></span></span></span>，计算 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>x</mi><mo>×</mo><mi>y</mi><msub><mo stretchy="false">]</mo><mtext>原</mtext></msub></mrow><annotation encoding="application/x-tex">[x\times y]_\text{原}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">原</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，符号位为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>⊕</mo><mn>1</mn><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">0\oplus 1=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊕</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，数值位为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1110</mn><mo>×</mo><mn>1101</mn></mrow><annotation encoding="application/x-tex">1110 \times 1101</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1110</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1101</span></span></span></span> 的原码乘法运算结果 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10110110</mn></mrow><annotation encoding="application/x-tex">10110110</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10110110</span></span></span></span>。</p><p><strong>可以优化吗？分块思想</strong>。由于是逐位运算，因此我们需要进行 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 次相乘再相加的操作，时间复杂度为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>。现有的优化方案就是逐 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 位运算，那么时间复杂度就可以优化为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mfrac><mi>n</mi><mi>k</mi></mfrac><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(\frac{n}{k})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.095em;vertical-align:-0.345em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span></span></span></span>。</p><h4 id="补码乘法运算">补码乘法运算</h4><p><strong>怎么算的</strong>？如何在已知 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><msub><mo stretchy="false">]</mo><mtext>补</mtext></msub></mrow><annotation encoding="application/x-tex">[X]_{\text{补}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">补</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>Y</mi><msub><mo stretchy="false">]</mo><mtext>补</mtext></msub></mrow><annotation encoding="application/x-tex">[Y]_{\text{补}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">补</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的情况下，计算 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>×</mo><mi>Y</mi><msub><mo stretchy="false">]</mo><mtext>补</mtext></msub></mrow><annotation encoding="application/x-tex">[X\times Y]_{\text{补}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">补</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>？由于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>×</mo><mi>Y</mi><msub><mo stretchy="false">]</mo><mtext>补</mtext></msub><mo mathvariant="normal">≠</mo><mo stretchy="false">[</mo><mi>X</mi><msub><mo stretchy="false">]</mo><mtext>补</mtext></msub><mo>×</mo><mo stretchy="false">[</mo><mi>Y</mi><msub><mo stretchy="false">]</mo><mtext>补</mtext></msub></mrow><annotation encoding="application/x-tex">[X\times Y]_{\text{补}} \ne [X]_{\text{补}} \times [Y]_{\text{补}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">补</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">补</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">补</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，因此补码乘法不能直接使用原码乘法的算法，需要我们重新设计运算方法，这里引入 Booth 算法。</p>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-6749b3d7" role="button" aria-expanded="false" aria-controls="collapse-6749b3d7">        <div class="fold-arrow">▶</div>推导过程      </div>      <div class="fold-collapse collapse" id="collapse-6749b3d7">        <div class="fold-content">          <p>布斯算法的本质是将符号位与数值位一起运算，也就是对有符号数的一次性运算算法。如下推导：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409291113812.png" alt="推导"></p><p>进而可以得到关于「真值」的部分积递推公式：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub><mo>=</mo><mo stretchy="false">[</mo><msub><mi>P</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><mo stretchy="false">(</mo><msub><mi>y</mi><mrow><mi>i</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>−</mo><msub><mi>y</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mi>X</mi><mo stretchy="false">]</mo><mo>≫</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">P_{i} = [P_{i-1} + (y_{i-2}-y_{i-1})X] \gg 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≫</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></p><p>于是可以得到关于「补码」的部分积递推公式：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mo stretchy="false">[</mo><msub><mi>P</mi><mi>i</mi></msub><msub><mo stretchy="false">]</mo><mtext>补</mtext></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo stretchy="false">[</mo><msub><mi>P</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><mo stretchy="false">(</mo><msub><mi>y</mi><mrow><mi>i</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>−</mo><msub><mi>y</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mi>X</mi><msub><mo stretchy="false">]</mo><mtext>补</mtext></msub><mo>≫</mo><mn>1</mn></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mo stretchy="false">[</mo><msub><mi>P</mi><mi>i</mi></msub><msub><mo stretchy="false">]</mo><mtext>补</mtext></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo stretchy="false">[</mo><msub><mi>P</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mo stretchy="false">]</mo><mtext>补</mtext></msub><mo>≫</mo><mn>1</mn><mo>+</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mi>y</mi><mrow><mi>i</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>−</mo><msub><mi>y</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mi>X</mi><msub><mo stretchy="false">]</mo><mtext>补</mtext></msub><mo>≫</mo><mn>1</mn></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align}[P_{i}]_{\text{补}} &amp;= [P_{i-1} + (y_{i-2}-y_{i-1})X]_{\text{补}} \gg 1 \\[P_{i}]_{\text{补}} &amp;= [P_{i-1}]_{\text{补}} \gg 1 + [(y_{i-2}-y_{i-1})X]_{\text{补}} \gg 1\end{align}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">[</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">补</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">[</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">补</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">补</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≫</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">补</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≫</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">[(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">补</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≫</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span><span class="tag"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.75em;"><span class="pstrut" style="height:2.84em;"></span><span class="eqn-num"></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:2.84em;"></span><span class="eqn-num"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></p><p>显然的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mrow><mi>i</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>−</mo><msub><mi>y</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">y_{i-2}-y_{i-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span> 只有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">-1,0,1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span></span></span></span> 共 3 种情况，因此我们只需要知道 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mo stretchy="false">[</mo><mi>X</mi><msub><mo stretchy="false">]</mo><mtext>补</mtext></msub></mrow><annotation encoding="application/x-tex">-[X]_{\text{补}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">补</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><msub><mo stretchy="false">]</mo><mtext>补</mtext></msub></mrow><annotation encoding="application/x-tex">[X]_{\text{补}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">补</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 即可利用移位和加法快速运算。例如下面的算例：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409291137346.png" alt="算例"></p>        </div>      </div>    </div><p><strong>有哪些应用场景？很多</strong>。由于计算机中的数据都是以补码形式存储，因此补码乘法的使用场景更加广泛。</p><p><strong>可以优化吗？分块思想</strong>。与原码乘法的优化方案类似。</p><h4 id="整数除法运算">整数除法运算</h4><h2 id="四、指令系统">四、指令系统</h2><h2 id="五、中央处理器">五、中央处理器</h2><h2 id="六、指令流水线">六、指令流水线</h2><h2 id="七、存储器层次结构">七、存储器层次结构</h2><h3 id="7-4-存储器的数据校验">7.4 存储器的数据校验</h3><p>数据在信道传送的过程中，可能会因为各种噪声或者硬件原因出现错误，我们有必要进行数据的检错与纠错。本目我们主要介绍「<strong>数据检错中的奇偶校验法</strong>」策略。其实逻辑很简单，先看数据校验的流程图：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409180822743.png" alt="数据校验流程图"></p><p>如上图所示，校验的根本逻辑就是：比对「信源和信宿数据的二进制码」在某种转换规则下的转换代码。而所谓的奇偶校验就是定义了「某种规则」为「比对数据的二进制码中 1 的个数」。即通过判断校验单元（校验位+原始数据）和传输后数据中 1 的个数的奇偶性是否相同，来判断数据传送是否正确，进而达到检错的功能。有两个问题：</p><ol><li><strong>这合理吗</strong>？显然不具备绝对正确性。因为奇偶校验的根本逻辑是比对数据传输前后的 1 的数量的奇偶是否。反例有很多：<ul><li>首先我们不能保证校验码传输前后是否会发生错误。</li><li>其次针对校验逻辑，奇偶校验都不能检测出偶数位数据传输错误，并且，如果校验位前后不同则表明数据传输的过程中一定发生了错误，但是不能知道多少位 (1,3,5,…2n-1) 发生了错误，并且也不知道哪几位发生了错误。</li></ul></li><li><strong>怎么计算二进制码中 1 的个数的奇偶性</strong>？很简单，直接把所有的位异或起来即可。</li></ol><p><strong>码距是什么</strong>？所谓码距就是两个二进制代码（码字）中不同位的个数。在奇偶校验的码制下，两个合法码字的最小距离显然是 2。</p><p>参考：<a href="https://blog.csdn.net/Dontla/article/details/103627008">什么是奇偶校验原理？奇校验、偶校验、校验位（单比特奇偶校验、两维奇偶校验（矩阵校验或交叉奇偶校验））</a></p><h2 id="八、系统互连及输入输出组织">八、系统互连及输入输出组织</h2><h2 id="九、并行处理系统">九、并行处理系统 *</h2><p>不做要求，略。</p>]]></content>
+    <content type="html"><![CDATA[<h2 id="计算机组成原理">计算机组成原理</h2><h2 id="前言">前言</h2><p>学科地位：</p><table><thead><tr><th style="text-align:center">主讲教师</th><th style="text-align:center">学分配额</th><th style="text-align:center">学科类别</th></tr></thead><tbody><tr><td style="text-align:center">闫文珠</td><td style="text-align:center">3.5</td><td style="text-align:center">自发课</td></tr></tbody></table><p>成绩组成：</p><table><thead><tr><th style="text-align:center">作业+实验</th><th style="text-align:center">期末</th></tr></thead><tbody><tr><td style="text-align:center">50%</td><td style="text-align:center">50%</td></tr></tbody></table><p>教材情况：</p><table><thead><tr><th style="text-align:center">课程名称</th><th style="text-align:center">选用教材</th><th style="text-align:center">版次</th><th style="text-align:center">作者</th><th style="text-align:center">出版社</th><th style="text-align:center">ISBN 号</th></tr></thead><tbody><tr><td style="text-align:center">计算机组成原理</td><td style="text-align:center">《计算机组成与系统结构》</td><td style="text-align:center">3</td><td style="text-align:center">袁春风</td><td style="text-align:center">清华大学出版社</td><td style="text-align:center">978-7-302-59988-3</td></tr></tbody></table><p>最基本的知识普及：</p>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-d6ca2bab" role="button" aria-expanded="false" aria-controls="collapse-d6ca2bab">        <div class="fold-arrow">▶</div>最基本的知识普及      </div>      <div class="fold-collapse collapse" id="collapse-d6ca2bab">        <div class="fold-content">          <ul><li><a href="https://www.bilibili.com/video/BV1xQ4y1b7JS/">【硬核科普】从零开始认识主板</a></li><li><a href="https://www.bilibili.com/video/BV1xE421j7Uv/">【硬核科普】从零开始认识显卡</a></li></ul><p><strong>主板（Motherboard）</strong></p><ul><li><strong>作用</strong>：主板是所有硬件的连接平台，它连接并协调 CPU、内存、存储设备、显卡、网络设备和其他外部接口。主板上还包含芯片组，负责管理数据传输和设备间的通信。</li><li><strong>与其他部分的关系</strong>：主板是各个硬件组件的核心枢纽，确保数据能够在各个组件之间高效传输和通信。</li></ul><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409102038540.png" alt="机箱视角中的主板"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409102039187.png" alt="主板的 6 大区域"></p><p><strong>输入/输出设备（I/O Devices）</strong></p><ul><li><strong>作用</strong>：输入设备（如键盘、鼠标、触摸屏）用于用户与计算机的交互。输出设备（如显示器、打印机）用于显示计算机的操作结果。网络设备（如网卡、Wi-Fi 模块）则允许电脑连接到网络。</li><li><strong>与其他部分的关系</strong>：输入设备通过主板将用户的操作传递给 CPU 进行处理，输出设备则从 CPU 或显卡接收处理后的数据进行展示。</li></ul><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409102039284.png" alt="I/O 区域"></p><p><strong>中央处理器（CPU）</strong></p><ul><li><strong>作用</strong>：CPU 是电脑的大脑，负责执行计算任务和指令。它从内存中读取指令，进行运算，然后将结果写回内存。现代 CPU 具有多个核心，可以并行处理多项任务，提高计算效率。</li><li><strong>与其他部分的关系</strong>：CPU 与内存和存储设备密切配合。它从内存中获取数据和指令，然后进行处理，最终将结果写回内存或存储设备。</li></ul><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409102040689.png" alt="CPU 区域"></p><p><strong>内存（RAM）</strong></p><ul><li><strong>作用</strong>：内存是计算机中的临时存储器，用于存储当前运行的程序和数据。它的访问速度极快，但数据在断电后会丢失。内存用于支持 CPU 的高速操作。</li><li><strong>与其他部分的关系</strong>：内存是 CPU 执行任务时的工作区域，存储从存储设备（如硬盘）中读取的数据和指令。内存的数据会在需要时被写回到存储设备中。</li></ul><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409102040651.png" alt="内存区域"></p><p><strong>显卡（GPU）</strong></p><ul><li><strong>作用</strong>：显卡负责处理图形计算任务，特别是在图像和视频渲染、游戏以及某些并行计算任务中（如深度学习）。高性能显卡能够加速这些计算任务。</li><li><strong>与其他部分的关系</strong>：显卡通过主板与 CPU 和内存通信，从内存中读取图形数据，并进行处理，然后将渲染结果输出到显示器。</li></ul><p><strong>存储设备（Storage Devices）</strong></p><ul><li><strong>作用</strong>：存储设备如机械硬盘 hard disk (HDD)、固态硬盘 solid state disk (SSD)，用于长期保存数据和程序，即使在电脑断电后数据也不会丢失。SSD 的速度比传统硬盘更快，因此越来越常用。</li><li><strong>与其他部分的关系</strong>：存储设备保存操作系统、应用程序和用户数据。程序执行时，数据会从存储设备加载到内存中供 CPU 使用。</li></ul><p><strong>网卡（Network Interface Card, NIC）</strong></p><ul><li><strong>作用</strong>：网卡是计算机与网络之间的桥梁，负责处理计算机与网络间的数据通信。它将计算机的数据转换为网络信号，以太网网卡通过有线或无线方式连接局域网（LAN）或互联网。</li><li><strong>与其他部分的关系</strong>：网卡通过主板与南桥芯片组或 PCIe 总线连接，接收来自 CPU 的数据，经过处理后通过网络接口发送到外部网络。同时，它也接收来自网络的数据，传递给 CPU 进行处理，确保计算机能够与其他网络设备进行通信。</li></ul><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409102040086.png" alt="扩展区域"></p><p><strong>南桥芯片组（Southbridge）</strong></p><ul><li><strong>作用</strong>：南桥芯片组是主板上的一个重要芯片，为了避免过多的模块与 CPU 直连导致主板排线困难以及布局问题，产生了南桥芯片组用于负责管理较慢的外围设备的连接和数据传输。包括 USB 接口、SATA 接口、网络接口、音频设备和其他 I/O 设备。南桥还处理 BIOS、系统时钟、硬盘和光驱的控制。</li><li><strong>与其他部分的关系</strong>：南桥通过主板与 CPU 和北桥芯片组（或直接与 CPU）连接，接收来自 CPU 的指令，并将其转化为外设可以理解的信号。它还通过 I/O 总线连接各种外设，确保数据在外围设备与系统之间顺畅传输。</li></ul><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409102040366.png" alt="南桥芯片组区域"></p><p><strong>BIOS（Basic Input/Output System）</strong></p><ul><li><strong>作用</strong>：BIOS 是主板的固件程序，它存储在主板的 BIOS 芯片里。负责在计算机启动时进行硬件初始化，指导所有的硬件运行，并启动操作系统。</li><li><strong>与其他部分的关系</strong>：BIOS 在开机时检查并初始化硬件，然后引导操作系统，从而开始系统的工作。</li></ul><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409102040350.png" alt="BIOS 芯片"></p>        </div>      </div>    </div><p>为什么要学这门课？</p><blockquote><p>上上学期学习了《数字逻辑电路》，云里雾里；上学期学习了《计算机系统基础》，继续云里雾里。本学期开始学习《计算机组成原理》？如果用一句话来概括我对数字逻辑电路的理解，大概可以这样说：原来计算机是一个由“门电路”、“导线”和“时钟脉冲”组成的机器。如果用一句话来概括我对计算机系统基础的理解，大概可以这样说：原来计算机所有的活动都是由翻译过来的 01 序列驱动的。</p><p>现在我们让 AI 模仿上面我对课程理解的概括逻辑，用一句话来概括计算机组成原理，ta 是这样回答的：<strong>原来计算机的高效运作是通过硬件架构的精妙设计，将复杂的指令和数据流转化为有序的电子信号运动来实现的</strong>。</p></blockquote><p>会收获什么？</p><blockquote><p>对计算机整体有一个宏观的把握，了解 CPU、存储和 I/O 设备的工作逻辑，尤其要熟悉单核 CPU 的工作逻辑。最后能够利用 verilog 硬件描述语言从逻辑上实现一个单核 CPU。</p></blockquote><p>注：</p><blockquote><p><strong>本书 ISA 采用 MIPS 指令系统</strong>。之前学习的 <a href="https://blog.dwj601.cn/GPA/4th-term/SysBasic/#3-2-IA-32-%E6%8C%87%E4%BB%A4%E7%B3%BB%E7%BB%9F%E6%A6%82%E8%BF%B0">《计算机系统基础》</a> 中 ISA 采用的是 IA-32 指令系统，也就是大名鼎鼎的 x86-64 指令系统的前身。</p><p>为了形式化地描述 CPU 的运行逻辑，需要使用寄存器传送级 (register transfer level, 简称 RTL) 语言。<strong>本书 RTL 语言有以下规定</strong>：R[r] 表示通用寄存器 r 的内容，M[addr] 表示存储单元 addr 的内容；M[R[r]] 表示寄存器 r 的内容所指存储单元的内容；PC 表示 PC 的内容，M[PC] 表示 PC 所指存储单元的内容；SEXT[imm] 表示对 imm 进行符号扩展，ZEXT[imm] 表示对 imm 进行零扩展；传送方向用 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>←</mo></mrow><annotation encoding="application/x-tex">\leftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">←</span></span></span></span> 表示，即传送源在右，传送目的在左。</p></blockquote><h2 id="一、计算机系统概述">一、计算机系统概述</h2><p>注：本章与《计算机系统基础》的第 1 章重复，详见 <a href="https://blog.dwj601.cn/GPA/4th-term/SysBasic/#%E7%AC%AC1%E7%AB%A0-%E8%AE%A1%E7%AE%97%E6%9C%BA%E7%B3%BB%E7%BB%9F%E6%A6%82%E8%BF%B0">https://blog.dwj601.cn/GPA/4th-term/SysBasic/#第1章-计算机系统概述</a></p><p>主要掌握 <strong>冯诺依曼状态机</strong> 和 <strong>计算机性能度量</strong> 两个知识点。</p><h2 id="二、数据的机器级表示">二、数据的机器级表示</h2><p>注：本章与《计算机系统基础》的第 2 章重复，详见 <a href="https://blog.dwj601.cn/GPA/4th-term/SysBasic/#%E7%AC%AC2%E7%AB%A0-%E6%95%B0%E6%8D%AE%E7%9A%84%E6%9C%BA%E5%99%A8%E7%BA%A7%E8%A1%A8%E7%A4%BA%E4%B8%8E%E5%A4%84%E7%90%86">https://blog.dwj601.cn/GPA/4th-term/SysBasic/#第2章-数据的机器级表示与处理</a></p><p>主要掌握 <strong>数值/非数值数据的表示</strong>、<strong>数据宽度</strong>、<strong>存储对齐</strong> 和 <strong>纠/检错</strong> 四个知识点。</p><p>数据的纠错/检错见 7.4 节。</p><h2 id="三、运算方法和运算部件">三、运算方法和运算部件</h2><p>本章我们讲讲 CPU 中的 <strong>算数逻辑单元（Arithmetic and Logic Unit，简称 ALU）</strong>。ALU 在计算机中的地位大致如下图所示：</p><pre><code class=" mermaid">graph RL    实现功能    执行程序    机器指令    subgraph ALU    direction RL    算数运算逻辑    subgraph 运算部件    direction LR    加法器    移位器    end    end        运算部件 --&gt;|硬件支持| 算数运算逻辑    ALU --&gt; 机器指令    机器指令 --&gt; 执行程序    执行程序 --&gt; 实现功能</code></pre><p>也就有了这样的学习路线：涉及运算的机器指令（MIPS 指令系统） <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> 运算部件支持（逻辑电路图） <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> 算数运算逻辑（算法设计）</p><h3 id="3-1-涉及运算的机器指令">3.1 涉及运算的机器指令</h3><p><strong>都有哪些涉及运算的机器指令</strong>？机器指令的种类有很多，我们在计算机系统基础中已经学习到了比如：数据传输指令、控制指令、跳转指令等等。由于本章讲解的是算数运算，因此我们主要学习涉及到运算的机器指令。我们将所有涉及到运算的机器指令抽丝剥茧，凝练出了以下共 11 条指令：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409252041301.png" alt="11 条涉及运算指令的 RTL 描述"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409252041551.png" alt="11 条涉及运算指令的 RTL 描述 - 续"></p><p>不难发现，上述 11 种除了最后一个跳转指令，其余的 10 条运算指令只有 3 大类：加减法、按位或、小于置 1。</p><h3 id="3-2-运算部件支持">3.2 运算部件支持</h3><p><strong>有了上述的程序运算需求，如何设计出对应的硬件来支持呢？CPU 中的 ALU 单元就是负责运算的部件</strong>。ALU 内部的逻辑结构如下图所示，其输入有两个运算数 A 和 B 以及一个控制输入 ALUctr：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409250813229.png" alt="ALU 实现"></p><p>ALUctr 有 4 个控制信号，下面借书中原文分别解释 4 个控制信号对应的逻辑：</p><blockquote><p>SUBctr 用来控制 ALU 执行加法还是减法运算。当 SUBctr=1 时，做减法；当 SUBctr=0 时，做加法。</p><p>OPctr 用来控制选择哪种运算的结果作为 Result 输出。因为所实现的 11 条指令中只可能有加/减、按位或、小于置 1 这 3 大类运算，所以 OPctr 有两位。</p><p>OVctr 用来控制是否要进行溢出判断。当 OVctr=1 时，进行溢出判断，此时，若结果发生溢出，则溢出标志 Overflow 为 1，当 OVctr=0 时，无须溢出判断，此时，即使结果发生溢出，溢出标志 Overflow 也不为 1。</p><p>SIGctr 信号控制 ALU 是执行「带符号整数比较小于置 1」还是执行「无符号数比较小于置 1」的功能。当 SIGctr=0，执行「无符号数比较小于置 1」的功能；当 SIGctr=1 时，执行「带符号整数比较小于置 1」的功能。</p></blockquote><p>完美实现了所需的 3 大类运算逻辑。</p><p><strong>ALU 是如何控制以精准实现上述 11 条指令的？本质上就是通过 ALUctr 来实现运算控制的</strong>。除了最后一个跳转，难道 10 个指令就一定需要 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⌈</mo><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo><mo stretchy="false">⌉</mo><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">\lceil \log_2(10) \rceil =4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⌈</span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.207em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">10</span><span class="mclose">)⌉</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">4</span></span></span></span> 位进行信号选择控制吗？并不是，其中一些虽然功能不同，但是运算逻辑相同：</p><ul><li>指令 addiu、lw、sw 和 beq 转移目标地址计算的 ALU 控制信号取值一样，都是进行加法运算并不判溢出，记为 addu 操作.</li><li>指令 subu 和 beq 判 0 操作的 ALU 控制信号可看成一样，都做减法运算并不判溢出，记为 subu 操作。</li></ul><p>下表详细解释了每一条指令的运算类型与 ALUctr 取值之间的关系：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409261336957.png" alt="运算类型与 ALUctr 取值之间的关系"></p><p>因此，这 11 条指令可以归纳为 7 种操作：addu、add、or、subu、sub、sltu、slt，也就是说 ALUctr 只需要 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">⌈</mo><msub><mrow><mi>log</mi><mo>⁡</mo></mrow><mn>2</mn></msub><mo stretchy="false">(</mo><mn>7</mn><mo stretchy="false">)</mo><mo stretchy="false">⌉</mo><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">\lceil \log_2(7) \rceil =3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⌈</span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.207em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">7</span><span class="mclose">)⌉</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span></span></span></span> 个选择控制位！如下表所示，列出了 ALUctr 的选择控制逻辑：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409261337035.png" alt="ALUctr 的选择控制逻辑"></p><p><strong>为什么可以这样控制</strong>？</p><p>加减运算控制（SUBctr）：很显然，加法（SUBctr=0），减法（SUBctr=1）。</p><p>输出内容控制（OPctr）：很显然， 3 大类运算就对应 3 个取值。</p><p>溢出判断控制（OVctr）：很显然，有符号运算需要溢出判断，其他运算都不需要。</p><p>小于置一控制（SIGctr）：这个逻辑比较有意思。为什么无符号数比大小时 <code>Less = Carry ^ SUBctr</code>；有符号数比大小时 <code>Less = Sign ^ Overflow</code> 呢？首先我们知道，比大小的本质是使用加法器做减法运算，那么 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>−</mo><mi>B</mi><mo>=</mo><mi>A</mi><mo>+</mo><msub><mi>B</mi><mtext>补</mtext></msub><mo>=</mo><mi>A</mi><mo>+</mo><mo stretchy="false">(</mo><mo>∼</mo><mi>B</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A-B=A+B_{\text{补}}=A+(\sim B +1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">补</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span>。</p><ul><li>对于无符号数的比大小逻辑。由于是减法，SUBctr 一定是 1，因此此时的输出其实可以进一步归纳为 <code>Less = Carry ^ 1 = !Carry</code>。也就是说我们只需要分析加法运算的进位结果即可，而这就是之前学过的无符号整数加减运算逻辑了。如果 A 严格小于 B，则 B 的补码与 A 相加后不会产生进位，此时 <code>!Carry = !0 = 1</code> 表示 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>&lt;</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A&lt;B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>；如果 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>≥</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A\ge B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>，则 B 的补码与 A 相加后就会超过无符号整数表示的范围产生进位，此时 <code>!Carry = !1 = 0</code> 表示 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>≥</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \ge B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>。很巧妙的逻辑。</li><li>对于有符号数的比大小逻辑。<ul><li>如果运算没有溢出，即 <code>Overflow=0</code>，此时 A 与 B 的正负一定相同。A 与 B 的比大小结果可以直接根据加法器运算结果的符号位来确定。如果运算结果是负的，即 <code>Sign=1</code>，那么显然 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>&lt;</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A&lt;B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>；反之如果运算结果是正的，即 <code>Sign=0</code>，那么显然 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>≥</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A\ge B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>。</li><li>如果运算发生溢出，即 <code>Overflow=1</code>，此时 A 与 B 的正负一定不同。但我们不知道谁正谁负，根据有符号整数加减运算的溢出符号判定逻辑可知：<ul><li>若 A 为正数 B 为负数。溢出发生时运算结果一定是负数（正溢出），即 <code>Sign=1</code>，此时 <code>Less = Sign ^ OverFlow = 0</code>，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>≥</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \ge B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>。</li><li>若 A 为负数 B 为正数，溢出发生时运算结果一定是正数（负溢出），即 <code>Sign=0</code>，此时 <code>Less = Sign ^ OverFlow = 1</code>，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>&lt;</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A&lt; B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>。</li></ul></li></ul></li></ul><h3 id="3-3-算数运算逻辑">3.3 算数运算逻辑</h3><p>程序的运算需求有了（算数运算），能进行基本运算的硬件也设计出来了（ALU）。如何利用已有的运算部件来巧妙地设计算法以高效地实现我们常见的数学运算呢？让我们一探究竟！</p><h4 id="整数加减运算">整数加减运算</h4><p>不重复造轮子，见上学期记的笔记：<a href="https://blog.dwj601.cn/GPA/4th-term/SysBasic/#2-7-4-%E6%95%B4%E6%95%B0%E5%8A%A0%E5%87%8F%E8%BF%90%E7%AE%97">https://blog.dwj601.cn/GPA/4th-term/SysBasic/#2-7-4-整数加减运算</a></p><h4 id="原码乘法运算">原码乘法运算</h4><p><strong>怎么算的？按照手算乘法的逻辑进行运算</strong>。但是在计算机内部进行了一定的改进，有三点：</p><ol><li>并不是全部算完每一步的乘法结果后再一次性累加，而是使用一个"局部变量"保存前缀和（部分积）；</li><li>由于每一位的乘法结果都是在最高位进行的，因此我们不是对当前的乘法运算结果左移一位，而是将前面计算出的前缀和右移一位；</li><li>由于单步乘法运算时只有 0 和 1，显然若当前为 0 则对答案的贡献也为 0，因此当乘法位为 0 时只需要前缀和右移一位即可，而不需要执行相加操作。</li></ol>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-3fe975e9" role="button" aria-expanded="false" aria-controls="collapse-3fe975e9">        <div class="fold-arrow">▶</div>推导过程      </div>      <div class="fold-collapse collapse" id="collapse-3fe975e9">        <div class="fold-content">          <p>其实算法过程很简单，就是模拟了乘法运算的过程，这里就不罗列了。只不过其中有一些关于运算部件的巧妙利用。比如将每次部分积右移后多出来的一位存放到 Y 中，反正 Y 右移后的最后一位已经没用并且舍弃掉了，前面空出来的一位正好就用来存储部分积的最后一位。</p><p>我们将二进制位从低到高的下标从 1 开始计数，进位位记作 C，部分积记作 P，乘数位记作 Y，则有这样的递推式：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>P</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><mi>X</mi><msub><mi>y</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>≫</mo><mn>1</mn><mo separator="true">,</mo><mspace width="1em"></mspace><msub><mi>P</mi><mn>0</mn></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">P_i = (P_{i-1}+Xy_i) \gg 1,\quad P_0=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≫</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p><p>模拟过程如下：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409291051084.png" alt="算例"></p>        </div>      </div>    </div><p><strong>有哪些应用场景？浮点数尾数运算</strong>。在浮点数的尾数运算中，对数值位直接使用原码乘法即可，符号位就是两个乘数的符号位相异或的结果，例如：设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>x</mi><msub><mo stretchy="false">]</mo><mtext>原</mtext></msub><mo>=</mo><mn>0.1110</mn></mrow><annotation encoding="application/x-tex">[x]_\text{原}=0.1110</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal">x</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">原</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0.1110</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>y</mi><msub><mo stretchy="false">]</mo><mtext>原</mtext></msub><mo>=</mo><mn>1.1101</mn></mrow><annotation encoding="application/x-tex">[y]_\text{原}=1.1101</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">原</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1.1101</span></span></span></span>，计算 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>x</mi><mo>×</mo><mi>y</mi><msub><mo stretchy="false">]</mo><mtext>原</mtext></msub></mrow><annotation encoding="application/x-tex">[x\times y]_\text{原}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">原</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，符号位为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>⊕</mo><mn>1</mn><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">0\oplus 1=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊕</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，数值位为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1110</mn><mo>×</mo><mn>1101</mn></mrow><annotation encoding="application/x-tex">1110 \times 1101</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1110</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1101</span></span></span></span> 的原码乘法运算结果 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10110110</mn></mrow><annotation encoding="application/x-tex">10110110</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10110110</span></span></span></span>。</p><p><strong>可以优化吗？分块思想</strong>。由于是逐位运算，因此我们需要进行 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 次相乘再相加的操作，时间复杂度为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>。现有的优化方案就是逐 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 位运算，那么时间复杂度就可以优化为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mfrac><mi>n</mi><mi>k</mi></mfrac><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(\frac{n}{k})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.095em;vertical-align:-0.345em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span></span></span></span>。</p><h4 id="补码乘法运算">补码乘法运算</h4><p><strong>怎么算的</strong>？如何在已知 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><msub><mo stretchy="false">]</mo><mtext>补</mtext></msub></mrow><annotation encoding="application/x-tex">[X]_{\text{补}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">补</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>Y</mi><msub><mo stretchy="false">]</mo><mtext>补</mtext></msub></mrow><annotation encoding="application/x-tex">[Y]_{\text{补}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">补</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的情况下，计算 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>×</mo><mi>Y</mi><msub><mo stretchy="false">]</mo><mtext>补</mtext></msub></mrow><annotation encoding="application/x-tex">[X\times Y]_{\text{补}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">补</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>？由于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>×</mo><mi>Y</mi><msub><mo stretchy="false">]</mo><mtext>补</mtext></msub><mo mathvariant="normal">≠</mo><mo stretchy="false">[</mo><mi>X</mi><msub><mo stretchy="false">]</mo><mtext>补</mtext></msub><mo>×</mo><mo stretchy="false">[</mo><mi>Y</mi><msub><mo stretchy="false">]</mo><mtext>补</mtext></msub></mrow><annotation encoding="application/x-tex">[X\times Y]_{\text{补}} \ne [X]_{\text{补}} \times [Y]_{\text{补}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">补</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">补</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">补</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，因此补码乘法不能直接使用原码乘法的算法，需要我们重新设计运算方法，这里引入 Booth 算法。</p>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-6749b3d7" role="button" aria-expanded="false" aria-controls="collapse-6749b3d7">        <div class="fold-arrow">▶</div>推导过程      </div>      <div class="fold-collapse collapse" id="collapse-6749b3d7">        <div class="fold-content">          <p>布斯算法的本质是将符号位与数值位一起运算，也就是对有符号数的一次性运算算法。如下推导：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409291113812.png" alt="推导"></p><p>进而可以得到关于「真值」的部分积递推公式：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>P</mi><mi>i</mi></msub><mo>=</mo><mo stretchy="false">[</mo><msub><mi>P</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><mo stretchy="false">(</mo><msub><mi>y</mi><mrow><mi>i</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>−</mo><msub><mi>y</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mi>X</mi><mo stretchy="false">]</mo><mo>≫</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">P_{i} = [P_{i-1} + (y_{i-2}-y_{i-1})X] \gg 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≫</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></p><p>于是可以得到关于「补码」的部分积递推公式：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mo stretchy="false">[</mo><msub><mi>P</mi><mi>i</mi></msub><msub><mo stretchy="false">]</mo><mtext>补</mtext></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo stretchy="false">[</mo><msub><mi>P</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><mo stretchy="false">(</mo><msub><mi>y</mi><mrow><mi>i</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>−</mo><msub><mi>y</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mi>X</mi><msub><mo stretchy="false">]</mo><mtext>补</mtext></msub><mo>≫</mo><mn>1</mn></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mo stretchy="false">[</mo><msub><mi>P</mi><mi>i</mi></msub><msub><mo stretchy="false">]</mo><mtext>补</mtext></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo stretchy="false">[</mo><msub><mi>P</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mo stretchy="false">]</mo><mtext>补</mtext></msub><mo>≫</mo><mn>1</mn><mo>+</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mi>y</mi><mrow><mi>i</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>−</mo><msub><mi>y</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mi>X</mi><msub><mo stretchy="false">]</mo><mtext>补</mtext></msub><mo>≫</mo><mn>1</mn></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align}[P_{i}]_{\text{补}} &amp;= [P_{i-1} + (y_{i-2}-y_{i-1})X]_{\text{补}} \gg 1 \\[P_{i}]_{\text{补}} &amp;= [P_{i-1}]_{\text{补}} \gg 1 + [(y_{i-2}-y_{i-1})X]_{\text{补}} \gg 1\end{align}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">[</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">补</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">[</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">补</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">补</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≫</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">补</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≫</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">[(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">补</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≫</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span><span class="tag"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.75em;"><span class="pstrut" style="height:2.84em;"></span><span class="eqn-num"></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:2.84em;"></span><span class="eqn-num"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></p><p>显然的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mrow><mi>i</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>−</mo><msub><mi>y</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">y_{i-2}-y_{i-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span> 只有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">-1,0,1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span></span></span></span> 共 3 种情况，因此我们只需要知道 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mo stretchy="false">[</mo><mi>X</mi><msub><mo stretchy="false">]</mo><mtext>补</mtext></msub></mrow><annotation encoding="application/x-tex">-[X]_{\text{补}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">补</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><msub><mo stretchy="false">]</mo><mtext>补</mtext></msub></mrow><annotation encoding="application/x-tex">[X]_{\text{补}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">补</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 即可利用移位和加法快速运算。例如下面的算例：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409291137346.png" alt="算例"></p>        </div>      </div>    </div><p><strong>有哪些应用场景？很多</strong>。由于计算机中的数据都是以补码形式存储，因此补码乘法的使用场景更加广泛。</p><p><strong>可以优化吗？分块思想</strong>。与原码乘法的优化方案类似。</p><h4 id="整数除法运算">整数除法运算</h4><h2 id="四、指令系统">四、指令系统</h2><h2 id="五、中央处理器">五、中央处理器</h2><h2 id="六、指令流水线">六、指令流水线</h2><h2 id="七、存储器层次结构">七、存储器层次结构</h2><h3 id="7-4-存储器的数据校验">7.4 存储器的数据校验</h3><p>数据在信道传送的过程中，可能会因为各种噪声或者硬件原因出现错误，我们有必要进行数据的检错与纠错。本目我们主要介绍「<strong>数据检错中的奇偶校验法</strong>」策略。其实逻辑很简单，先看数据校验的流程图：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409180822743.png" alt="数据校验流程图"></p><p>如上图所示，校验的根本逻辑就是：比对「信源和信宿数据的二进制码」在某种转换规则下的转换代码。而所谓的奇偶校验就是定义了「某种规则」为「比对数据的二进制码中 1 的个数」。即通过判断校验单元（校验位+原始数据）和传输后数据中 1 的个数的奇偶性是否相同，来判断数据传送是否正确，进而达到检错的功能。有两个问题：</p><ol><li><strong>这合理吗</strong>？显然不具备绝对正确性。因为奇偶校验的根本逻辑是比对数据传输前后的 1 的数量的奇偶是否。反例有很多：<ul><li>首先我们不能保证校验码传输前后是否会发生错误。</li><li>其次针对校验逻辑，奇偶校验都不能检测出偶数位数据传输错误，并且，如果校验位前后不同则表明数据传输的过程中一定发生了错误，但是不能知道多少位 (1,3,5,…2n-1) 发生了错误，并且也不知道哪几位发生了错误。</li></ul></li><li><strong>怎么计算二进制码中 1 的个数的奇偶性</strong>？很简单，直接把所有的位异或起来即可。</li></ol><p><strong>码距是什么</strong>？所谓码距就是两个二进制代码（码字）中不同位的个数。在奇偶校验的码制下，两个合法码字的最小距离显然是 2。</p><p>参考：<a href="https://blog.csdn.net/Dontla/article/details/103627008">什么是奇偶校验原理？奇校验、偶校验、校验位（单比特奇偶校验、两维奇偶校验（矩阵校验或交叉奇偶校验））</a></p><h2 id="八、系统互连及输入输出组织">八、系统互连及输入输出组织</h2><h2 id="九、并行处理系统">九、并行处理系统 *</h2><p>不做要求，略。</p>]]></content>
     
     
     <categories>
@@ -493,7 +493,7 @@
     <link href="/GPA/5th-term/OperatingSystem/"/>
     <url>/GPA/5th-term/OperatingSystem/</url>
     
-    <content type="html"><![CDATA[<h2 id="操作系统">操作系统</h2><p>学科地位：</p><table><thead><tr><th style="text-align:center">主讲教师</th><th style="text-align:center">学分配额</th><th style="text-align:center">学科类别</th></tr></thead><tbody><tr><td style="text-align:center">段博佳</td><td style="text-align:center">4+1</td><td style="text-align:center">专业课</td></tr></tbody></table><p>成绩组成：</p><table><thead><tr><th style="text-align:center">理论课</th><th style="text-align:center"></th><th style="text-align:center">实验课</th><th style="text-align:center"></th></tr></thead><tbody><tr><td style="text-align:center">平时</td><td style="text-align:center">10%</td><td style="text-align:center">小组作业</td><td style="text-align:center">100%</td></tr><tr><td style="text-align:center">期中</td><td style="text-align:center">30%</td><td style="text-align:center"></td><td style="text-align:center"></td></tr><tr><td style="text-align:center">期末</td><td style="text-align:center">60%</td><td style="text-align:center"></td><td style="text-align:center"></td></tr></tbody></table><p>教材情况：</p><table><thead><tr><th style="text-align:center">课程名称</th><th style="text-align:center">选用教材</th><th style="text-align:center">版次</th><th style="text-align:center">作者</th><th style="text-align:center">出版社</th><th style="text-align:center">ISBN 号</th></tr></thead><tbody><tr><td style="text-align:center">操作系统原理</td><td style="text-align:center">《操作系统教程》</td><td style="text-align:center">6</td><td style="text-align:center">骆斌 葛季栋 费翔林</td><td style="text-align:center">高等教育出版社</td><td style="text-align:center">978-7-04-055304-8</td></tr></tbody></table><p>学习资源：</p><ul><li>🖥 <a href="https://space.bilibili.com/202224425/channel/collectiondetail?sid=2237004">合集·[完结] 2024 南京大学 “操作系统：设计与实现” (蒋炎岩)</a></li><li>📑 <a href="https://jyywiki.cn/OS/2024/">蒋炎岩课程配套讲义</a></li></ul><p>最难崩的一集：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202408301623261.png" alt="最难崩的一集"></p><p>为什么要学这门课？</p><blockquote><p>目前无论是考研还是互联网行业求职，操作系统似乎一直深受青睐。在对操作系统有了一点粗浅的认知后，我做出了这样的人机类比，尝试解释为什么操作系统会有如今这般重要的地位。</p><p>对于任何一种具备交互、计算功能的电子计算机而言。不同的 <strong>模块</strong> 就是不同的器官，承担了各自的责任和义务（毋庸置疑 CPU 就是心脏）；源源不断的 <strong>电能</strong> 就是循环不止的血液，确保所有的细胞能及时的进行呼吸以正常工作；<strong>主板</strong> 则承担了骨架与神经的指责，搭载了所有的模块并确保数据可以在不同的模块之间自由的流通；而 <strong>操作系统</strong> 则肩负起肌肉的责任，从 CPU 的心肌泵动开始让整台计算器健康地运作。</p></blockquote><p>会收获什么？</p><blockquote><p>能详细地解释下面的过程中电脑都发生了什么：按下电源键，网络连接，蓝牙连接，启动网易云音乐播放器软件并开始播放，启动谷歌浏览器并进入力扣网站，启动 Pycharm 集成开发环境软件编写代码并运行测试，关闭 Pycharm，关闭谷歌浏览器，关闭网易云，断开蓝牙，断开 WIFI 连接，关机。</p><p>记得多想想：是什么？为什么？会怎么样？当然也可以想想：可以比它更优吗？比如：什么是【操作系统】？为什么会有【操作系统】？有了【操作系统】会怎么样吗？可以用别的策略代替吗？上述的【操作系统】可以替换为任意一个名词或短语。</p></blockquote><h2 id="绪论">绪论</h2><p>本文的行文大纲是基于硬件进行的。毕竟操作系统并非实体，本质是一个运行在硬件之上的软件，只不过这个软件的权限比较高可以直接调用分配硬件资源而已。全文围绕以下 4 个硬件模块的 <strong>各自功能</strong> 以及 <strong>之间的交互</strong> 展开：</p><pre><code class=" mermaid">---title: What can Operating System do?---graph LR    subgraph CPU    direction LR    进程管理    资源调度    并发执行    end    subgraph RAM    内存管理    虚拟内存    end    subgraph I/O    设备管理    中断处理    缓存管理    end    subgraph SSD    文件系统    存储分配    end    subgraph NIC    网络与通信管理    end    CPU --&gt; RAM    CPU --&gt; I/O    CPU --&gt; SSD    CPU -.-&gt; NIC</code></pre><p><strong>操作系统的定义是什么</strong>？对于硬件。操作系统封装所有的硬件资源；对于用户和软件。操作系统将封装好的硬件资源分配给软件调用，同时为用户提供可视化界面。</p><p><strong>操作系统是怎么管理资源的</strong>？主要有三个技术。资源复用、资源虚拟化和资源抽象。后两个技术都是从第一个资源复用衍生而来，个人认为很多东西的解释都被复杂化了。</p><ul><li>比如，资源复用。其实根本没有复用，这只是相对于一开始的时候，一台计算机只能一次运行一个程序而言的。现在一台计算机需要同时运行很多程序，怎么办呢？既然程序运行需要计算资源，那我直接把计算机中的资源划分成很多份不就好了。当然了，除了内存进行空间上的划分，时间上也可以划分，比如 CPU，于是就有了资源复用中「空分复用」和「时分复用」两个理念。</li><li>资源虚拟化听着玄乎，其实就是名词学术化的资源复用。比如，我将物理世界中的硬件，通过操作系统的划分与调度，将其转化为逻辑世界中的多个资源，也就是所谓的将「单一物理资源」虚拟化为「多个逻辑资源」的理念。</li><li>资源抽象也很好理解，就是一种底层封装，顶层调用的思想。比如，对于每一个进程而言，都获得了操作系统为 ta 单独分配的 CPU 状态集和内存空间，而无需关心这些硬件资源是怎么分配给 ta 的。比如，对于每一个文件而言，都是对物理存储空间的抽象。</li></ul><p><strong>操作系统有什么特点</strong>？并发性、共享性和异步性，都很好理解。并发就是多个进程任务同时进行，但其实是操作系统的一个小把戏，让 CPU 在极短的时间间隔内频繁地切换执行不同的进程；共享是并发的衍生，我们知道程序执行是需要消耗资源的，而资源有限，因此我们可以让所有的进程共享同一块资源；而异步就是指一个任务在等待其他任务执行时，不会阻塞整个系统，等到其他任务执行结束或者切换到当前任务时，可以继续 ta 的执行。<strong>所以这么折腾到底是为了啥</strong>？可以从两个角度出发：</p><ol><li>对于一个任务而言：这样做可以 <strong>提升资源利用率</strong>。因为一个进程在执行时，任务会有很多，不仅仅需要 CPU 的状态计算，可能还需要和别的设备进行数据交互，我们知道 CPU 的计算效率远高于诸如 I/O 之类的设备，这就会让系统停下来等待进程执行其他的任务。我们就可以利用系统停下来等待的这段时间，把这个进程的其他任务给执行了。同时，如果一个任务可以被划分为多个子任务，也可以并发以提升资源利用率。</li><li>对于多个任务而言：可以 <strong>提升系统响应效率</strong>。多个用户同时提交请求？是一个一个响应，最快的极快，最慢的很慢。还是牺牲一点速度，大家都慢一点但是都没那么慢，同时响应？显然是后者。</li></ol><p><strong>操作系统是怎么对上层提供服务的</strong>？一句话概括就是，操作系统将封装好的应用程序接口 (Application Program Interface, 简称 API) 给用户或应用程序调用。例如下面的两个流程图：</p><pre><code class=" mermaid">---title: 用户调用系统资源---graph RL    subgraph User[调用者]    direction LR    用户逻辑    界面    解释器    用户逻辑 --&gt; 界面    界面 --&gt; 解释器    end    subgraph OS[OS]    内核    end    subgraph Source[硬件资源]    内存    end    User --&gt;|API| OS    OS --&gt;|调用| Source    Source --&gt;|反馈| OS    OS --&gt;|反馈| User</code></pre><pre><code class=" mermaid">---title: 应用程度调用系统资源---graph RL    subgraph User[调用者]    应用程度逻辑    end    subgraph OS[OS]    内核    end    subgraph Source[硬件资源]    内存    end    User --&gt;|API| OS    OS --&gt;|调用| Source    Source --&gt;|反馈| OS    OS --&gt;|反馈| User</code></pre><p><strong>为什么让操作系统对下层抽象进而为上层提供服务</strong>？因为下层的硬件十分繁杂，并且不同厂商生产的硬件可能会有使用上的不同，通过操作系统对下层硬件的封装，不仅可以隐藏复杂的硬件资源调度，可以统一出一种普适性更高的规则供上层服务使用。</p><p><strong>操作系统的内核是什么？操作系统的其他部分是什么？为什么要这样区分？</strong></p><ul><li>首先我们有必要了解「用户态」和「内核态」的概念。这两种状态定义了应用程序的指令执行权限，当程序运行在用户态时，不可以执行一些关键的指令，例如中断、网络通信等；但当程序运行在核心态时，可以执行所有指令。不难发现隔离化的运行模式相比于所有应用程序都有最高指令执行权限来说，可以进一步提升程序运行时系统的安全性。于是乎，操作系统的内核概念应运而生。</li><li>知道了指令执行权限隔离的程序运行理念后，再来理解操作系统的内核就很显然了。我们知道操作系统本质上就是一系列的系统软件，我们将操作系统中直接与硬件交互的软件划分到一起并称为「内核」，代表性内核软件可以实现的功能如「中断处理，进程管理」。操作系统剩余的部分就被划分为「非内核」部分，用于和应用程序直接交互，代表性非内核软件可以实现的功能形如「提供系统调用接口，基本计算」。当程序运行在用户态时，仅可以和操作系统的非内核部分交互；当切换到核心态时，就可以操作系统包括内核的所有部分交互。</li></ul><pre><code class=" mermaid">block-beta  columns 2    %% 上层  user["用户"]:1 app["应用程序"]:1    %% OS  nonkernel&lt;["封装调用，基本计算\n「OS 非内核」"]&gt;(up,down):2  kernel&lt;["中断处理，进程管理\n「OS 内核」"]&gt;(up,down):2    %% 底层  source["硬件"]:2</code></pre><h2 id="并发">并发</h2><h3 id="进程控制与管理">进程控制与管理</h3><p>CPU 的核心关键是并发程序，为此每一个硬件设计与相应 OS 的运行逻辑，都和 <strong>进程管理、资源调度、并发</strong> 息息相关。</p><p><strong>进程和程序的区别是什么？动态性</strong>。程序是一个指令序列，当程序载入进程时就转换为了进程。</p><p>五态模型：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409241450175.png" alt="五态模型 - 状态转换"></p><p>七态模型：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409242052428.png" alt="七态模型 - 状态转换"></p><p><strong>如何准确描述一个进程？通过进程映象</strong>。我们引入进程映象的概念。所谓进程映象就是某一时刻的进程状态，可以描述为以下 4 个部分：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409270845459.png" alt="进程映象"></p><p><strong>操作系统如何跟踪每一个进程？通过 PCB</strong>。每一个进程都会有很多信息，OS 设计者将其统一存储在进程控制块 (Process Control Block, 简称 PCB) 中。每一个进程都有自己的 PCB，只要是跟进程管理相关的数据信息一定都存储于 PCB 中。PCB 中存储的内容主要有以下内容：</p><ul><li><p>标识信息</p><ul><li>进程标识 PID</li><li>进程组标识 ID</li></ul></li><li><p>现场信息</p><ul><li>程序计数器 Program Counter</li><li>CPU 寄存器：包括栈指针、通用寄存器等</li></ul></li><li><p>控制信息</p><ul><li>进程状态：三态、五态等等</li><li>CPU 调度信息：如优先级、调度队列指针、调度参数等</li><li>内存管理信息：依据内存管理的方式不同而不同，如基/限长寄存器、段/页表指针等</li><li>计费信息：如进程使用的 CPU 时间</li><li>I/O 状态信息：包括分配给该进程的 I/O 设备列表、打开文件列表等</li></ul></li></ul><p><strong>当有多个进程处于同一个状态时，操作系统如何访存不同的进程呢？通过等待队列存储 PCB 以及优先级调度算法实现</strong>。我们将状态相同的进程存放到同一个容器（如链表队列）中，操作系统拥有每一个容器的指针。容器中相同状态的不同进程可能会按照优先级的顺序链接。那么容器中存的是什么呢？从上文可以知道，每一个 PCB 都可以唯一确定一个进程，因此等待队列中只需要存储 PCB 即可。</p><p><strong>并发执行程序时，如何切换状态？如何继续运行原进程呢？通过中断/系统调用切换状态，通过 PCB 保存现场数据</strong>。PCB 中的寄存器可以在切换进程时保存必要的数据，确保进程在中断获得运行权后能够从刚才的状态继续运行。这个「触发中断、保存现场、处理中断、恢复现场、结束中断」的过程被称为进程上下文切换。如下图所示：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409270937298.png" alt="进程上下文切换"></p><p><strong>如果在执行中断任务时又中断了呢？这是不会发生的</strong>。很多操作是不允许被打断的，例如保存现场等和数据极其重要的操作。我们就需要最小操作单元，这被称作「原语」操作。</p><p><strong>线程是什么</strong>？线程是从进程引申出来的概念与技术。进程是资源分配和调度的最小单位，而线程是 CPU 调度的最小单位。一个进程下可以有多个线程，每一个线程共享当前进程的所有资源，线程只会有执行指令最基本的资源（PC、寄存器等）。</p><p><strong>为什么会有线程</strong>？当一个进程的任务很复杂以至于也需要并发时，线程就可以充当更细力度的并发单元。也就是说线程的出现是为了解决越来越复杂的进程多任务并发问题的。</p><h3 id="互斥">互斥</h3><h3 id="同步">同步</h3><h3 id="死锁">死锁</h3><h2 id="虚拟化">虚拟化</h2><h2 id="内核">内核</h2><h3 id="中断-系统调用-上下文切换">中断 | 系统调用 | 上下文切换</h3><h3 id="处理器调度">处理器调度</h3><h2 id="持久化">持久化</h2>]]></content>
+    <content type="html"><![CDATA[<h2 id="操作系统">操作系统</h2><h2 id="前言">前言</h2><p>学科地位：</p><table><thead><tr><th style="text-align:center">主讲教师</th><th style="text-align:center">学分配额</th><th style="text-align:center">学科类别</th></tr></thead><tbody><tr><td style="text-align:center">段博佳</td><td style="text-align:center">4+1</td><td style="text-align:center">专业课</td></tr></tbody></table><p>成绩组成：</p><table><thead><tr><th style="text-align:center">理论课</th><th style="text-align:center"></th><th style="text-align:center">实验课</th><th style="text-align:center"></th></tr></thead><tbody><tr><td style="text-align:center">平时</td><td style="text-align:center">10%</td><td style="text-align:center">小组作业</td><td style="text-align:center">100%</td></tr><tr><td style="text-align:center">期中</td><td style="text-align:center">30%</td><td style="text-align:center"></td><td style="text-align:center"></td></tr><tr><td style="text-align:center">期末</td><td style="text-align:center">60%</td><td style="text-align:center"></td><td style="text-align:center"></td></tr></tbody></table><p>教材情况：</p><table><thead><tr><th style="text-align:center">课程名称</th><th style="text-align:center">选用教材</th><th style="text-align:center">版次</th><th style="text-align:center">作者</th><th style="text-align:center">出版社</th><th style="text-align:center">ISBN 号</th></tr></thead><tbody><tr><td style="text-align:center">操作系统原理</td><td style="text-align:center">《操作系统教程》</td><td style="text-align:center">6</td><td style="text-align:center">骆斌 葛季栋 费翔林</td><td style="text-align:center">高等教育出版社</td><td style="text-align:center">978-7-04-055304-8</td></tr></tbody></table><p>学习资源：</p><ul><li>🖥 <a href="https://space.bilibili.com/202224425/channel/collectiondetail?sid=2237004">合集·[完结] 2024 南京大学 “操作系统：设计与实现” (蒋炎岩)</a></li><li>📑 <a href="https://jyywiki.cn/OS/2024/">蒋炎岩课程配套讲义</a></li></ul><p>最难崩的一集：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202408301623261.png" alt="最难崩的一集"></p><p>为什么要学这门课？</p><blockquote><p>目前无论是考研还是互联网行业求职，操作系统似乎一直深受青睐。在对操作系统有了一点粗浅的认知后，我做出了这样的人机类比，尝试解释为什么操作系统会有如今这般重要的地位。</p><p>对于任何一种具备交互、计算功能的电子计算机而言。不同的 <strong>模块</strong> 就是不同的器官，承担了各自的责任和义务（毋庸置疑 CPU 就是心脏）；源源不断的 <strong>电能</strong> 就是循环不止的血液，确保所有的细胞能及时的进行呼吸以正常工作；<strong>主板</strong> 则承担了骨架与神经的指责，搭载了所有的模块并确保数据可以在不同的模块之间自由的流通；而 <strong>操作系统</strong> 则肩负起肌肉的责任，从 CPU 的心肌泵动开始让整台计算器健康地运作。</p></blockquote><p>会收获什么？</p><blockquote><p>能详细地解释下面的过程中电脑都发生了什么：按下电源键，网络连接，蓝牙连接，启动网易云音乐播放器软件并开始播放，启动谷歌浏览器并进入力扣网站，启动 Pycharm 集成开发环境软件编写代码并运行测试，关闭 Pycharm，关闭谷歌浏览器，关闭网易云，断开蓝牙，断开 WIFI 连接，关机。</p><p>记得多想想：是什么？为什么？会怎么样？当然也可以想想：可以比它更优吗？比如：什么是【操作系统】？为什么会有【操作系统】？有了【操作系统】会怎么样吗？可以用别的策略代替吗？上述的【操作系统】可以替换为任意一个名词或短语。</p></blockquote><h2 id="绪论">绪论</h2><p>本文的行文大纲是基于硬件进行的。毕竟操作系统并非实体，本质是一个运行在硬件之上的软件，只不过这个软件的权限比较高可以直接调用分配硬件资源而已。全文围绕以下 4 个硬件模块的 <strong>各自功能</strong> 以及 <strong>之间的交互</strong> 展开：</p><pre><code class=" mermaid">---title: What can Operating System do?---graph LR    subgraph CPU    direction LR    进程管理    资源调度    并发执行    end    subgraph RAM    内存管理    虚拟内存    end    subgraph I/O    设备管理    中断处理    缓存管理    end    subgraph SSD    文件系统    存储分配    end    subgraph NIC    网络与通信管理    end    CPU --&gt; RAM    CPU --&gt; I/O    CPU --&gt; SSD    CPU -.-&gt; NIC</code></pre><p><strong>操作系统的定义是什么</strong>？对于硬件。操作系统封装所有的硬件资源；对于用户和软件。操作系统将封装好的硬件资源分配给软件调用，同时为用户提供可视化界面。</p><p><strong>操作系统是怎么管理资源的</strong>？主要有三个技术。资源复用、资源虚拟化和资源抽象。后两个技术都是从第一个资源复用衍生而来，个人认为很多东西的解释都被复杂化了。</p><ul><li>比如，资源复用。其实根本没有复用，这只是相对于一开始的时候，一台计算机只能一次运行一个程序而言的。现在一台计算机需要同时运行很多程序，怎么办呢？既然程序运行需要计算资源，那我直接把计算机中的资源划分成很多份不就好了。当然了，除了内存进行空间上的划分，时间上也可以划分，比如 CPU，于是就有了资源复用中「空分复用」和「时分复用」两个理念。</li><li>资源虚拟化听着玄乎，其实就是名词学术化的资源复用。比如，我将物理世界中的硬件，通过操作系统的划分与调度，将其转化为逻辑世界中的多个资源，也就是所谓的将「单一物理资源」虚拟化为「多个逻辑资源」的理念。</li><li>资源抽象也很好理解，就是一种底层封装，顶层调用的思想。比如，对于每一个进程而言，都获得了操作系统为 ta 单独分配的 CPU 状态集和内存空间，而无需关心这些硬件资源是怎么分配给 ta 的。比如，对于每一个文件而言，都是对物理存储空间的抽象。</li></ul><p><strong>操作系统有什么特点</strong>？并发性、共享性和异步性，都很好理解。并发就是多个进程任务同时进行，但其实是操作系统的一个小把戏，让 CPU 在极短的时间间隔内频繁地切换执行不同的进程；共享是并发的衍生，我们知道程序执行是需要消耗资源的，而资源有限，因此我们可以让所有的进程共享同一块资源；而异步就是指一个任务在等待其他任务执行时，不会阻塞整个系统，等到其他任务执行结束或者切换到当前任务时，可以继续 ta 的执行。<strong>所以这么折腾到底是为了啥</strong>？可以从两个角度出发：</p><ol><li>对于一个任务而言：这样做可以 <strong>提升资源利用率</strong>。因为一个进程在执行时，任务会有很多，不仅仅需要 CPU 的状态计算，可能还需要和别的设备进行数据交互，我们知道 CPU 的计算效率远高于诸如 I/O 之类的设备，这就会让系统停下来等待进程执行其他的任务。我们就可以利用系统停下来等待的这段时间，把这个进程的其他任务给执行了。同时，如果一个任务可以被划分为多个子任务，也可以并发以提升资源利用率。</li><li>对于多个任务而言：可以 <strong>提升系统响应效率</strong>。多个用户同时提交请求？是一个一个响应，最快的极快，最慢的很慢。还是牺牲一点速度，大家都慢一点但是都没那么慢，同时响应？显然是后者。</li></ol><p><strong>操作系统是怎么对上层提供服务的</strong>？一句话概括就是，操作系统将封装好的应用程序接口 (Application Program Interface, 简称 API) 给用户或应用程序调用。例如下面的两个流程图：</p><pre><code class=" mermaid">---title: 用户调用系统资源---graph RL    subgraph User[调用者]    direction LR    用户逻辑    界面    解释器    用户逻辑 --&gt; 界面    界面 --&gt; 解释器    end    subgraph OS[OS]    内核    end    subgraph Source[硬件资源]    内存    end    User --&gt;|API| OS    OS --&gt;|调用| Source    Source --&gt;|反馈| OS    OS --&gt;|反馈| User</code></pre><pre><code class=" mermaid">---title: 应用程度调用系统资源---graph RL    subgraph User[调用者]    应用程度逻辑    end    subgraph OS[OS]    内核    end    subgraph Source[硬件资源]    内存    end    User --&gt;|API| OS    OS --&gt;|调用| Source    Source --&gt;|反馈| OS    OS --&gt;|反馈| User</code></pre><p><strong>为什么让操作系统对下层抽象进而为上层提供服务</strong>？因为下层的硬件十分繁杂，并且不同厂商生产的硬件可能会有使用上的不同，通过操作系统对下层硬件的封装，不仅可以隐藏复杂的硬件资源调度，可以统一出一种普适性更高的规则供上层服务使用。</p><p><strong>操作系统的内核是什么？操作系统的其他部分是什么？为什么要这样区分？</strong></p><ul><li>首先我们有必要了解「用户态」和「内核态」的概念。这两种状态定义了应用程序的指令执行权限，当程序运行在用户态时，不可以执行一些关键的指令，例如中断、网络通信等；但当程序运行在核心态时，可以执行所有指令。不难发现隔离化的运行模式相比于所有应用程序都有最高指令执行权限来说，可以进一步提升程序运行时系统的安全性。于是乎，操作系统的内核概念应运而生。</li><li>知道了指令执行权限隔离的程序运行理念后，再来理解操作系统的内核就很显然了。我们知道操作系统本质上就是一系列的系统软件，我们将操作系统中直接与硬件交互的软件划分到一起并称为「内核」，代表性内核软件可以实现的功能如「中断处理，进程管理」。操作系统剩余的部分就被划分为「非内核」部分，用于和应用程序直接交互，代表性非内核软件可以实现的功能形如「提供系统调用接口，基本计算」。当程序运行在用户态时，仅可以和操作系统的非内核部分交互；当切换到核心态时，就可以操作系统包括内核的所有部分交互。</li></ul><pre><code class=" mermaid">block-beta  columns 2    %% 上层  user["用户"]:1 app["应用程序"]:1    %% OS  nonkernel&lt;["封装调用，基本计算\n「OS 非内核」"]&gt;(up,down):2  kernel&lt;["中断处理，进程管理\n「OS 内核」"]&gt;(up,down):2    %% 底层  source["硬件"]:2</code></pre><h2 id="并发">并发</h2><h3 id="进程控制与管理">进程控制与管理</h3><p>CPU 的核心关键是并发程序，为此每一个硬件设计与相应 OS 的运行逻辑，都和 <strong>进程管理、资源调度、并发</strong> 息息相关。</p><h4 id="进程的定义">进程的定义</h4><p><strong>进程和程序的区别是什么？动态性</strong>。程序是一个指令序列，当程序载入进程时就转换为了进程。</p><h4 id="进程的状态">进程的状态</h4><p>五态模型：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409241450175.png" alt="五态模型 - 状态转换"></p><p>七态模型：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409242052428.png" alt="七态模型 - 状态转换"></p><h4 id="进程的描述">进程的描述</h4><p><strong>如何准确描述一个进程？通过进程映象</strong>。我们引入进程映象的概念。所谓进程映象就是某一时刻的进程状态，可以描述为以下 4 个部分：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409270845459.png" alt="进程映象"></p><h4 id="进程的组织">进程的组织</h4><p><strong>操作系统如何跟踪每一个进程？通过 PCB</strong>。每一个进程都会有很多信息，OS 设计者将其统一存储在进程控制块 (Process Control Block, 简称 PCB) 中。每一个进程都有自己的 PCB，只要是跟进程管理相关的数据信息一定都存储于 PCB 中。PCB 中存储的内容主要有以下内容：</p><ul><li><p>标识信息</p><ul><li>进程标识 PID</li><li>进程组标识 ID</li></ul></li><li><p>现场信息</p><ul><li>程序计数器 Program Counter</li><li>CPU 寄存器：包括栈指针、通用寄存器等</li></ul></li><li><p>控制信息</p><ul><li>进程状态：三态、五态等等</li><li>CPU 调度信息：如优先级、调度队列指针、调度参数等</li><li>内存管理信息：依据内存管理的方式不同而不同，如基/限长寄存器、段/页表指针等</li><li>计费信息：如进程使用的 CPU 时间</li><li>I/O 状态信息：包括分配给该进程的 I/O 设备列表、打开文件列表等</li></ul></li></ul><p><strong>当有多个进程处于同一个状态时，操作系统如何访存不同的进程呢？通过等待队列存储 PCB 以及优先级调度算法实现</strong>。我们将状态相同的进程存放到同一个容器（如链表队列）中，操作系统拥有每一个容器的指针。容器中相同状态的不同进程可能会按照优先级的顺序链接。那么容器中存的是什么呢？从上文可以知道，每一个 PCB 都可以唯一确定一个进程，因此等待队列中只需要存储 PCB 即可。</p><h4 id="进程的切换">进程的切换</h4><p><strong>并发执行程序时，如何切换状态？如何继续运行原进程呢？通过中断/系统调用切换状态，通过 PCB 保存现场数据</strong>。PCB 中的寄存器可以在切换进程时保存必要的数据，确保进程在中断获得运行权后能够从刚才的状态继续运行。这个「触发中断、保存现场、处理中断、恢复现场、结束中断」的过程被称为进程上下文切换。如下图所示：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409270937298.png" alt="进程上下文切换"></p><h4 id="进程的最小执行单元">进程的最小执行单元</h4><p><strong>如果在执行中断任务时又中断了呢？这是不会发生的</strong>。很多操作是不允许被打断的我们就需要最小操作单元，这被称作「原语」操作。例如更新进程信息（修改 PCB 中的信息）、更新进程状态（修改 PCB 所在的状态队列）、分配/回收资源。</p><h4 id="线程">线程</h4><p><strong>线程是什么</strong>？线程是从进程引申出来的概念与技术。进程是资源分配和调度的最小单位，而线程是 CPU 调度的最小单位。一个进程下可以有多个线程，每一个线程共享当前进程的所有资源，线程只会有执行指令最基本的资源（PC、寄存器等）。</p><p><strong>为什么会有线程</strong>？当一个进程的任务很复杂以至于也需要并发时，线程就可以充当更细力度的并发单元。也就是说线程的出现是为了解决越来越复杂的进程多任务并发问题的。</p><p><strong>多线程的几种模式是什么？分别有用户级线程、内核级线程和混合级线程</strong>。其中用户级线程 (User Level Thread) 的多线程逻辑顾名思义，就是由程序开发者定义，便于解决「逻辑并行」的问题。内核级线程 (Kernel Level Thread) 的多线程则由系统调用决定，便于解决「物理并行」的问题。</p><h3 id="互斥">互斥</h3><h3 id="同步">同步</h3><h3 id="死锁">死锁</h3><h2 id="虚拟化">虚拟化</h2><h2 id="内核">内核</h2><h3 id="中断-系统调用-上下文切换">中断 | 系统调用 | 上下文切换</h3><h3 id="处理器调度">处理器调度</h3><p><strong>处理器调度的三级模式</strong>。分为高级调度、中级调度和低级调度。其中：</p><ul><li>高级调度是决定哪些进程从外存对换到内存中，也就是分配内存资源。</li><li>中级调度是决定哪些进程从内存对换到外存中。</li><li>低级调度是决定哪些内存中的进程可以获得合适的 CPU 计算资源。</li></ul><p><strong>处理器调度的评价指标</strong>。</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202410081443847.png" alt="处理器调度的评价指标"></p><p><strong>处理器调度的七种算法</strong>。</p><h2 id="持久化">持久化</h2>]]></content>
     
     
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@@ -514,7 +514,7 @@
     <link href="/GPA/5th-term/NeuralNetworkAndDeepLearning/"/>
     <url>/GPA/5th-term/NeuralNetworkAndDeepLearning/</url>
     
-    <content type="html"><![CDATA[<h2 id="神经网络与深度学习">神经网络与深度学习</h2><h2 id="前言">前言</h2><p>学科地位：</p><table><thead><tr><th style="text-align:center">主讲教师</th><th style="text-align:center">学分配额</th><th style="text-align:center">学科类别</th></tr></thead><tbody><tr><td style="text-align:center">宋歌</td><td style="text-align:center">3</td><td style="text-align:center">自发课</td></tr></tbody></table><p>成绩组成：</p><table><thead><tr><th style="text-align:center">小组作业</th><th style="text-align:center">个人大作业</th></tr></thead><tbody><tr><td style="text-align:center">40%</td><td style="text-align:center">60%</td></tr></tbody></table><p>教材情况：</p><table><thead><tr><th style="text-align:center">课程名称</th><th style="text-align:center">选用教材</th><th style="text-align:center">版次</th><th style="text-align:center">作者</th><th style="text-align:center">出版社</th><th style="text-align:center">ISBN 号</th></tr></thead><tbody><tr><td style="text-align:center">神经网络与深度学习</td><td style="text-align:center">《神经网络与深度学习》</td><td style="text-align:center">1</td><td style="text-align:center">邱锡鹏</td><td style="text-align:center">机械工业出版社</td><td style="text-align:center">978-7-111-64968-7</td></tr></tbody></table><p>学习资源：</p><ul><li>课程官网：<a href="https://nndl.github.io/">https://nndl.github.io/</a></li></ul><p>为什么要学这门课？</p><blockquote><p>还记得 NJU 的 jyy 老师在上 OS 时说过的一句话，“我们现在学习的微积分是 300 年前的人类智慧结晶，何不再学学 50 年前的人类智慧结晶呢？”印象深刻。当一切都可以用层层嵌套的简单函数模拟时，人类社会必将发生翻天覆地的变化！</p></blockquote><p>会收获什么？</p><blockquote><p>如何学习特征？如何优化调参？为什么要这样做？背后的原理是什么？</p></blockquote><h2 id="概述">概述</h2><h3 id="绪论">绪论</h3><p><strong>表示学习是什么</strong>？与传统的特征工程目的一致，为了得到数据中的更好的特征。不同的是，特征工程中的策略都是可控的方式，而表示学习就是利用深度学习从数据中学习高层的有效特征。</p><p><strong>深度学习是什么</strong>？我们知道机器学习就是在手动处理完特征后，构建对应的模型 <strong>预测输出</strong>。而深度学习就是将机器学习的手动特征工程也用模型进行 <strong>表示学习</strong> 来学习出有效特征，然后继续构建模型 <strong>预测输出</strong>。如下图所示：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409121508150.png" alt="深度学习的数据处理流程"></p><p><strong>为什么会有深度学习</strong>？最简单的一点就是，很多特征我们根本没法定义一种表示规则来表示特征，比如说对于图像，怎么定义复杂的图像的特征呢？比如说对于音频，又怎么定义复杂的音频的特征呢？没办法，我们直接学特征！</p><p><strong>神经网络是什么</strong>？就是万千模型中的一种，仅此而已。</p><p><strong>为什么用神经网络进行深度学习</strong>？有了上面对深度学习定义的理解，可以发现其中最具有挑战性的特点就是，模型怎么知道什么才是好特征？什么是不好的特征？神经网络可以很好的解决这个问题。通过由浅到深层层神经元的特征提取，越深的神经元就可以学习更高语义的特征。说的高大上一点就是，神经网络可以很好的解决深度学习中的「贡献度分配」问题。</p><h3 id="机器学习概述">机器学习概述</h3><p>数据 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> 模型 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> 学习准则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> 优化算法</p><h3 id="线性模型">线性模型</h3><p>学习任务：回归、分类。</p><p>学习准则：</p><ul><li>经验风险最小化：通过 <strong>最小化训练集上的损失函数</strong> 来学习模型参数。</li><li>结构风险最小化：引入 <strong>正则化</strong> 来控制模型的复杂度，避免过拟合。</li><li>最大似然估计：通过 <strong>最大化模型在给定数据上的似然函数</strong> 来估计模型参数。</li><li>最大后验估计：结合了似然函数和 <strong>先验</strong> 分布，在贝叶斯框架下估计模型参数。</li></ul><p>优化算法：最小二乘法、梯度下降法。</p><h2 id="有监督学习">有监督学习</h2><h3 id="基础神经网络模型">基础神经网络模型</h3><h4 id="前馈神经网络">前馈神经网络</h4><h4 id="卷积神经网络">卷积神经网络</h4><h4 id="循环神经网络">循环神经网络</h4><h3 id="记忆与注意力机制">记忆与注意力机制</h3><h3 id="网络优化与正则化">网络优化与正则化</h3><h2 id="无监督学习">无监督学习</h2><h3 id="概率图模型">概率图模型</h3><h3 id="深度生成模型">深度生成模型</h3><h2 id="强化学习">强化学习</h2><h3 id="深度强化学习">深度强化学习</h3><h2 id="个人大作业">个人大作业</h2><ol><li>背景介绍：做好国内外研究现状的调研。</li><li>项目理解：深入对模型与代码的理解。</li><li>充分实验：对比实验、验证实验（消融实验）、参数实验（验证模型对不同的参数的敏感性）。</li><li>项目总结。</li></ol>]]></content>
+    <content type="html"><![CDATA[<h2 id="神经网络与深度学习">神经网络与深度学习</h2><h2 id="前言">前言</h2><p>学科地位：</p><table><thead><tr><th style="text-align:center">主讲教师</th><th style="text-align:center">学分配额</th><th style="text-align:center">学科类别</th></tr></thead><tbody><tr><td style="text-align:center">宋歌</td><td style="text-align:center">3</td><td style="text-align:center">自发课</td></tr></tbody></table><p>成绩组成：</p><table><thead><tr><th style="text-align:center">小组作业</th><th style="text-align:center">个人大作业</th></tr></thead><tbody><tr><td style="text-align:center">40%</td><td style="text-align:center">60%</td></tr></tbody></table><p>教材情况：</p><table><thead><tr><th style="text-align:center">课程名称</th><th style="text-align:center">选用教材</th><th style="text-align:center">版次</th><th style="text-align:center">作者</th><th style="text-align:center">出版社</th><th style="text-align:center">ISBN 号</th></tr></thead><tbody><tr><td style="text-align:center">神经网络与深度学习</td><td style="text-align:center">《神经网络与深度学习》</td><td style="text-align:center">1</td><td style="text-align:center">邱锡鹏</td><td style="text-align:center">机械工业出版社</td><td style="text-align:center">978-7-111-64968-7</td></tr></tbody></table><p>学习资源：</p><ul><li>课程官网：<a href="https://nndl.github.io/">https://nndl.github.io/</a></li></ul><p>为什么要学这门课？</p><blockquote><p>还记得 NJU 的 jyy 老师在上 OS 时说过的一句话，“我们现在学习的微积分是 300 年前的人类智慧结晶，何不再学学 50 年前的人类智慧结晶呢？”印象深刻。当一切都可以用层层嵌套的简单函数模拟时，人类社会必将发生翻天覆地的变化！</p></blockquote><p>会收获什么？</p><blockquote><p>如何学习特征？如何优化调参？为什么要这样做？背后的原理是什么？</p></blockquote><h2 id="概述">概述</h2><h3 id="绪论">绪论</h3><p><strong>表示学习是什么</strong>？与传统的特征工程目的一致，为了得到数据中的更好的特征。不同的是，特征工程中的策略都是可控的方式，而表示学习就是利用深度学习从数据中学习高层的有效特征。</p><p><strong>深度学习是什么</strong>？我们知道机器学习就是在手动处理完特征后，构建对应的模型 <strong>预测输出</strong>。而深度学习就是将机器学习的手动特征工程也用模型进行 <strong>表示学习</strong> 来学习出有效特征，然后继续构建模型 <strong>预测输出</strong>。如下图所示：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202409121508150.png" alt="深度学习的数据处理流程"></p><p><strong>为什么会有深度学习</strong>？最简单的一点就是，很多特征我们根本没法定义一种表示规则来表示特征，比如说对于图像，怎么定义复杂的图像的特征呢？比如说对于音频，又怎么定义复杂的音频的特征呢？没办法，我们直接学特征！</p><p><strong>神经网络是什么</strong>？就是万千模型中的一种，仅此而已。</p><p><strong>为什么用神经网络进行深度学习</strong>？有了上面对深度学习定义的理解，可以发现其中最具有挑战性的特点就是，模型怎么知道什么才是好特征？什么是不好的特征？神经网络可以很好的解决这个问题。通过由浅到深层层神经元的特征提取，越深的神经元就可以学习更高语义的特征。说的高大上一点就是，神经网络可以很好的解决深度学习中的「贡献度分配」问题。</p><h3 id="机器学习概述">机器学习概述</h3><p>数据 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> 模型 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> 学习准则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> 优化算法</p><h3 id="线性模型">线性模型</h3><p>学习任务：分类、回归。</p><p>学习准则：</p><ul><li>经验风险最小化：通过 <strong>最小化训练集上的损失函数</strong> 来学习模型参数。例如：线性回归中的最小二乘损失。</li><li>结构风险最小化：引入 <strong>正则化</strong> 来控制模型的复杂度，避免过拟合。</li><li>最大似然估计：通过 <strong>最大化模型在给定数据上的似然函数</strong> 来估计模型参数。例如二分类 logistic 和多分类 softmax 中的交叉熵损失。</li><li>最大后验估计：结合了似然函数和 <strong>先验</strong> 分布，在贝叶斯框架下估计模型参数。</li></ul><p>优化算法：最小二乘法、梯度下降法、拟牛顿法。</p><h2 id="有监督学习">有监督学习</h2><h3 id="基础神经网络模型">基础神经网络模型</h3><h4 id="前馈神经网络">前馈神经网络</h4><h4 id="卷积神经网络">卷积神经网络</h4><h4 id="循环神经网络">循环神经网络</h4><h3 id="记忆与注意力机制">记忆与注意力机制</h3><h3 id="网络优化与正则化">网络优化与正则化</h3><h2 id="无监督学习">无监督学习</h2><h3 id="概率图模型">概率图模型</h3><h3 id="深度生成模型">深度生成模型</h3><h2 id="强化学习">强化学习</h2><h3 id="深度强化学习">深度强化学习</h3><h2 id="个人大作业">个人大作业</h2><ol><li>背景介绍：做好国内外研究现状的调研。</li><li>项目理解：深入对模型与代码的理解。</li><li>充分实验：对比实验、验证实验（消融实验）、参数实验（验证模型对不同的参数的敏感性）。</li><li>项目总结。</li></ol>]]></content>
     
     
     <categories>
@@ -888,16 +888,16 @@
   
   
   <entry>
-    <title>road-map-guide</title>
-    <link href="/RoadMap/road-map-guide/"/>
-    <url>/RoadMap/road-map-guide/</url>
+    <title>operation-guide</title>
+    <link href="/Operation/operation-guide/"/>
+    <url>/Operation/operation-guide/</url>
     
-    <content type="html"><![CDATA[<h2 id="学习路线指南">学习路线指南</h2><p>参考：<a href="https://www.codefather.cn/%E5%AD%A6%E4%B9%A0%E8%B7%AF%E7%BA%BF/">https://www.codefather.cn/学习路线/</a></p><p>注：打星 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo>∗</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(*)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">∗</span><span class="mclose">)</span></span></span></span> 内容为上述链接未涉及到的路线，侵权必删</p><h2 id="计算机基础">计算机基础</h2><h3 id="计算机网络">计算机网络</h3><p>参考：<a href="https://www.code-nav.cn/post/1640588119619551233">https://www.code-nav.cn/post/1640588119619551233</a></p><h3 id="操作系统">操作系统</h3><p>参考：<a href="https://www.code-nav.cn/post/1640587909942099969">https://www.code-nav.cn/post/1640587909942099969</a></p><h2 id="软件开发通用">软件开发通用</h2><h3 id="企业项目研发流程">企业项目研发流程 *</h3><p>大纲：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202402140035678.png" alt="大纲"></p><p>内容：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202402140035679.png" alt="内容1"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202402140035680.png" alt="内容2"></p><h3 id="Git-GitHub">Git &amp; GitHub</h3><p>个人博客：<a href="http://blog.dwj601.cn/DevTools/Git/git-learning-record/">http://blog.dwj601.cn/DevTools/Git/git-learning-record/</a></p><h3 id="Linux">Linux</h3><p>参考：<a href="https://www.codefather.cn/linux%E5%AD%A6%E4%B9%A0%E8%B7%AF%E7%BA%BF-by-%E7%A8%8B%E5%BA%8F%E5%91%98%E9%B1%BC%E7%9A%AE/">https://www.codefather.cn/linux学习路线-by-程序员鱼皮/</a></p><h3 id="设计模式">设计模式</h3><p>参考：<a href="https://www.codefather.cn/%E8%AE%BE%E8%AE%A1%E6%A8%A1%E5%BC%8F%E5%AD%A6%E4%B9%A0%E8%B7%AF%E7%BA%BF-by-%E7%A8%8B%E5%BA%8F%E5%91%98%E9%B1%BC%E7%9A%AE/">https://www.codefather.cn/设计模式学习路线-by-程序员鱼皮/</a></p><h3 id="软件工程">软件工程 *</h3><p>大纲：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202402140035690.png" alt="大纲"></p><p>内容：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202402140035691.png" alt="内容"></p><h2 id="后端开发通用">后端开发通用</h2><h3 id="SQL">SQL</h3><p>参考：<a href="https://www.codefather.cn/sql%E5%85%8D%E8%B4%B9%E5%AE%9E%E6%88%98%E8%87%AA%E5%AD%A6%E7%BD%91%E7%AB%99-by-%E7%A8%8B%E5%BA%8F%E5%91%98%E9%B1%BC%E7%9A%AE/">https://www.codefather.cn/sql免费实战自学网站-by-程序员鱼皮/</a></p><h3 id="MySQL">MySQL *</h3><p>大纲：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202402140035694.png" alt="大纲"></p><p>内容：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202402140035695.png" alt="内容"></p><h3 id="Redis">Redis *</h3><p>大纲：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202402140035696.png" alt="大纲"></p><p>内容：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202402140035697.png" alt="内容"></p><h2 id="小程序">小程序 *</h2><p>大纲：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202402132345834.png" alt="大纲"></p><p>内容：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202402132347941.png" alt="内容"></p><h2 id="Python">Python</h2><p>参考：<a href="https://www.codefather.cn/python%E5%AD%A6%E4%B9%A0%E8%B7%AF%E7%BA%BF-by-%E7%A8%8B%E5%BA%8F%E5%91%98%E9%B1%BC%E7%9A%AE/">https://www.codefather.cn/python学习路线-by-程序员鱼皮/</a></p><h2 id="C">C++</h2><p>参考：<a href="https://www.codefather.cn/c-%E5%AD%A6%E4%B9%A0%E8%B7%AF%E7%BA%BF-by-%E7%A8%8B%E5%BA%8F%E5%91%98%E9%B1%BC%E7%9A%AE/">https://www.codefather.cn/c-学习路线-by-程序员鱼皮/</a></p>]]></content>
+    <content type="html"><![CDATA[<h2 id="运维指南">运维指南</h2><h2 id="前言">前言</h2><p>本文旨在介绍运维的基本理念和路径规划。正在不断完善中。</p><p>云计算运维的相关知识包括但不限于：</p><ol><li>Linux 基础：常用命令、文件及用户管理、文本处理、Vim 工具使用。</li><li>网络基础：网络基础知识、TCP/IP 协议、七层网络模型、Linux 网络管理与配置。</li><li>服务器运维：SSH 远程连接、文件上传下载、Nginx 和 MySQL 服务器搭建、LVS 负载均衡配置以及服务器优化经验。</li><li>自动化运维：Shell 脚本编写、自动化运维工具 Ansible 的使用。</li></ol><h2 id="Linux-基础">Linux 基础</h2><h2 id="网络基础">网络基础</h2><h2 id="服务器运维">服务器运维</h2>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-71de71b4" role="button" aria-expanded="false" aria-controls="collapse-71de71b4">        <div class="fold-arrow">▶</div>基本概念      </div>      <div class="fold-collapse collapse" id="collapse-71de71b4">        <div class="fold-content">          <p>服务器 server 就是类似于一台台电脑，量产后在全球各地通过网络为用户提供高质量的便捷服务。在学习服务器相关知识之前，有必要先了解计算机网络相关的概念，有助于后续理解代理系统等的运作逻辑与开发逻辑。在正式开始之前，首先罗列一下最基本的概念：</p><ul><li><p>云服务器：</p><ul><li>实例：一个云服务器相当于一个抽象的类，在其中购买配置了指定的实例后相当于实例化一个类，从而一个云服务器对应一个实例</li><li>ip 地址：一台云服务器对应唯一的一个 ip 地址</li><li>备案：所谓的网站备案其实是对云服务器进行备案。当前的形式是，对于指向中国大陆 ip 的云服务器需要备案，如果指向的是非中国大陆的 ip，就不需要备案了。一般而言，中文的指向 HK，英文的指向 UK</li><li>端口：一台电脑能做的肯定不止一种，ip 地址端口的概念就好比一台电脑可以有不同的功能模块，不同的端口对应服务器不同的数据通道，开发者可以通过不同的端口开发不同的功能，用户可以通过不同的端口使用一台服务器上的不同服务</li><li><a href="https://www.bt.cn/new/download.html">宝塔面板</a>：由于 linux 操作系统的命令行操作方式与 windows 的图形化界面操作方式差别较大，对于习惯图形化操作的用户不友好，故有一系列 <strong>图形化管理</strong> linux 操作系统的工具，宝塔面板就是其中的一个代表</li></ul></li><li><p>域名：</p><ul><li>概念：代替 ip 地址访问的一种更加容易记忆与推广的媒介，通过 DNS 服务器将域名与 ip 地址进行绑定后，用户通过域名 domain 即可访问相应的服务器的资源</li><li>顶级域名：即 <strong>域名+后缀</strong> 的组合</li><li>二级域名：即 <strong>主机名+域名+后缀</strong> 的组合</li></ul></li><li><p>协议：</p><ul><li>概念：服务器与用户进行数据传输的一种约定规则</li><li>http：传统的数据传输协议，默认服务器 80 端口进行访问</li><li>https：即 http + ssl，传统协议 + ssl 加密证书，默认服务器 443 端口访问</li><li>ssl：即 Secret Sockets Layer 安全套件层，用于加密服务器与用户之间传输的数据。其中证书文件为 pem 文件，密钥文件为 key 文件</li></ul></li><li><p>网站框架：</p><ul><li>目前两款主流的网站框架分别为 LNMP 与 LAMP，都是用于搭建 Web 服务器环境的 <strong>软件堆栈</strong></li><li>LNMP：表示使用 Linux 操作系统、Nginx 作为 Web 服务器、MySQL 作为数据库、PHP 作为服务器端脚本语言的技术堆栈</li><li>LAMP：表示使用 Linux 操作系统、Apache 作为 Web 服务器、MySQL 作为数据库、PHP 作为服务器端脚本语言的技术堆栈</li></ul></li></ul>        </div>      </div>    </div><h2 id="自动化运维">自动化运维</h2><h2 id="参考">参考</h2><p><a href="https://developer.aliyun.com/learning/roadmap/linux">Linux 运维学习路线 - 阿里云</a></p><p><a href="https://www.runoob.com/linux/linux-tutorial.html">菜鸟 Linux 教程</a></p><p><a href="https://www.bilibili.com/video/BV16A4y1X7vg/">搭建一个自己的网站？看这个就够了！</a></p>]]></content>
     
     
     <categories>
       
-      <category>RoadMap</category>
+      <category>Operation</category>
       
     </categories>
     
@@ -907,16 +907,16 @@
   
   
   <entry>
-    <title>operation-guide</title>
-    <link href="/Operation/operation-guide/"/>
-    <url>/Operation/operation-guide/</url>
+    <title>road-map-guide</title>
+    <link href="/RoadMap/road-map-guide/"/>
+    <url>/RoadMap/road-map-guide/</url>
     
-    <content type="html"><![CDATA[<h2 id="运维指南">运维指南</h2><h2 id="前言">前言</h2><p>本文旨在介绍运维的基本理念和路径规划。正在不断完善中。</p><p>云计算运维的相关知识包括但不限于：</p><ol><li>Linux 基础：常用命令、文件及用户管理、文本处理、Vim 工具使用。</li><li>网络基础：网络基础知识、TCP/IP 协议、七层网络模型、Linux 网络管理与配置。</li><li>服务器运维：SSH 远程连接、文件上传下载、Nginx 和 MySQL 服务器搭建、LVS 负载均衡配置以及服务器优化经验。</li><li>自动化运维：Shell 脚本编写、自动化运维工具 Ansible 的使用。</li></ol><h2 id="Linux-基础">Linux 基础</h2><h2 id="网络基础">网络基础</h2><h2 id="服务器运维">服务器运维</h2>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-71de71b4" role="button" aria-expanded="false" aria-controls="collapse-71de71b4">        <div class="fold-arrow">▶</div>基本概念      </div>      <div class="fold-collapse collapse" id="collapse-71de71b4">        <div class="fold-content">          <p>服务器 server 就是类似于一台台电脑，量产后在全球各地通过网络为用户提供高质量的便捷服务。在学习服务器相关知识之前，有必要先了解计算机网络相关的概念，有助于后续理解代理系统等的运作逻辑与开发逻辑。在正式开始之前，首先罗列一下最基本的概念：</p><ul><li><p>云服务器：</p><ul><li>实例：一个云服务器相当于一个抽象的类，在其中购买配置了指定的实例后相当于实例化一个类，从而一个云服务器对应一个实例</li><li>ip 地址：一台云服务器对应唯一的一个 ip 地址</li><li>备案：所谓的网站备案其实是对云服务器进行备案。当前的形式是，对于指向中国大陆 ip 的云服务器需要备案，如果指向的是非中国大陆的 ip，就不需要备案了。一般而言，中文的指向 HK，英文的指向 UK</li><li>端口：一台电脑能做的肯定不止一种，ip 地址端口的概念就好比一台电脑可以有不同的功能模块，不同的端口对应服务器不同的数据通道，开发者可以通过不同的端口开发不同的功能，用户可以通过不同的端口使用一台服务器上的不同服务</li><li><a href="https://www.bt.cn/new/download.html">宝塔面板</a>：由于 linux 操作系统的命令行操作方式与 windows 的图形化界面操作方式差别较大，对于习惯图形化操作的用户不友好，故有一系列 <strong>图形化管理</strong> linux 操作系统的工具，宝塔面板就是其中的一个代表</li></ul></li><li><p>域名：</p><ul><li>概念：代替 ip 地址访问的一种更加容易记忆与推广的媒介，通过 DNS 服务器将域名与 ip 地址进行绑定后，用户通过域名 domain 即可访问相应的服务器的资源</li><li>顶级域名：即 <strong>域名+后缀</strong> 的组合</li><li>二级域名：即 <strong>主机名+域名+后缀</strong> 的组合</li></ul></li><li><p>协议：</p><ul><li>概念：服务器与用户进行数据传输的一种约定规则</li><li>http：传统的数据传输协议，默认服务器 80 端口进行访问</li><li>https：即 http + ssl，传统协议 + ssl 加密证书，默认服务器 443 端口访问</li><li>ssl：即 Secret Sockets Layer 安全套件层，用于加密服务器与用户之间传输的数据。其中证书文件为 pem 文件，密钥文件为 key 文件</li></ul></li><li><p>网站框架：</p><ul><li>目前两款主流的网站框架分别为 LNMP 与 LAMP，都是用于搭建 Web 服务器环境的 <strong>软件堆栈</strong></li><li>LNMP：表示使用 Linux 操作系统、Nginx 作为 Web 服务器、MySQL 作为数据库、PHP 作为服务器端脚本语言的技术堆栈</li><li>LAMP：表示使用 Linux 操作系统、Apache 作为 Web 服务器、MySQL 作为数据库、PHP 作为服务器端脚本语言的技术堆栈</li></ul></li></ul>        </div>      </div>    </div><h2 id="自动化运维">自动化运维</h2><h2 id="参考">参考</h2><p><a href="https://developer.aliyun.com/learning/roadmap/linux">Linux 运维学习路线 - 阿里云</a></p><p><a href="https://www.runoob.com/linux/linux-tutorial.html">菜鸟 Linux 教程</a></p><p><a href="https://www.bilibili.com/video/BV16A4y1X7vg/">搭建一个自己的网站？看这个就够了！</a></p>]]></content>
+    <content type="html"><![CDATA[<h2 id="学习路线指南">学习路线指南</h2><p>参考：<a href="https://www.codefather.cn/%E5%AD%A6%E4%B9%A0%E8%B7%AF%E7%BA%BF/">https://www.codefather.cn/学习路线/</a></p><p>注：打星 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo>∗</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(*)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">∗</span><span class="mclose">)</span></span></span></span> 内容为上述链接未涉及到的路线，侵权必删</p><h2 id="计算机基础">计算机基础</h2><h3 id="计算机网络">计算机网络</h3><p>参考：<a href="https://www.code-nav.cn/post/1640588119619551233">https://www.code-nav.cn/post/1640588119619551233</a></p><h3 id="操作系统">操作系统</h3><p>参考：<a href="https://www.code-nav.cn/post/1640587909942099969">https://www.code-nav.cn/post/1640587909942099969</a></p><h2 id="软件开发通用">软件开发通用</h2><h3 id="企业项目研发流程">企业项目研发流程 *</h3><p>大纲：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202402140035678.png" alt="大纲"></p><p>内容：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202402140035679.png" alt="内容1"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202402140035680.png" alt="内容2"></p><h3 id="Git-GitHub">Git &amp; GitHub</h3><p>个人博客：<a href="http://blog.dwj601.cn/DevTools/Git/git-learning-record/">http://blog.dwj601.cn/DevTools/Git/git-learning-record/</a></p><h3 id="Linux">Linux</h3><p>参考：<a href="https://www.codefather.cn/linux%E5%AD%A6%E4%B9%A0%E8%B7%AF%E7%BA%BF-by-%E7%A8%8B%E5%BA%8F%E5%91%98%E9%B1%BC%E7%9A%AE/">https://www.codefather.cn/linux学习路线-by-程序员鱼皮/</a></p><h3 id="设计模式">设计模式</h3><p>参考：<a href="https://www.codefather.cn/%E8%AE%BE%E8%AE%A1%E6%A8%A1%E5%BC%8F%E5%AD%A6%E4%B9%A0%E8%B7%AF%E7%BA%BF-by-%E7%A8%8B%E5%BA%8F%E5%91%98%E9%B1%BC%E7%9A%AE/">https://www.codefather.cn/设计模式学习路线-by-程序员鱼皮/</a></p><h3 id="软件工程">软件工程 *</h3><p>大纲：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202402140035690.png" alt="大纲"></p><p>内容：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202402140035691.png" alt="内容"></p><h2 id="后端开发通用">后端开发通用</h2><h3 id="SQL">SQL</h3><p>参考：<a href="https://www.codefather.cn/sql%E5%85%8D%E8%B4%B9%E5%AE%9E%E6%88%98%E8%87%AA%E5%AD%A6%E7%BD%91%E7%AB%99-by-%E7%A8%8B%E5%BA%8F%E5%91%98%E9%B1%BC%E7%9A%AE/">https://www.codefather.cn/sql免费实战自学网站-by-程序员鱼皮/</a></p><h3 id="MySQL">MySQL *</h3><p>大纲：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202402140035694.png" alt="大纲"></p><p>内容：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202402140035695.png" alt="内容"></p><h3 id="Redis">Redis *</h3><p>大纲：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202402140035696.png" alt="大纲"></p><p>内容：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202402140035697.png" alt="内容"></p><h2 id="小程序">小程序 *</h2><p>大纲：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202402132345834.png" alt="大纲"></p><p>内容：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202402132347941.png" alt="内容"></p><h2 id="Python">Python</h2><p>参考：<a href="https://www.codefather.cn/python%E5%AD%A6%E4%B9%A0%E8%B7%AF%E7%BA%BF-by-%E7%A8%8B%E5%BA%8F%E5%91%98%E9%B1%BC%E7%9A%AE/">https://www.codefather.cn/python学习路线-by-程序员鱼皮/</a></p><h2 id="C">C++</h2><p>参考：<a href="https://www.codefather.cn/c-%E5%AD%A6%E4%B9%A0%E8%B7%AF%E7%BA%BF-by-%E7%A8%8B%E5%BA%8F%E5%91%98%E9%B1%BC%E7%9A%AE/">https://www.codefather.cn/c-学习路线-by-程序员鱼皮/</a></p>]]></content>
     
     
     <categories>
       
-      <category>Operation</category>
+      <category>RoadMap</category>
       
     </categories>
     
@@ -947,11 +947,11 @@
   
   
   <entry>
-    <title>PyAlgo</title>
-    <link href="/GPA/4th-term/PyAlgo/"/>
-    <url>/GPA/4th-term/PyAlgo/</url>
+    <title>OptMethod</title>
+    <link href="/GPA/4th-term/OptMethod/"/>
+    <url>/GPA/4th-term/OptMethod/</url>
     
-    <content type="html"><![CDATA[<h2 id="算法设计与分析">算法设计与分析</h2><h2 id="前言">前言</h2><p>学科地位：</p><table><thead><tr><th style="text-align:center">主讲教师</th><th style="text-align:center">学分配额</th><th style="text-align:center">学科类别</th></tr></thead><tbody><tr><td style="text-align:center">段博佳</td><td style="text-align:center">3</td><td style="text-align:center">自发课</td></tr></tbody></table><p>成绩组成：</p><table><thead><tr><th style="text-align:center">平时（5次作业+签到）</th><th style="text-align:center">期末（书本+讲义）</th></tr></thead><tbody><tr><td style="text-align:center">60%</td><td style="text-align:center">40%</td></tr></tbody></table><p>教材情况：</p><table><thead><tr><th style="text-align:center">课程名称</th><th style="text-align:center">选用教材</th><th style="text-align:center">版次</th><th style="text-align:center">作者</th><th style="text-align:center">出版社</th><th style="text-align:center">ISBN号</th></tr></thead><tbody><tr><td style="text-align:center">算法设计与分析</td><td style="text-align:center">《算法设计与分析(python版)》</td><td style="text-align:center">1</td><td style="text-align:center">王秋芬</td><td style="text-align:center">清华大学出版社</td><td style="text-align:center">978-7-302-57072-1</td></tr></tbody></table><h2 id="一、贪心">一、贪心</h2><p>Dijkstra、Prim、Kruskal、Huffman</p><h2 id="二、分治">二、分治</h2><p>二分、归并、快排（第 k 个数）</p><h2 id="三、动规">三、动规</h2><ul><li>区间dp：矩阵连乘</li><li>网格图dp：LCS</li><li>背包：完全背包、多重背包</li></ul><h2 id="四、深搜">四、深搜</h2><ul><li>排列树：TSP、图染色、调度、n 皇后</li><li>子集树：01背包</li></ul><h2 id="考后碎碎念">考后碎碎念</h2><p>考的中规中矩，但是忘带修正带导致卷面一坨hh。按照试卷顺序罗列一下：表达式树、大根堆、Dijkstra、图染色、第 k 个数、Kruskal、完全背包。</p><p>试卷一看就是 wq 出的，但还是用 python 写的代码。有一说一这次考试还加深了我对快排的理解，尤其是在让我给出分治后的序列时hh。硬要按照上面四个标签分类的话大概就是：</p><ul><li>贪心：Dijkstra、Kruskal</li><li>分治：第 k 个数</li><li>动规：完全背包</li><li>深搜：表达式树、大根堆、图染色</li></ul>]]></content>
+    <content type="html"><![CDATA[<h2 id="最优化方法">最优化方法</h2><h2 id="前言">前言</h2><p>学科地位：</p><table><thead><tr><th style="text-align:center">主讲教师</th><th style="text-align:center">学分配额</th><th style="text-align:center">学科类别</th></tr></thead><tbody><tr><td style="text-align:center">王启春</td><td style="text-align:center">4</td><td style="text-align:center">专业课</td></tr></tbody></table><p>成绩组成：</p><table><thead><tr><th style="text-align:center">平时（作业+考勤）</th><th style="text-align:center">期中（大作业）</th><th style="text-align:center">期末（闭卷）</th></tr></thead><tbody><tr><td style="text-align:center">20%</td><td style="text-align:center">30%</td><td style="text-align:center">50%</td></tr></tbody></table><p>教材情况：</p><table><thead><tr><th style="text-align:center">课程名称</th><th style="text-align:center">选用教材</th><th style="text-align:center">版次</th><th style="text-align:center">作者</th><th style="text-align:center">出版社</th><th style="text-align:center">ISBN号</th></tr></thead><tbody><tr><td style="text-align:center">最优化算法</td><td style="text-align:center">《最优化方法》</td><td style="text-align:center">2</td><td style="text-align:center">孙文瑜</td><td style="text-align:center">高等教育出版社</td><td style="text-align:center">978-7-04-029763-8</td></tr></tbody></table><h2 id="第一章-基本概念">第一章 基本概念</h2><h3 id="1-1-最优化问题简介">1.1 最优化问题简介</h3><p>本目主要讲解最优化问题的一些分类，下附脑图（由 Xmind 软件制作）：</p>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-cb469c68" role="button" aria-expanded="false" aria-controls="collapse-cb469c68">        <div class="fold-arrow">▶</div>分类脑图      </div>      <div class="fold-collapse collapse" id="collapse-cb469c68">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403091214182.png" alt="分类脑图"></p>        </div>      </div>    </div><h3 id="1-2-凸集和凸函数">1.2 凸集和凸函数</h3><div class="note note-light">            <p>由于本书介绍的最优化求解方法一般只适用于求解<strong>局部最优解</strong>，那么如何确定<strong>全局最优解</strong>呢？以及如何确定<strong>唯一的全局最优解</strong>呢？本目揭晓答案：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">(</mo><mtext>唯一</mtext><mo stretchy="false">)</mo><mtext>全局最优解</mtext><mo>=</mo><mtext>局部最优解</mtext><mo>+</mo><mo stretchy="false">(</mo><mtext>严格</mtext><mo stretchy="false">)</mo><mtext>凸目标函数</mtext><mo>+</mo><mtext>凸可行域</mtext></mrow><annotation encoding="application/x-tex">(\text{唯一})\text{全局最优解}=\text{局部最优解}+(\text{严格})\text{凸目标函数}+\text{凸可行域}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord text"><span class="mord cjk_fallback">唯一</span></span><span class="mclose">)</span><span class="mord text"><span class="mord cjk_fallback">全局最优解</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord text"><span class="mord cjk_fallback">局部最优解</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord text"><span class="mord cjk_fallback">严格</span></span><span class="mclose">)</span><span class="mord text"><span class="mord cjk_fallback">凸目标函数</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord cjk_fallback">凸可行域</span></span></span></span></span></span></p><p>其中凸可行域包含于凸集，凸目标函数包含于凸函数，凸目标函数+凸可行域的问题称为凸规划问题。因此本目将分别介绍凸集、凸函数和凸规划三个概念。</p>          </div><h4 id="1-2-1-凸集">1.2.1 凸集</h4><p><strong>凸集的定义</strong>：若集合中任意两点的线性组合都包含于集合，则称该集合为凸集。符号化即：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="normal">∀</mi><mspace width="1em"></mspace><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>∈</mo><mi>D</mi><mtext>&nbsp;</mtext><mo>∧</mo><mtext>&nbsp;</mtext><mi>λ</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>s</mi><mi mathvariant="normal">.</mi><mi>t</mi><mi mathvariant="normal">.</mi><mspace width="1em"></mspace><mi>λ</mi><mi>x</mi><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>λ</mi><mo stretchy="false">)</mo><mi>y</mi><mo>∈</mo><mi>D</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\forall \quad x,y \in D \ \land \ \lambda \in [0,1] \\s.t. \quad \lambda x+(1-\lambda)y \in D \\\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∀</span><span class="mspace" style="margin-right:1em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mspace">&nbsp;</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∧</span><span class="mspace">&nbsp;</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">λ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">]</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="mord">.</span><span class="mord mathnormal">t</span><span class="mord">.</span><span class="mspace" style="margin-right:1em;"></span><span class="mord mathnormal">λ</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">λ</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p><strong>凸集的性质</strong>：设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>D</mi><mn>2</mn></msub><mo>⊂</mo><msup><mi>R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">D_1,D_2 \subset R^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> 均为凸集（下列 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo separator="true">,</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x,y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> 均表示向量），则：</p><ol><li>两凸集的交 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mn>1</mn></msub><mo>∩</mo><msub><mi>D</mi><mn>2</mn></msub><mo>=</mo><mo stretchy="false">{</mo><mi>x</mi><mtext>&nbsp;</mtext><mi mathvariant="normal">∣</mi><mtext>&nbsp;</mtext><mi>x</mi><mo>∈</mo><msub><mi>D</mi><mn>1</mn></msub><mo>∧</mo><mi>x</mi><mo>∈</mo><msub><mi>D</mi><mn>2</mn></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">D_1 \cap D_2 = \{x\ |\ x \in D_1 \land x \in D_2\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal">x</span><span class="mspace">&nbsp;</span><span class="mord">∣</span><span class="mspace">&nbsp;</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span> 是凸集</li><li>两凸集的和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mn>1</mn></msub><mo>+</mo><msub><mi>D</mi><mn>2</mn></msub><mo>=</mo><mo stretchy="false">{</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mtext>&nbsp;</mtext><mi mathvariant="normal">∣</mi><mtext>&nbsp;</mtext><mi>x</mi><mo>∈</mo><msub><mi>D</mi><mn>1</mn></msub><mo separator="true">,</mo><mi>y</mi><mo>∈</mo><msub><mi>D</mi><mn>2</mn></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">D_1 + D_2 = \{x,y\ |\ x \in D_1 , y \in D_2\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace">&nbsp;</span><span class="mord">∣</span><span class="mspace">&nbsp;</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span> 是凸集</li><li>两凸集的差 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mn>1</mn></msub><mo>−</mo><msub><mi>D</mi><mn>2</mn></msub><mo>=</mo><mo stretchy="false">{</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mtext>&nbsp;</mtext><mi mathvariant="normal">∣</mi><mtext>&nbsp;</mtext><mi>x</mi><mo>∈</mo><msub><mi>D</mi><mn>1</mn></msub><mo separator="true">,</mo><mi>y</mi><mo>∈</mo><msub><mi>D</mi><mn>2</mn></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">D_1 - D_2 = \{x,y\ |\ x \in D_1 , y \in D_2\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace">&nbsp;</span><span class="mord">∣</span><span class="mspace">&nbsp;</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span> 是凸集</li><li>对于任意非零实数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span>，集合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><msub><mi>D</mi><mn>1</mn></msub><mo>=</mo><mo stretchy="false">{</mo><mi>α</mi><mi>x</mi><mtext>&nbsp;</mtext><mi mathvariant="normal">∣</mi><mtext>&nbsp;</mtext><mi>x</mi><mo>∈</mo><msub><mi>D</mi><mn>1</mn></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\alpha D_1 = \{ \alpha x \ |\ x \in D_1 \}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal">αx</span><span class="mspace">&nbsp;</span><span class="mord">∣</span><span class="mspace">&nbsp;</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span>​ 是凸集</li></ol>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-8f34ca46" role="button" aria-expanded="false" aria-controls="collapse-8f34ca46">        <div class="fold-arrow">▶</div>证明      </div>      <div class="fold-collapse collapse" id="collapse-8f34ca46">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403101941905.png" alt="凸集的性质证明"></p>        </div>      </div>    </div><p><strong>凸集的应用</strong></p><ol><li><p>刻画可行域</p><ul><li><p>凸组合定义</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>设&nbsp;</mtext><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mo separator="true">,</mo><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msup><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></msup><mo>∈</mo><msup><mi>R</mi><mi>n</mi></msup><mo separator="true">,</mo><mtext>且&nbsp;</mtext><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>p</mi></munderover><msub><mi>α</mi><mi>i</mi></msub><mo>=</mo><mn>1</mn><mo stretchy="false">(</mo><msub><mi>a</mi><mi>i</mi></msub><mo>≥</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>s</mi><mi mathvariant="normal">.</mi><mi>t</mi><mi mathvariant="normal">.</mi><mspace width="1em"></mspace><mi>x</mi><mo>=</mo><msub><mi>α</mi><mn>1</mn></msub><msup><mi>x</mi><mn>1</mn></msup><mo>+</mo><msub><mi>α</mi><mn>2</mn></msub><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>α</mi><mi>p</mi></msub><msup><mi>x</mi><mi>p</mi></msup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>则称&nbsp;</mtext><mi>x</mi><mtext>&nbsp;为向量&nbsp;</mtext><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mo separator="true">,</mo><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msup><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></msup><mtext>&nbsp;的凸组合</mtext></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\text{设}\ x^{(1)},x^{(2)}, \cdots, x^{(p)} \in R^n ,\text{且}\ \sum_{i=1}^p \alpha_i = 1(a_i \ge 0)  \\s.t. \quad x = \alpha_1x^1 + \alpha_2x^2 + \cdots + \alpha_px^p  \\\text{则称}\ x\ \text{为向量}\ x^{(1)},x^{(2)}, \cdots, x^{(p)}\ \text{的凸组合}\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:6.3983em;vertical-align:-2.9491em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.4491em;"><span style="top:-5.4491em;"><span class="pstrut" style="height:3.6985em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">设</span></span><span class="mspace">&nbsp;</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">2</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight">p</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord text"><span class="mord cjk_fallback">且</span></span><span class="mspace">&nbsp;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6985em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3471em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span><span style="top:-3.0074em;"><span class="pstrut" style="height:3.6985em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="mord">.</span><span class="mord mathnormal">t</span><span class="mord">.</span><span class="mspace" style="margin-right:1em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span></span></span></span></span></span></span><span style="top:-1.4094em;"><span class="pstrut" style="height:3.6985em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">则称</span></span><span class="mspace">&nbsp;</span><span class="mord mathnormal">x</span><span class="mspace">&nbsp;</span><span class="mord text"><span class="mord cjk_fallback">为向量</span></span><span class="mspace">&nbsp;</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">2</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight">p</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace">&nbsp;</span><span class="mord text"><span class="mord cjk_fallback">的凸组合</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.9491em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li><li><p>凸组合定理：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo>∈</mo><msup><mi>R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">D \in R^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> 是凸集的充分必要条件是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 中任取 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 个点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mi>i</mi></msup><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x^i(1,2,\cdots m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0747em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span> 的凸组合任属于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span>，即：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><msub><mi>α</mi><mi>i</mi></msub><msub><mi>x</mi><mi>i</mi></msub><mo>∈</mo><mi>D</mi><mrow><mo fence="true">(</mo><msub><mi>α</mi><mi>i</mi></msub><mo>≥</mo><mn>0</mn><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mi>m</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><msub><mi>α</mi><mi>i</mi></msub><mo>=</mo><mn>1</mn><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\sum_{i=1}^m \alpha_ix_i \in D\left( \alpha_i \ge 0(i=1,2,\cdots,m),\sum_{i=1}^m \alpha_i = 1 \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.0277em;vertical-align:-1.2777em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">(</span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size4">)</span></span></span></span></span></span></span></p>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-e771bc53" role="button" aria-expanded="false" aria-controls="collapse-e771bc53">        <div class="fold-arrow">▶</div>凸组合定理证明      </div>      <div class="fold-collapse collapse" id="collapse-e771bc53">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403102025159.png" alt="凸组合定理证明"></p>        </div>      </div>    </div></li></ul></li><li><p>分析最优解的最优性条件</p><ul><li><p>超平面定义（凸集分离定理）：设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>D</mi><mn>2</mn></msub><mo>⊂</mo><msup><mi>R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">D_1,D_2 \subset R^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> 为两非空凸集，若存在非零向量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo>∈</mo><msup><mi>R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\alpha \in R^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> 和实数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span>，使得</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>D</mi><mn>1</mn></msub><mo>⊂</mo><msup><mi>H</mi><mo>+</mo></msup><mo>=</mo><mo stretchy="false">{</mo><mi>x</mi><mo>∈</mo><msup><mi>R</mi><mi>n</mi></msup><mtext>&nbsp;</mtext><mi mathvariant="normal">∣</mi><mtext>&nbsp;</mtext><msup><mi>α</mi><mi>T</mi></msup><mi>x</mi><mo>≥</mo><mi>β</mi><mo stretchy="false">}</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>D</mi><mn>2</mn></msub><mo>⊂</mo><msup><mi>H</mi><mo>−</mo></msup><mo>=</mo><mo stretchy="false">{</mo><mi>x</mi><mo>∈</mo><msup><mi>R</mi><mi>n</mi></msup><mtext>&nbsp;</mtext><mi mathvariant="normal">∣</mi><mtext>&nbsp;</mtext><msup><mi>α</mi><mi>T</mi></msup><mi>x</mi><mo>≤</mo><mi>β</mi><mo stretchy="false">}</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}D_1 \subset H^+ = \{ x \in R^n \ | \ \alpha^T x \ge \beta\} \\D_2 \subset H^- = \{ x \in R^n \ | \ \alpha^T x \le \beta\}\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.1027em;vertical-align:-1.3013em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8013em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8213em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">{</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace">&nbsp;</span><span class="mord">∣</span><span class="mspace">&nbsp;</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mclose">}</span></span></span><span style="top:-2.3587em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8213em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">−</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">{</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace">&nbsp;</span><span class="mord">∣</span><span class="mspace">&nbsp;</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mclose">}</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3013em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>则称超平面 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi><mo>=</mo><mo stretchy="false">{</mo><mi>x</mi><mo>∈</mo><msup><mi>R</mi><mi>n</mi></msup><mtext>&nbsp;</mtext><mi mathvariant="normal">∣</mi><mtext>&nbsp;</mtext><msup><mi>α</mi><mi>T</mi></msup><mi>x</mi><mo>=</mo><mi>β</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">H = \{ x \in R^n \ | \ \alpha^Tx=\beta \}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0913em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace">&nbsp;</span><span class="mord">∣</span><span class="mspace">&nbsp;</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mclose">}</span></span></span></span> <strong>分离</strong>集合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">D_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">D_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，<strong>严格分离</strong>时上述不等式无法取等</p></li><li><p>投影定理：设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo>∈</mo><msup><mi>R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">D \in R^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> 是非空闭凸集，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>∈</mo><msup><mi>R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">y \in R^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> 但 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo mathvariant="normal">∉</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">y \notin D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mord"><span class="mrel">∈</span></span><span class="mord vbox"><span class="thinbox"><span class="llap"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="inner"><span class="mord"><span class="mord">/</span><span class="mspace" style="margin-right:0.0556em;"></span></span></span><span class="fix"></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span>，则</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mtext>存在唯一的点&nbsp;</mtext><mover accent="true"><mi>x</mi><mo stretchy="true">‾</mo></mover><mo>∈</mo><mi>D</mi><mo separator="true">,</mo><mtext>使得集合</mtext><mi>D</mi><mtext>到点</mtext><mi>y</mi><mtext>的距离最小</mtext></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mover accent="true"><mi>x</mi><mo stretchy="true">‾</mo></mover><mo>∈</mo><mi>D</mi><mtext>是点</mtext><mi>y</mi><mtext>到集合D的最短距离点的充分必要条件为</mtext><mo>:</mo><mi mathvariant="normal">∀</mi><mi>x</mi><mo>∈</mo><mi>D</mi><mo separator="true">,</mo><mo>&lt;</mo><mi>x</mi><mo>−</mo><mover accent="true"><mi>x</mi><mo stretchy="true">‾</mo></mover><mo separator="true">,</mo><mi>y</mi><mo>−</mo><mover accent="true"><mi>x</mi><mo stretchy="true">‾</mo></mover><mo>&gt;</mo><mo>≤</mo><mn>0</mn></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align*}    (1)&amp; \text{存在唯一的点} \ \overline x \in D,\text{使得集合}D\text{到点} y \text{的距离最小} \\    (2)&amp; \overline x \in D \text{是点} y \text{到集合D的最短距离点的充分必要条件为}:\forall x \in D,&lt;x-\overline x,y - \overline x&gt; \le 0\end{align*}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">2</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord text"><span class="mord cjk_fallback">存在唯一的点</span></span><span class="mspace">&nbsp;</span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6306em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span></span></span><span style="top:-3.5506em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord text"><span class="mord cjk_fallback">使得集合</span></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord text"><span class="mord cjk_fallback">到点</span></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mord text"><span class="mord cjk_fallback">的距离最小</span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6306em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span></span></span><span style="top:-3.5506em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord text"><span class="mord cjk_fallback">是点</span></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mord text"><span class="mord cjk_fallback">到集合</span><span class="mord">D</span><span class="mord cjk_fallback">的最短距离点的充分必要条件为</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">∀</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6306em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span></span></span><span style="top:-3.5506em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6306em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span></span></span><span style="top:-3.5506em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ul></li></ol><h4 id="1-2-2-凸函数">1.2.2 凸函数</h4><p>凸函数的定义：设函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在凸集 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上有定义</p><ul><li>若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∀</mi><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>∈</mo><mi>D</mi><mtext>&nbsp;和&nbsp;</mtext><mi>λ</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\forall x,y \in D\ \text{和}\ \lambda \in [0,1]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">∀</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mspace">&nbsp;</span><span class="mord text"><span class="mord cjk_fallback">和</span></span><span class="mspace">&nbsp;</span><span class="mord mathnormal">λ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">]</span></span></span></span> 有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>λ</mi><mi>x</mi><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>λ</mi><mo stretchy="false">)</mo><mi>y</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>λ</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>λ</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(\lambda x + (1-\lambda)y) \le \lambda f(x) + (1-\lambda)f(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">λ</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">λ</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">λ</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">λ</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span>，则称 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 是凸集 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上的凸函数</li><li>若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∀</mi><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>∈</mo><mi>D</mi><mtext>&nbsp;和&nbsp;</mtext><mi>λ</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\forall x,y \in D\ \text{和}\ \lambda \in (0,1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">∀</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mspace">&nbsp;</span><span class="mord text"><span class="mord cjk_fallback">和</span></span><span class="mspace">&nbsp;</span><span class="mord mathnormal">λ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>λ</mi><mi>x</mi><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>λ</mi><mo stretchy="false">)</mo><mi>y</mi><mo stretchy="false">)</mo><mo>&lt;</mo><mi>λ</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>λ</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(\lambda x + (1-\lambda)y) &lt; \lambda f(x) + (1-\lambda)f(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">λ</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">λ</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">λ</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">λ</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span>，则称 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 是凸集 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上的严格凸函数</li></ul><p>凸函数的性质：</p><ol><li>如果 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span> 是定义在凸集 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上的凸函数，实数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\alpha \ge 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">\alpha f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span> 也是凸集 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上的凸函数</li><li>如果 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>f</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">f_1,f_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是定义在凸集 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上的凸函数，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mn>1</mn></msub><mo>+</mo><msub><mi>f</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">f_1+f_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 也是凸集 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上的凸函数</li><li>如果 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f_i(x)(i=1,2,\cdots,m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span> 是非空凸集 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上的凸函数，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>max</mi><mo>⁡</mo></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>m</mi></mrow></msub><mi mathvariant="normal">∣</mi><msub><mi>f</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">f(x) = \max_{1 \le i \le m} |f_i(x)|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop"><span class="mop">max</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mrel mtight">≤</span><span class="mord mathnormal mtight">i</span><span class="mrel mtight">≤</span><span class="mord mathnormal mtight">m</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2452em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord">∣</span></span></span></span> 也是凸集 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上的凸函数</li><li>如果 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f_i(x)(i=1,2,\cdots,m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span> 是非空凸集 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上的凸函数，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><msub><mi>α</mi><mi>i</mi></msub><msub><mi>f</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="1em"></mspace><mo stretchy="false">(</mo><msub><mi>α</mi><mi>i</mi></msub><mo>≥</mo><mn>0</mn><mo stretchy="false">)</mo></mstyle></mrow><annotation encoding="application/x-tex">f(x) = \displaystyle \sum_{i=1}^m \alpha_i f_i(x)\quad(\alpha_i \ge 0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:1em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span> 也是凸集 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span>​ 上的凸函数</li></ol>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-dcc9b38c" role="button" aria-expanded="false" aria-controls="collapse-dcc9b38c">        <div class="fold-arrow">▶</div>凸函数的性质证明      </div>      <div class="fold-collapse collapse" id="collapse-dcc9b38c">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403112030190.png" alt="第1、2条"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403112030160.png" alt="第4条"></p>        </div>      </div>    </div><p><strong>凸函数的判定定理</strong>：</p><ol><li><p>函数值角度：函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">R^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> 上的凸函数的充分必要条件是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∀</mi><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>∈</mo><msup><mi>R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\forall x,y \in R^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">∀</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span>，单变量函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>α</mi><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(\alpha)=f(x + \alpha y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">ϕ</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> 是关于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 的凸函数</p>     <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-d7d004d8" role="button" aria-expanded="false" aria-controls="collapse-d7d004d8">        <div class="fold-arrow">▶</div>凸函数的判别定理证明：函数值角度      </div>      <div class="fold-collapse collapse" id="collapse-d7d004d8">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403131019868.png" alt="凸函数的判别定理证明 - 函数值角度"></p>        </div>      </div>    </div></li><li><p>一阶梯度角度：设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 是定义在非空开凸集 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上的可微函数，则：</p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上凸函数的充分必要条件是：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>≥</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mi>T</mi></msup><mo stretchy="false">(</mo><mi>y</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mspace width="1em"></mspace><mi mathvariant="normal">∀</mi><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>∈</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">f(y) \ge f(x)+\nabla f(x)^T(y-x), \quad \forall x,y \in D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0913em;vertical-align:-0.25em;"></span><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∀</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span></li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上严格凸函数的充分必要条件是：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>&gt;</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mi>T</mi></msup><mo stretchy="false">(</mo><mi>y</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mspace width="1em"></mspace><mi mathvariant="normal">∀</mi><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>∈</mo><mi>D</mi><mo separator="true">,</mo><mspace width="1em"></mspace><mi>x</mi><mo mathvariant="normal">≠</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">f(y) &gt; f(x)+\nabla f(x)^T(y-x), \quad \forall x,y \in D,\quad x \ne y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0913em;vertical-align:-0.25em;"></span><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∀</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span></li></ul>     <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-9d403b87" role="button" aria-expanded="false" aria-controls="collapse-9d403b87">        <div class="fold-arrow">▶</div>凸函数的判别定理证明：一阶导数角度      </div>      <div class="fold-collapse collapse" id="collapse-9d403b87">        <div class="fold-content">          <p>无需掌握证明，但是为了便于理解性记忆，可以从<strong>二次凸函数</strong>进行辅助理解记忆。<br><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403131018032.png" alt="凸函数的判别定理证明 - 一阶导数角度"></p>        </div>      </div>    </div></li><li><p>二阶梯度角度：设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 是定义在非空开凸集 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上的二阶可微函数，则：</p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上凸函数的充分必要条件是：海塞矩阵正半定</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上严格凸函数的充分必要条件是：海塞矩阵正定？其实不一定，例如 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mn>4</mn></msup></mrow><annotation encoding="application/x-tex">x^4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span> 是严格凸函数，但是其海塞矩阵是正半定的并非正定。因此更严谨的定义应该是：海塞矩阵正定的是严格凸函数，严格凸函数的海塞矩阵是正半定的。</li></ul></li></ol><h4 id="1-2-3-凸规划（个人补充）">1.2.3 凸规划（个人补充）</h4><p>本目为个人补充内容，用于整合上述 1.2.1 与 1.2.2 内容。我们知道，学习凸集和凸函数的终极目标是为了求解凸规划问题，凸规划问题可以简述为 <strong>凸可行域 + 凸目标函数 + 局部最优解 = 全局最优解</strong>，那么如何证明这个定理是正确的呢？局部最优解求出来以后，目标函数也确定为凸函数以后，如何确定可行域是凸集呢？下面揭晓：</p><p><strong>证明凸规划问题的正确性</strong></p><ol><li><p>定理：在可行域是凸集，目标函数非严格凸的情况下，局部最优解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">x^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span>​ 也是全局最优解</p>     <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-01c9e6ee" role="button" aria-expanded="false" aria-controls="collapse-01c9e6ee">        <div class="fold-arrow">▶</div>证明（反证法）      </div>      <div class="fold-collapse collapse" id="collapse-01c9e6ee">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403122016285.png" alt="证明（反证法）"></p>        </div>      </div>    </div></li><li><p>定理：在可行域是凸集，目标函数是严格凸的情况下，局部最优解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">x^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span> 也是唯一的全局最优解</p>     <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-858dd40d" role="button" aria-expanded="false" aria-controls="collapse-858dd40d">        <div class="fold-arrow">▶</div>证明（反证法）      </div>      <div class="fold-collapse collapse" id="collapse-858dd40d">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403122017101.png" alt="证明（反证法）"></p>        </div>      </div>    </div></li></ol><p><strong>确定可行域是否为凸集</strong></p><ol><li><p>定理：若约束条件 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">c_i(x) \ge 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 中每一个约束函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c_i(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 都是凹函数，则可行域 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span> 是凸集</p>     <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-4ecb9cd4" role="button" aria-expanded="false" aria-controls="collapse-4ecb9cd4">        <div class="fold-arrow">▶</div>证明      </div>      <div class="fold-collapse collapse" id="collapse-4ecb9cd4">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403122019899.png" alt="证明"></p>        </div>      </div>    </div></li><li><p>定理：若约束条件 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≤</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">c_i(x) \le 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 中每一个约束函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c_i(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 都是凸函数，则可行域 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span> 是凸集</p>     <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-8354fb68" role="button" aria-expanded="false" aria-controls="collapse-8354fb68">        <div class="fold-arrow">▶</div>证明      </div>      <div class="fold-collapse collapse" id="collapse-8354fb68">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403122019572.png" alt="证明"></p>        </div>      </div>    </div></li><li><p>定理：若约束条件中每一个约束函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c_i(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 都恒等于零，则可行域 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span> 是凸集</p>     <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-b528b2da" role="button" aria-expanded="false" aria-controls="collapse-b528b2da">        <div class="fold-arrow">▶</div>证明      </div>      <div class="fold-collapse collapse" id="collapse-b528b2da">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403122019751.png" alt="证明"></p>        </div>      </div>    </div></li></ol><h3 id="1-3-最优性条件">1.3 最优性条件</h3><div class="note note-light">            <p>最优性条件是指最优化问题的最优解所必须满足的条件，本目介绍求解<strong>无约束最优化问题</strong>的一阶必要条件和二阶必要条件，再补充介绍二阶充分条件以及针对凸规划问题的二阶充分必要条件。最后简单介绍求解<strong>等式约束最优化问题</strong>的拉格朗日乘子法。</p>          </div><h4 id="1-3-1-下降方向">1.3.1 下降方向</h4><p>在开始之前先简单介绍一下「下降方向」这个概念。</p><ul><li><p><strong>下降方向</strong>定义：设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 为定义在空间 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">R^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> 的连续函数，点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>x</mi><mo>ˉ</mo></mover><mo>∈</mo><msup><mi>R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\bar x \in R^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6069em;vertical-align:-0.0391em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">x</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span>，若对于方向 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi><mo>∈</mo><msup><mi>R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">s \in R^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> 存在数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>δ</mi><mo>&gt;</mo></mrow><annotation encoding="application/x-tex">\delta &gt;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.03785em;">δ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span></span></span></span> 0 使</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mover accent="true"><mi>x</mi><mo>ˉ</mo></mover><mo>+</mo><mi>α</mi><mi>s</mi><mo stretchy="false">)</mo><mo>&lt;</mo><mi>f</mi><mo stretchy="false">(</mo><mover accent="true"><mi>x</mi><mo>ˉ</mo></mover><mo stretchy="false">)</mo><mo separator="true">,</mo><mspace width="1em"></mspace><mi mathvariant="normal">∀</mi><mi>α</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mi>δ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(\bar x+\alpha s) &lt; f(\bar x),\quad \forall \alpha \in (0,\delta)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">x</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord mathnormal">s</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">x</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∀</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03785em;">δ</span><span class="mclose">)</span></span></span></span></span></p><p>成立，则称 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">s</span></span></span></span> 为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>x</mi><mo>ˉ</mo></mover></mrow><annotation encoding="application/x-tex">\bar x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5678em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">x</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">ˉ</span></span></span></span></span></span></span></span></span></span> 处的一个下降方向。在 点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>x</mi><mo>ˉ</mo></mover></mrow><annotation encoding="application/x-tex">\bar x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5678em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">x</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">ˉ</span></span></span></span></span></span></span></span></span></span> 处的所有下降方向的全体记为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mover accent="true"><mi>x</mi><mo>ˉ</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(\bar x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">x</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></p></li><li><p><strong>下降方向</strong>定理：设函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>x</mi><mo>ˉ</mo></mover></mrow><annotation encoding="application/x-tex">\bar x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5678em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">x</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">ˉ</span></span></span></span></span></span></span></span></span></span> 处连续可微，如存在非零向量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi><mo>∈</mo><msup><mi>R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">s  \in R^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> 使</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><mover accent="true"><mi>x</mi><mo>ˉ</mo></mover><msup><mo stretchy="false">)</mo><mi>T</mi></msup><mi>s</mi><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\nabla f(\bar x)^Ts &lt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1413em;vertical-align:-0.25em;"></span><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">x</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p><p>成立，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">s</span></span></span></span> 是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>x</mi><mo>ˉ</mo></mover></mrow><annotation encoding="application/x-tex">\bar x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5678em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">x</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">ˉ</span></span></span></span></span></span></span></span></span></span> 处的一个下降方向</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-b02d0016" role="button" aria-expanded="false" aria-controls="collapse-b02d0016">        <div class="fold-arrow">▶</div>证明      </div>      <div class="fold-collapse collapse" id="collapse-b02d0016">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403131639082.jpg" alt="证明"></p>        </div>      </div>    </div></li></ul><h4 id="1-3-2-充分必要条件">1.3.2 充分必要条件</h4><p>设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo>:</mo><mi>D</mi><mo>⊂</mo><msup><mi>R</mi><mi>n</mi></msup><mo>→</mo><msup><mi>R</mi><mn>1</mn></msup></mrow><annotation encoding="application/x-tex">f:D \subset R^n \to R^1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span> 是定义在开集 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上的函数，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup><mo>∈</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">x^* \in D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0391em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 是该函数的局部极小点，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 的一阶导数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(x)=\nabla f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>，二阶导数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi mathvariant="normal">∇</mi><mn>2</mn></msup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G(x)=\nabla ^2f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">G</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord">∇</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>。</p><ul><li><p><strong>一阶必要条件</strong>。若目标函数在定义域 D 上连续可微，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">x^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span> 是局部极小点的一阶必要条件为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><msup><mi>x</mi><mo>∗</mo></msup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">g(x^*)=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7387em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p></li><li><p><strong>二阶必要条件</strong>。若目标函数在定义域 D 上二阶连续可微，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">x^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span> 是局部极小点的二阶必要条件为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><msup><mi>x</mi><mo>∗</mo></msup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mspace width="1em"></mspace><mi>G</mi><mo stretchy="false">(</mo><msup><mi>x</mi><mo>∗</mo></msup><mo stretchy="false">)</mo><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">g(x^*)=0,\quad G(x^*)\ge 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7387em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">G</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7387em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p></li><li><p><strong>二阶充分条件</strong>。若目标函数在定义域 D 上二阶连续可微，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">x^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span> 是<strong>严格</strong>局部极小点的二阶充分条件为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><msup><mi>x</mi><mo>∗</mo></msup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mspace width="1em"></mspace><mi>G</mi><mo stretchy="false">(</mo><msup><mi>x</mi><mo>∗</mo></msup><mo stretchy="false">)</mo><mtext>是正定的</mtext></mrow><annotation encoding="application/x-tex">g(x^*)=0,\quad G(x^*) \text{是正定的}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7387em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">G</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7387em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord text"><span class="mord cjk_fallback">是正定的</span></span></span></span></span></span></p></li><li><p><strong>充分必要条件（凸最优性定理）</strong>。在无约束最优化问题中，如果目标函数是凸的，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">x^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span> 是全局极小点的充分必要条件是：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><msup><mi>x</mi><mo>∗</mo></msup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">g(x^*)=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7387em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p></li></ul><h4 id="1-3-3-拉格朗日乘子法">1.3.3 拉格朗日乘子法</h4><p>如果现在最优化问题不是单纯的<strong>无约束最优化问题</strong>，而是增设了等式约束的<strong>等式约束最优化问题</strong>，如何求解呢？我们引入 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Lagrange</mtext></mrow><annotation encoding="application/x-tex">\text{Lagrange}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">Lagrange</span></span></span></span></span> 乘子法，将所有的等式约束都转化到目标函数中得到一个等价的无约束最优化问题，即：</p><p>对于这样的等式约束最优化问题：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>min</mi><mo>⁡</mo><mspace width="1em"></mspace></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>s.t.</mtext><mspace width="1em"></mspace></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msub><mi>c</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mi>m</mi><mi mathvariant="normal">.</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\min\quad&amp; f(x)\\\text{s.t.}\quad&amp;c_i(x)=0,i=1,2,\cdots,m.\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">min</span><span class="mspace" style="margin-right:1em;"></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">s.t.</span></span><span class="mspace" style="margin-right:1em;"></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mord">.</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>引入拉格朗日乘子将其转化为无约束最优化问题，进而利用上述无约束最优化问题的求解策略进行求解：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>λ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><msub><mi>λ</mi><mi>i</mi></msub><msub><mi>c</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(x,\lambda) = f(x) - \sum_{i=1}^m\lambda_ic_i(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">λ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></p><h3 id="1-4-最优化方法概述">1.4 最优化方法概述</h3><p>现实生活中，对于常见问题的建模往往是极其复杂的，为了求得最优解，我们需要对建立的模型进行微分算子的求解。但是问题是，尽管我们知道最优解一定存在于微分算子的数值解中，但是我们往往不能很快的计算出其数值解，因此我们需要采用别的方法进行计算。最常用的方法就是本书介绍的迭代法。</p><p>大概就两步：确定初值开始判断，如果符合条件的约束则停止迭代得到最终的解；如果不符合约束则对当前迭代值赋予修正量并继续迭代判断。直到找到最终的解。接下来将对 5 个专有名词进行解释，分别为：初始点的选取、迭代点好坏的判定、收敛速度、迭代的终止条件、修正量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>s</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">s^{(k)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.888em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span> 的确定。</p><h4 id="1-4-1-初始点的选取">1.4.1 初始点的选取</h4><p>初始点的选取取决于算法的收敛性能。</p><ul><li>如果一个算法可以做到全局收敛，则初始点的选取是任意的</li><li>如果一个算法只能局部收敛，则初始点的选取往往需要尽可能的接近最优解，而这往往是困难的，因为我们并不知道的最优解是什么。此时可以采用借鉴之前的经验来选择初始点</li></ul><h4 id="1-4-2-迭代点好坏的判定">1.4.2 迭代点好坏的判定</h4><p>判定一个迭代点的好坏时，我们往往会设计一个评价函数，而评价函数的设计取决于约束问题的种类。</p><ul><li>对于无约束最优化问题。我们可以直接将目标函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 设计为评价函数。很显然，如果一个迭代点对应的的函数值比之前点对应的函数值更小，则可以说明当前迭代点由于之前的点</li><li>对于有约束最优化问题。此时我们不仅仅需要考虑函数值的大小，也要考虑当前迭代点的可行程度（离可行域的距离）。因此此类最优化问题往往既包含目标函数，也包含约束函数。</li></ul><h4 id="1-4-3-收敛速度">1.4.3 收敛速度</h4><p>若当前算法一定收敛，我们还需要判断收敛的速度，接下来介绍收敛的速度。</p><p>假设</p><ul><li>设向量序列 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup><mo>⊂</mo><msup><mi>R</mi><mi>n</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{ x^{(k)} \subset R^n \}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.138em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span> 收敛于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">x^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span>​</li></ul><p>定义</p><ol><li>误差序列：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>e</mi><mi>k</mi></msub><mo>=</mo><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup><mo>−</mo><msup><mi>x</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">e_k = x^{(k)} - x^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9713em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span></li><li>收敛率表达式：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>k</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mfrac><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><msub><mi>e</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi></mrow><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><msub><mi>e</mi><mi>k</mi></msub><mi mathvariant="normal">∣</mi><msup><mi mathvariant="normal">∣</mi><mi>r</mi></msup></mrow></mfrac><mo>=</mo><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \lim_{k \to \infty} \frac{|| e_{k+1} ||}{||e_k||^r} = C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.3479em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">→</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7521em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣∣</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord">∣</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5904em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣∣</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mord">∣∣</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>，并称序列 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{x^{(k)}\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.138em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span> 的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span> 阶以 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> 为因子收敛于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">x^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span></li></ol><p>线性收敛</p><ul><li>定义：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>0</mn><mo>&lt;</mo><mi>C</mi><mo>&lt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">r=1,0 &lt; C&lt;1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></li><li>性质：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> 越小，算法收敛越快</li></ul><p>超线性收敛</p><ul><li><p>定义：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mi>C</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">r=1,C=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></p></li><li><p>定理：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mo>&gt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">r&gt;1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>​ 时的算法均为超线性收敛算法</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-7f0bbd51" role="button" aria-expanded="false" aria-controls="collapse-7f0bbd51">        <div class="fold-arrow">▶</div>证明      </div>      <div class="fold-collapse collapse" id="collapse-7f0bbd51">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403201331819.png" alt="证明"></p>        </div>      </div>    </div></li></ul><h4 id="1-4-4-迭代的终止条件">1.4.4 迭代的终止条件</h4><ul><li><p>对于收敛速度比较慢的算法：</p><p>根据一阶必要条件，我们知道最优解一定取在微分算子为 0 的向量上，因此我们以下式为终止条件是一种可行的选择</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mo>≤</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">||\nabla f(x^{(k)}|| \le \epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.188em;vertical-align:-0.25em;"></span><span class="mord">∣∣∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mord">∣∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ϵ</span></span></span></span></span></p><p>但是问题在于这对收敛速度很快的算法不使用，如下图的无约束最优化问题。已知两个局部最优解分别为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>x</mi><mn>1</mn><mo>∗</mo></msubsup><mo separator="true">,</mo><msubsup><mi>x</mi><mn>2</mn><mo>∗</mo></msubsup></mrow><annotation encoding="application/x-tex">x_1^*,x_2^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9368em;vertical-align:-0.2481em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-2.4519em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-2.4519em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span></span></span></span>，迭代的两个解分别为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>x</mi><mo stretchy="true">‾</mo></mover><mn>1</mn></msub><mo separator="true">,</mo><msub><mover accent="true"><mi>x</mi><mo stretchy="true">‾</mo></mover><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\overline{x}_1,\overline{x}_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.825em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6306em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span></span></span><span style="top:-3.5506em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6306em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span></span></span><span style="top:-3.5506em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，可以看出：尽管 2 号点相对于 1 号点更靠近局部最优解，但是由于 2 号点的梯度更大，明显不如 1 号点更加局部最优，因此利用微分算子作为迭代的终止条件不适用于收敛速度快的算法</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403201330908.png" alt="无约束最优化问题"></p></li><li><p>对于超线性收敛的算法：</p><p>一个理想的收敛终止条件为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup><mo>−</mo><msup><mi>x</mi><mo>∗</mo></msup><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mo>≤</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">||x^{(k)} - x^*|| \le \epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.138em;vertical-align:-0.25em;"></span><span class="mord">∣∣</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mord">∣∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ϵ</span></span></span></span>，但是由于最优解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">x^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span> 是未知的，因此该方案不可行，同理 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msup><mi>x</mi><mo>∗</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x^*)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 也是未知的，因此 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mi>f</mi><mo stretchy="false">(</mo><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><msup><mi>x</mi><mo>∗</mo></msup><mo stretchy="false">)</mo><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mo>≤</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">||f(x^{(k)}) - f(x^*)|| \le \epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.138em;vertical-align:-0.25em;"></span><span class="mord">∣∣</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord">∣∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ϵ</span></span></span></span> 也是不可行的。那么有没有什么替代的方案呢？答案是有的。</p><ol><li><p>方案一：利用<strong>函数解序列</strong>进行替代。即以下式作为迭代的终止条件</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mo>−</mo><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mo>≤</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">|| x^{(k+1)} - x^{(k)} || \le \epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.188em;vertical-align:-0.25em;"></span><span class="mord">∣∣</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.188em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mord">∣∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ϵ</span></span></span></span></span></p>     <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-f8eb5676" role="button" aria-expanded="false" aria-controls="collapse-f8eb5676">        <div class="fold-arrow">▶</div>证明      </div>      <div class="fold-collapse collapse" id="collapse-f8eb5676">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403201425856.jpg" alt="证明"></p>        </div>      </div>    </div></li><li><p>方案二：利用<strong>函数值序列</strong>进行替代。即以下式作为迭代的终止条件</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mi>f</mi><mo stretchy="false">(</mo><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mo>≤</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">|| f(x^{(k+1)}) - f(x^{(k)}) || \le \epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.188em;vertical-align:-0.25em;"></span><span class="mord">∣∣</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.188em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord">∣∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ϵ</span></span></span></span></span></p>     <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-1f855ea6" role="button" aria-expanded="false" aria-controls="collapse-1f855ea6">        <div class="fold-arrow">▶</div>证明      </div>      <div class="fold-collapse collapse" id="collapse-1f855ea6">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403201425473.jpg" alt="证明"></p>        </div>      </div>    </div></li></ol></li><li><p>一般情况下，对于上述超线性算法的判断收敛的方法，只用其中一种往往不适当。此时一般使用两种方法集成的思路进行判断。</p></li></ul><h4 id="1-4-5-修正量的确定">1.4.5 修正量的确定</h4><p>本书介绍的迭代修正值都是使得当前迭代后的值小于上一个状态的函数值，我们称这类使评价函数值下降的算法为<strong>单调下降算法</strong>。至于如何得出修正量，我们往往通过求解一个相对简单易于求解的最优化问题（通常成为子问题），来计算获得修正量。</p><h2 id="第二章「约束最优化」线性规划">第二章「约束最优化」线性规划</h2><div class="note note-light">            <p>目标函数和约束函数都是线性函数。</p>          </div><h3 id="2-1-线性规划问题和基本性质">2.1 线性规划问题和基本性质</h3><h4 id="2-1-1-线性规划问题">2.1.1 线性规划问题</h4><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403211918859.png" alt="线性规划问题的一般形式"></p><h4 id="2-1-2-图解法">2.1.2 图解法</h4><p><strong>仅适用于二元变量</strong>的线性规划问题。我们将所有的约束条件全部画到平面直角坐标系中构成一个可行域，然后将目标函数作为一条直线进行平移，直到与可行域初次有交点，则该交点就是最优解对应的点。当然不一定会有交点，一共分为四种情况：</p>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-0f7ad755" role="button" aria-expanded="false" aria-controls="collapse-0f7ad755">        <div class="fold-arrow">▶</div>刚好只有一个最优解      </div>      <div class="fold-collapse collapse" id="collapse-0f7ad755">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403211917063.png" alt="刚好只有一个最优解"></p>        </div>      </div>    </div>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-2097dcc0" role="button" aria-expanded="false" aria-controls="collapse-2097dcc0">        <div class="fold-arrow">▶</div>有无穷多最优解      </div>      <div class="fold-collapse collapse" id="collapse-2097dcc0">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403211920849.png" alt="有无穷多最优解"></p>        </div>      </div>    </div>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-c681e611" role="button" aria-expanded="false" aria-controls="collapse-c681e611">        <div class="fold-arrow">▶</div>有无界解      </div>      <div class="fold-collapse collapse" id="collapse-c681e611">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403211921495.png" alt="有无界解"></p>        </div>      </div>    </div>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-75bcc499" role="button" aria-expanded="false" aria-controls="collapse-75bcc499">        <div class="fold-arrow">▶</div>无可行解 - 可行域为空集      </div>      <div class="fold-collapse collapse" id="collapse-75bcc499">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403211922747.png" alt="无可行解 - 可行域为空集"></p>        </div>      </div>    </div><h4 id="2-1-3-基本性质">2.1.3 基本性质</h4><ol><li><p>线性规划问题的可行域如果非空，则是一个凸集</p>     <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-da25d73d" role="button" aria-expanded="false" aria-controls="collapse-da25d73d">        <div class="fold-arrow">▶</div>证明      </div>      <div class="fold-collapse collapse" id="collapse-da25d73d">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403212109584.jpg" alt="证明"></p>        </div>      </div>    </div></li><li><p>如果线性规划问题有最优解，那么最优解可在可行域的顶点中确定</p></li><li><p>如果可行域有界且可行域只有有限个顶点，则问题的最优解必存在，并在这有限个顶点中确定</p></li><li><p>最优解可由最优顶点处的<strong>有效约束</strong>形成的方程组的解确定</p></li></ol><h4 id="2-1-4-线性规划的标准形">2.1.4 线性规划的标准形</h4><p>为什么要学习线性规划的标准形？</p><ul><li>我们要学习单纯形法来计算多维线性规划的最优解</li><li>单纯形法需要线性规划的具有所谓的标准形</li></ul><p>线性规划的标准形的定义：</p><ul><li>只含有线性等式约束和对变量的非负约束</li></ul><p>一般线性规划转化为标准形的方法：</p><ol><li>对于目标函数：需要将取 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>max</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\max</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mop">max</span></span></span></span> 通过<strong>取相反数</strong>转化为取 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>min</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\min</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mop">min</span></span></span></span></li><li>对于约束条件：需要将不等式约束<strong>松弛</strong>转化为等式约束<ul><li>对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≤</mo></mrow><annotation encoding="application/x-tex">\le</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≤</span></span></span></span> 的不等式约束，等式左边加上非负新变量，从而转化为等式</li><li>对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≥</mo></mrow><annotation encoding="application/x-tex">\ge</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≥</span></span></span></span> 的不等式约数，等式左边减去非负新变量，从而转化为等式</li></ul></li><li>对于无约束的变量：需要将其转化为<strong>两个新变量之差</strong>（可正可负），产生了一个新的等式，如果该无约束变量存在于目标函数中，还需要将目标函数中的该变量表示为两个新变量之差</li></ol>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-53f9dab5" role="button" aria-expanded="false" aria-controls="collapse-53f9dab5">        <div class="fold-arrow">▶</div>举个例子      </div>      <div class="fold-collapse collapse" id="collapse-53f9dab5">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403212111330.jpg" alt="一般线性规划转化为标准形 - 演示"></p>        </div>      </div>    </div><h4 id="2-1-5-基本可行解">2.1.5 基本可行解</h4><p>直接说结论：对于标准形的线性规划问题，其基本可行解就是凸集的顶点。</p><p>假设系数矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mrow><mi>m</mi><mo separator="true">,</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">A_{m, n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>。基本可行解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊂</mo></mrow><annotation encoding="application/x-tex">\subset</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">⊂</span></span></span></span> 基本解，而基本解的个数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≤</mo><msubsup><mi>C</mi><mi>n</mi><mi>m</mi></msubsup></mrow><annotation encoding="application/x-tex">\le C_{n}^{m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9303em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>（因为分块矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>B</mi><mrow><mi>m</mi><mo>×</mo><mi>m</mi></mrow></msub></mrow><annotation encoding="application/x-tex">B_{m \times m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2583em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">×</span><span class="mord mathnormal mtight">m</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span> 不一定可逆）。其中基本可行解需要满足解序列元素非负。如果<strong>基本变量</strong>向量对应的解中含有 0 元素，称其为退化的基本可行解，产生退化的基本可行解的原因是存在可删除的约数条件。</p><h4 id="2-1-6-最优解的性质">2.1.6 最优解的性质</h4><ul><li><p>若可行域有界，则线性规划问题的目标函数一定可以在其可行域的顶点上达到最优。</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-072cac81" role="button" aria-expanded="false" aria-controls="collapse-072cac81">        <div class="fold-arrow">▶</div>证明 - 反证法      </div>      <div class="fold-collapse collapse" id="collapse-072cac81">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403222103333.jpg" alt="证明"></p>        </div>      </div>    </div></li><li><p>有时，目标函数可能在多个顶点处达到最大，这时在这些顶点的<strong>凸组合</strong>上也达到最大值，这时线性规划问题有无限多个最优解。</p></li></ul><h3 id="2-2-单纯形法">2.2 单纯形法</h3><div class="note note-light">            <p>本目主要介绍一种常用的计算线性规划问题可行解的方法：单纯形法。其中，<strong>前三条</strong>分别介绍单纯性方法计算步骤的理论可行性，<strong>第四条</strong>具体介绍单纯形法的计算步骤与过程。</p>          </div><h4 id="2-2-1-初始基可行解的确定">2.2.1 初始基可行解的确定</h4><ol><li><strong>直接观察</strong>：直接看出系数矩阵中含有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>×</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">m \times m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 的单位矩阵，则选择该单位矩阵为初始基</li><li><strong>加松弛变量</strong>：当所有的约束条件都是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≤</mo></mrow><annotation encoding="application/x-tex">\le</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≤</span></span></span></span> 时，每一个约束条件都加一个非负的松弛变量，则这 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 个松弛变量的系数就对应了一个 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>×</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">m \times m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 的单位矩阵，选择其为初始基即可</li><li><strong>加非负的人工变量</strong>：求解方法与常规的线性规划问题不一样，见下述 2.2.3 条。<ul><li>对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo></mrow><annotation encoding="application/x-tex">=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">=</span></span></span></span> 约束，我们在约束条件的左边加上非负人工变量</li><li>对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≥</mo></mrow><annotation encoding="application/x-tex">\ge</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≥</span></span></span></span> 约束，我们先在约束条件的左边减去非负松弛变量，再在约束条件的左边加上非负的人工变量</li></ul></li></ol><h4 id="2-2-2-最优性检验">2.2.2 最优性检验</h4>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-73e17d47" role="button" aria-expanded="false" aria-controls="collapse-73e17d47">        <div class="fold-arrow">▶</div>符号说明      </div>      <div class="fold-collapse collapse" id="collapse-73e17d47">        <div class="fold-content">          <p>一、当前局面：已经计算出一个基本可行解</p><p>1.1 确定线性规划的目标</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403281940243.png" alt="确定线性规划的目标"></p><p>1.2 添加松弛变量</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403281942808.png" alt="添加松弛变量"></p><p>1.3 用非基变量线性表示基变量</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403281937210.png" alt="用非基变量线性表示基变量"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403281939336.png" alt="向量表示"></p><p>二、最优性检验</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403282010283.jpg" alt="最优性检验"></p>        </div>      </div>    </div><ol><li><p><strong>唯一最优解</strong>判别定理：对于<strong>目标函数求解最大值</strong>的情形。若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∀</mi><mi>j</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>m</mi><mo>+</mo><mn>1</mn><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\forall j \in [m+1, n]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">∀</span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">]</span></span></span></span> 有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>σ</mi><mi>j</mi></msub><mo>≤</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\sigma_j \le 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9221em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 则当前基本可行解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">x^{(k)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.888em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span> 为最优解。</p></li><li><p><strong>无穷多最优解</strong>判别定理：在满足第一条所有的检验数非正的情况下，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∃</mi><mi>j</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>m</mi><mo>+</mo><mn>1</mn><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\exist j \in [m+1, n]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">∃</span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">]</span></span></span></span>，有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>σ</mi><mi>j</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\sigma_j=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，则该线性规划问题有无穷多最优解。</p>     <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-0c8e4c38" role="button" aria-expanded="false" aria-controls="collapse-0c8e4c38">        <div class="fold-arrow">▶</div>证明      </div>      <div class="fold-collapse collapse" id="collapse-0c8e4c38">        <div class="fold-content">          <p>我们可以将任意一个检验数为 0 的非基变量与基变量进行置换得到新的一个基本可行解对应的目标函数值保持不变。于是下述凸组合内的可行解都是最优解</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>s</mi><mi mathvariant="normal">.</mi><mi>t</mi><mi mathvariant="normal">.</mi><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>k</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">∈</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo stretchy="false">[</mo><mi>m</mi><mo>+</mo><mn>1</mn><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">]</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>σ</mi><mi>k</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">=</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr></mtable></mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}&amp; x^{(k)} \\ &amp; s.t. \begin{cases} k &amp;\in&amp; [m+1, n] \\\sigma_k &amp;=&amp; 0\end{cases}\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.898em;vertical-align:-2.199em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.699em;"><span style="top:-5.511em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"></span></span><span style="top:-3.101em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.199em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.699em;"><span style="top:-5.511em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.101em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal">s</span><span class="mord">.</span><span class="mord mathnormal">t</span><span class="mord">.</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mrel">∈</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mrel">=</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mopen">[</span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">]</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.199em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>        </div>      </div>    </div></li><li><p><strong>无界解</strong>判别定理：pass</p></li><li><p><strong>无解</strong>判别定理：可行域为空集</p></li></ol><h4 id="2-2-3-计算新的基本可行解">2.2.3 计算新的基本可行解</h4><p>对于<strong>常规</strong>的线性规划问题（初始基本可行解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">x^{(0)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.888em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span> 可以直接找到）</p><ol><li>向量方程法<ul><li>确定入基变量：选择目标函数中<strong>系数最大的变量</strong>作为入基变量，即将其对应的系数列向量与出基变量对应的系数列向量进行置换</li><li>确定出基变量：线性表示基方程组以后，利用变量非负的特性找到入基变量刚好取 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 时对应的变量即为出基变量（示例的第二轮迭代中出基变量即为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>5</mn></msub></mrow><annotation encoding="application/x-tex">x_5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>）</li></ul></li><li>系数矩阵法<ul><li>pass</li></ul></li></ol><p>对于<strong>添加人工变量</strong>的线性规划问题（初始基本可行解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">x^{(0)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.888em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span> 不能直接找到，需要手动构造 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">x^{(0)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.888em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span>）</p><ol><li>大 M 法：pass</li><li>两阶段法<ul><li>第一阶段：pass</li><li>第二阶段：pass</li></ul></li></ol><h4 id="2-2-4-单纯形表法">2.2.4 单纯形表法</h4><p>向量方程法：</p>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-8c53e26b" role="button" aria-expanded="false" aria-controls="collapse-8c53e26b">        <div class="fold-arrow">▶</div>实战演练      </div>      <div class="fold-collapse collapse" id="collapse-8c53e26b">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403272056146.jpg" alt="单纯形法的求解步骤 - 1"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403272056440.jpg" alt="单纯形法的求解步骤 - 2"></p>        </div>      </div>    </div><p>单纯形表法：</p>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-766b3028" role="button" aria-expanded="false" aria-controls="collapse-766b3028">        <div class="fold-arrow">▶</div>实战演练      </div>      <div class="fold-collapse collapse" id="collapse-766b3028">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404171407440.jpg" alt="实战演练"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404171410846.jpg" alt="代码验算"></p>        </div>      </div>    </div><h3 id="2-3-对偶与对偶单纯形法">2.3 对偶与对偶单纯形法</h3><h4 id="2-3-1-确定对偶问题">2.3.1 确定对偶问题</h4><p>已知原问题的表达式，如何求解对偶问题的表达式？我们需掌握以下对偶转换规则：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404071144213.png" alt="转化规则"></p><p>从上图不难发现，其实就是对线性规划的三个部分进行转换：目标函数、m个约束条件、n个自变量，下面分别进行文字介绍：</p><ol><li>约束条件：从 m 个变为 n 个<ul><li><strong>常量矩阵</strong>中的元素为原问题中自变量的系数</li><li><strong>二元关系</strong>的变化取决于转化方向<ul><li>max to min：对偶问题约束条件的符号和原问题自变量的符号相同</li><li>min to max：对偶问题约束条件的符号和原问题自变量的符号相反</li></ul></li><li><strong>线性约束</strong>为原问题线性约束的转置</li></ul></li><li>自变量：从 n 个变为 m 个<ul><li><strong>二元关系</strong>的变化同样取决于转化方向<ul><li>max to min：对偶问题自变量的符号和原问题约束条件的符号相反</li><li>min to max：对偶问题自变量的符号和原问题约束条件的符号相同</li></ul></li></ul></li><li>目标函数：自元个数从 n 个变为 m 个<ul><li>变元的系数就是原问题中约束条件的常量矩阵对应的元素值</li></ul></li></ol>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-d8f9745d" role="button" aria-expanded="false" aria-controls="collapse-d8f9745d">        <div class="fold-arrow">▶</div>实战演练      </div>      <div class="fold-collapse collapse" id="collapse-d8f9745d">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202407081127115.png" alt="实例"></p>        </div>      </div>    </div><h4 id="2-3-2-对偶定理">2.3.2 对偶定理</h4><ul><li>对称性：对偶问题的对偶问题是原问题</li><li>无界性：若原问题是无界解，则对偶问题无可行解</li><li>对偶定理：若原问题有最优解，那么对偶问题也有最优解，且目标函数值相等</li><li>互补松弛性：若原问题最优解为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">x^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span>，对偶问题的最优解为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>y</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">y^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8831em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span>，则有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup><msub><mi>y</mi><mi>s</mi></msub><mo>=</mo><mn>0</mn><mo separator="true">,</mo><msup><mi>y</mi><mo>∗</mo></msup><msub><mi>x</mi><mi>s</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x^* y_s=0,y^*x_s=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8831em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8831em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>s</mi></msub><mo separator="true">,</mo><msub><mi>y</mi><mi>s</mi></msub></mrow><annotation encoding="application/x-tex">x_s,y_s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 分别为原问题的松弛变量和对偶问题的松弛变量</li></ul>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-59805afb" role="button" aria-expanded="false" aria-controls="collapse-59805afb">        <div class="fold-arrow">▶</div>实战演练      </div>      <div class="fold-collapse collapse" id="collapse-59805afb">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202407081259988.png" alt="问题"></p><p>对于上述问题，我们可以先写出其对偶问题：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202407081300732.png" alt="对偶问题"></p><p>显然的对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>y</mi><mo>∗</mo></msup><mo>=</mo><mo stretchy="false">(</mo><mfrac><mn>4</mn><mn>5</mn></mfrac><mo separator="true">,</mo><mfrac><mn>3</mn><mn>5</mn></mfrac><msup><mo stretchy="false">)</mo><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">y^*=(\frac{4}{5},\frac{3}{5})^T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8831em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mopen">(</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span></span></span>，上述 2,3,4 是取不到等号的，对应的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup><mo>=</mo><mo stretchy="false">(</mo><mo>−</mo><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mo>−</mo><msup><mo stretchy="false">)</mo><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">x^*=(-,0,0,0,-)^T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0913em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span></span></span>。还剩两个变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>5</mn></msub></mrow><annotation encoding="application/x-tex">x_1,x_5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 通过 1,5 两个不等式取等进行计算：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mn>3</mn><msub><mi>x</mi><mn>2</mn></msub><mo>=</mo><mn>4</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mn>2</mn><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub><mo>=</mo><mn>3</mn></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}x_1+3x_2=4\\2x_1+x_2=3\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">3</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">4</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>计算可得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn><mo separator="true">,</mo><msub><mi>x</mi><mn>5</mn></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x_1=1,x_5=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，于是原问题的最优解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">x^*=(1, 0, 0, 0, 1)^T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0913em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span></span></span></p>        </div>      </div>    </div><h4 id="2-3-3-对偶单纯形法">2.3.3 对偶单纯形法</h4>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-c81b1c05" role="button" aria-expanded="false" aria-controls="collapse-c81b1c05">        <div class="fold-arrow">▶</div>实战演练      </div>      <div class="fold-collapse collapse" id="collapse-c81b1c05">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404171409291.jpg" alt="实战演练"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404171411832.jpg" alt="代码验算"></p>        </div>      </div>    </div><h2 id="第三章-线性搜索">第三章 线性搜索</h2><div class="note note-light">            <p>本章介绍最优化问题中迭代解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><msub><mi>α</mi><mi>k</mi></msub><msub><mi>d</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">x_{k+1} = x_k + \alpha_kd_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 基于「线性搜索」的迭代方式。</p><p>假设搜索方向 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">d_{k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是一个定值且一定是下降方向，我们讨论步长因子 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>α</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_{k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的计算选择策略。分为两步：</p><ol><li>确定步长的搜索区间</li><li>通过精确 or 不精确算法来搜索合适的步长</li></ol><p>本章的内容分布：</p><ul><li>3.1 介绍<strong>确定初始搜索区间</strong>的「进退法」</li><li>3.2 介绍<strong>缩小搜索区间</strong>的「精确线性搜索算法」：0.618 法、斐波那契法、二分法、插值法</li><li>3.3 介绍<strong>缩小搜索区间</strong>的「不精确线性搜索算法」：Armijo 准则、Goldstein 准则、Wolfe 准则</li></ul>          </div><h3 id="3-1-确定初始搜索区间">3.1 确定初始搜索区间</h3><p>确定初始搜索区间 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[a,b]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mclose">]</span></span></span></span>，我们利用 <a href="https://zhuanlan.zhihu.com/p/109701879">进退法</a></p><h3 id="3-2-精确线性搜索算法">3.2 精确线性搜索算法</h3>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-d825d270" role="button" aria-expanded="false" aria-controls="collapse-d825d270">        <div class="fold-arrow">▶</div>预备知识：单峰函数定理      </div>      <div class="fold-collapse collapse" id="collapse-d825d270">        <div class="fold-content">          <p>在正式开始介绍之前，我们先了解<strong>单峰函数定理</strong>：</p><ul><li><p>定理：对于在区间 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[a,b]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mclose">]</span></span></span></span> 上的一个单峰函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup><mo>∈</mo><mo stretchy="false">[</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">x^* \in [a,b]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0391em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mclose">]</span></span></span></span> 是其极小点，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[a,b]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mclose">]</span></span></span></span> 上的任意两点，且 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>&lt;</mo><msub><mi>x</mi><mn>1</mn></msub><mo>&lt;</mo><msub><mi>x</mi><mn>2</mn></msub><mo>&lt;</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a&lt;x_1&lt;x_2&lt;b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span>，可以通过比较 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x_1),f(x_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 的值来确定点的保留和舍弃</p></li><li><p>迭代：</p><ol><li><p>若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>≥</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x_1) \ge f(x_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup><mo>∈</mo><mo stretchy="false">[</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">x^* \in [x_1,b]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0391em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mclose">]</span></span></span></span></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404121125038.png" alt="保留右区间"></p></li><li><p>若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>&lt;</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x_1) &lt; f(x_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup><mo>∈</mo><mo stretchy="false">[</mo><mi>a</mi><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">x^* \in [a,x_2]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0391em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span>​</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404121125961.png" alt="保留左区间"></p></li><li><p>若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x_1) = f(x_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup><mo>∈</mo><mo stretchy="false">[</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">x^* \in [x_1,x_2]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0391em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></p></li></ol></li></ul><p>于是迭代的关键就在于如何取点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，下面开始介绍三种取点方法。</p>        </div>      </div>    </div>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-649e6489" role="button" aria-expanded="false" aria-controls="collapse-649e6489">        <div class="fold-arrow">▶</div>计算代码：基于 python 实现 0.618 法、斐波那契法、二分法      </div>      <div class="fold-collapse collapse" id="collapse-649e6489">        <div class="fold-content">          <p>迭代计算代码：</p><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br><span class="line">82</span><br><span class="line">83</span><br><span class="line">84</span><br><span class="line">85</span><br><span class="line">86</span><br><span class="line">87</span><br><span class="line">88</span><br><span class="line">89</span><br><span class="line">90</span><br><span class="line">91</span><br><span class="line">92</span><br><span class="line">93</span><br><span class="line">94</span><br><span class="line">95</span><br><span class="line">96</span><br><span class="line">97</span><br><span class="line">98</span><br><span class="line">99</span><br><span class="line">100</span><br><span class="line">101</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">class</span> <span class="hljs-title class_">ExactLinearSearch</span>:<br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">__init__</span>(<span class="hljs-params">self, </span><br><span class="hljs-params">                 a: <span class="hljs-built_in">float</span>, b: <span class="hljs-built_in">float</span>, delta: <span class="hljs-built_in">float</span>, </span><br><span class="hljs-params">                 f: <span class="hljs-type">Callable</span>[[<span class="hljs-built_in">float</span>], <span class="hljs-built_in">float</span>], </span><br><span class="hljs-params">                 criterion: <span class="hljs-built_in">str</span>=<span class="hljs-string">"0.618"</span>, max_iter=<span class="hljs-number">100</span></span>) -&gt; <span class="hljs-literal">None</span>:<br>        <br>        <span class="hljs-variable language_">self</span>.a = a<br>        <span class="hljs-variable language_">self</span>.b = b<br>        <span class="hljs-variable language_">self</span>.delta = delta<br>        <span class="hljs-variable language_">self</span>.f = f<br>        <span class="hljs-variable language_">self</span>.max_iter = max_iter<br>        <br>        <span class="hljs-keyword">if</span> criterion == <span class="hljs-string">"0.618"</span>:<br>            new_a, new_b, x_star, count = <span class="hljs-variable language_">self</span>._f_goad()<br>        <span class="hljs-keyword">elif</span> criterion == <span class="hljs-string">"fibonacci"</span>:<br>            new_a, new_b, x_star, count = <span class="hljs-variable language_">self</span>._f_fibo()<br>        <span class="hljs-keyword">else</span>: <span class="hljs-comment"># criterion == "binary"</span><br>            new_a, new_b, x_star, count = <span class="hljs-variable language_">self</span>._f_binary()<br><br>        fixed = <span class="hljs-keyword">lambda</span> x, acc: <span class="hljs-built_in">round</span>(x, acc)<br>        <span class="hljs-built_in">print</span>(<span class="hljs-string">f"算法为：“<span class="hljs-subst">{criterion}</span>” 法"</span>)<br>        <span class="hljs-built_in">print</span>(<span class="hljs-string">f"共迭代：<span class="hljs-subst">{count}</span> 次"</span>)<br>        <span class="hljs-built_in">print</span>(<span class="hljs-string">f"左边界: <span class="hljs-subst">{fixed(new_a, <span class="hljs-number">4</span>)}</span>"</span>)<br>        <span class="hljs-built_in">print</span>(<span class="hljs-string">f"右边界: <span class="hljs-subst">{fixed(new_b, <span class="hljs-number">4</span>)}</span>"</span>)<br>        <span class="hljs-built_in">print</span>(<span class="hljs-string">f"最优解: <span class="hljs-subst">{fixed(x_star, <span class="hljs-number">4</span>)}</span>"</span>)<br>        <span class="hljs-built_in">print</span>(<span class="hljs-string">f"最优值: <span class="hljs-subst">{fixed(f(x_star), <span class="hljs-number">6</span>)}</span>\n"</span>)<br>        <br>    <br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">_f_goad</span>(<span class="hljs-params">self</span>) -&gt; <span class="hljs-type">Tuple</span>[<span class="hljs-built_in">float</span>, <span class="hljs-built_in">float</span>, <span class="hljs-built_in">float</span>, <span class="hljs-built_in">int</span>]:<br>        a, b, delta, f = <span class="hljs-variable language_">self</span>.a, <span class="hljs-variable language_">self</span>.b, <span class="hljs-variable language_">self</span>.delta, <span class="hljs-variable language_">self</span>.f<br>        count = <span class="hljs-number">0</span><br><br>        lam = a + <span class="hljs-number">0.382</span> * (b - a)<br>        mu = b - <span class="hljs-number">0.382</span> * (b - a)<br><br>        <span class="hljs-keyword">while</span> count &lt;= <span class="hljs-variable language_">self</span>.max_iter:<br>            phi_lam = f(lam)<br>            phi_mu = f(mu)<br><br>            <span class="hljs-keyword">if</span> phi_lam &lt;= phi_mu:<br>                b = mu<br>            <span class="hljs-keyword">else</span>:<br>                a = lam<br>            <br>            lam = a + <span class="hljs-number">0.382</span> * (b - a)<br>            mu = b - <span class="hljs-number">0.382</span> * (b - a)<br><br>            <span class="hljs-keyword">if</span> b - lam &lt;= delta:<br>                <span class="hljs-keyword">return</span> a, b, lam, count<br>            <span class="hljs-keyword">if</span> mu - a &lt;= delta:<br>                <span class="hljs-keyword">return</span> a, b, mu, count<br>            <br>            count += <span class="hljs-number">1</span><br>        <br>        <span class="hljs-keyword">return</span> a, b, lam <span class="hljs-keyword">if</span> f(lam) &lt;= f(mu) <span class="hljs-keyword">else</span> mu, count<br><br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">_f_fibo</span>(<span class="hljs-params">self</span>) -&gt; <span class="hljs-type">Tuple</span>[<span class="hljs-built_in">float</span>, <span class="hljs-built_in">float</span>, <span class="hljs-built_in">float</span>, <span class="hljs-built_in">int</span>]:<br>        a, b, delta, f, max_iter = <span class="hljs-variable language_">self</span>.a, <span class="hljs-variable language_">self</span>.b, <span class="hljs-variable language_">self</span>.delta, <span class="hljs-variable language_">self</span>.f, <span class="hljs-variable language_">self</span>.max_iter<br>        count = <span class="hljs-literal">None</span><br><br>        F = [<span class="hljs-number">0.0</span>] * max_iter<br>        F[<span class="hljs-number">1</span>] = F[<span class="hljs-number">2</span>] = <span class="hljs-number">1</span><br>        <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">3</span>, max_iter):<br>            F[i] = F[i - <span class="hljs-number">1</span>] + F[i - <span class="hljs-number">2</span>]<br>            <span class="hljs-keyword">if</span> F[i] &gt;= (b - a) / delta:<br>                count = i - <span class="hljs-number">2</span><br>                <span class="hljs-keyword">break</span><br>        <br>        <span class="hljs-keyword">if</span> count == <span class="hljs-literal">None</span>:<br>            ValueError(<span class="hljs-string">"区间过大或精度过高导致，找不到合适的迭代次数"</span>)<br><br>        lam, mu = a, b<br>        <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">3</span>, count + <span class="hljs-number">1</span>):<br><br>            lam = a + (<span class="hljs-number">1</span> - F[i - <span class="hljs-number">1</span>] / F[i]) * (b - a)<br>            mu = b - (<span class="hljs-number">1</span> - F[i - <span class="hljs-number">1</span>] / F[i]) * (b - a)<br><br>            <span class="hljs-keyword">if</span> f(lam) &lt;= f(mu):<br>                b = mu<br>            <span class="hljs-keyword">else</span>:<br>                a = lam<br><br>        <span class="hljs-keyword">return</span> a, b, lam <span class="hljs-keyword">if</span> f(lam) &lt;= f(mu) <span class="hljs-keyword">else</span> mu, count<br>    <br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">_f_binary</span>(<span class="hljs-params">self</span>) -&gt; <span class="hljs-type">Tuple</span>[<span class="hljs-built_in">float</span>, <span class="hljs-built_in">float</span>, <span class="hljs-built_in">float</span>, <span class="hljs-built_in">int</span>]:<br>        a, b, delta, f, max_iter = <span class="hljs-variable language_">self</span>.a, <span class="hljs-variable language_">self</span>.b, <span class="hljs-variable language_">self</span>.delta, <span class="hljs-variable language_">self</span>.f, <span class="hljs-variable language_">self</span>.max_iter<br>        count = <span class="hljs-literal">None</span><br>        <br>        count = np.ceil(np.log2((b - a) / delta)).astype(<span class="hljs-built_in">int</span>)<br><br>        <span class="hljs-keyword">if</span> count &gt; max_iter:<br>            ValueError(<span class="hljs-string">"区间过大或精度过高导致迭代次数过高"</span>)<br>    <br>        <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(count):<br>            c = (a + b) / <span class="hljs-number">2.0</span><br>            <span class="hljs-keyword">if</span> f(c) &gt;= <span class="hljs-number">0.0</span>:<br>                b = c<br>            <span class="hljs-keyword">else</span>:<br>                a = c<br><br>        <span class="hljs-keyword">return</span> a, b, c, count<br></code></pre></td></tr></tbody></table></figure><p>分别调用 0.618 法、斐波那契法、二分法进行迭代计算：</p><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br></pre></td><td class="code"><pre><code class="hljs python">a, b, delta = -<span class="hljs-number">1</span>, <span class="hljs-number">1</span>, <span class="hljs-number">0.01</span><br>f = <span class="hljs-keyword">lambda</span> x: np.exp(-x) + np.exp(x)  <span class="hljs-comment"># 原函数</span><br>calc_goad = ExactLinearSearch(a, b, delta, f, criterion=<span class="hljs-string">"0.618"</span>)<br>calc_fibo = ExactLinearSearch(a, b, delta, f, criterion=<span class="hljs-string">"fibonacci"</span>)<br><br>a, b, delta = -<span class="hljs-number">3</span>, <span class="hljs-number">6</span>, <span class="hljs-number">0.1</span><br>f = <span class="hljs-keyword">lambda</span> x: <span class="hljs-number">2</span> * x + <span class="hljs-number">2</span>               <span class="hljs-comment"># 导函数</span><br>calc_bina = ExactLinearSearch(a, b, delta, f, criterion=<span class="hljs-string">"binary"</span>)<br></code></pre></td></tr></tbody></table></figure><p>计算结果为：</p><figure class="highlight makefile"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br></pre></td><td class="code"><pre><code class="hljs makefile">算法为：“0.618” 法<br>共迭代：10 次<br><span class="hljs-section">左边界: -0.0031</span><br><span class="hljs-section">右边界: 0.0069</span><br><span class="hljs-section">最优解: 0.0007</span><br><span class="hljs-section">最优值: 2.000001</span><br><br>算法为：“fibonacci” 法<br>共迭代：11 次<br><span class="hljs-section">左边界: -0.0225</span><br><span class="hljs-section">右边界: 0.0</span><br><span class="hljs-section">最优解: -0.0139</span><br><span class="hljs-section">最优值: 2.000193</span><br><br>算法为：“binary” 法<br>共迭代：7 次<br><span class="hljs-section">左边界: -1.0312</span><br><span class="hljs-section">右边界: -0.9609</span><br><span class="hljs-section">最优解: -0.9609</span><br><span class="hljs-section">最优值: 0.078125</span><br></code></pre></td></tr></tbody></table></figure>        </div>      </div>    </div><h4 id="3-2-1-0-618-法">3.2.1 0.618 法</h4><p>基于「函数值」进行选点。</p><p>按照黄金分割的比例取点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，不断迭代判断 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x_1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 的值，直到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>b</mi><mi>k</mi></msub><mo>−</mo><msub><mi>λ</mi><mi>k</mi></msub><mo>&lt;</mo><mi>δ</mi></mrow><annotation encoding="application/x-tex">b_k-\lambda_k&lt;\delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03785em;">δ</span></span></span></span> 或 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>μ</mi><mi>k</mi></msub><mo>−</mo><msub><mi>a</mi><mi>k</mi></msub><mo>&lt;</mo><mi>δ</mi></mrow><annotation encoding="application/x-tex">\mu_k-a_k&lt;\delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03785em;">δ</span></span></span></span> 则结束迭代，取最优解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">x^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span> 为对应的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\lambda_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 或 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>μ</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\mu_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 即可。</p><h4 id="3-2-2-Fibonacci-法">3.2.2 Fibonacci 法</h4><p>基于「函数值」进行选点。</p><p>第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 次迭代的区间长度是上一个区间长度的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><msub><mi>F</mi><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msub><msub><mi>F</mi><mrow><mi>n</mi><mo>−</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mfrac></mrow><annotation encoding="application/x-tex">\frac{F_{n-k}}{F_{n-k+1}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4284em;vertical-align:-0.4925em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9359em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:-0.1389em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2107em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4575em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:-0.1389em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2107em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4925em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>​，即：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>b</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>a</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mfrac><msub><mi>F</mi><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msub><msub><mi>F</mi><mrow><mi>n</mi><mo>−</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mfrac><mo stretchy="false">(</mo><msub><mi>b</mi><mi>k</mi></msub><mo>−</mo><msub><mi>a</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">b_{k+1} - a_{k+1} = \frac{F_{n-k}}{F_{n-k+1}} (b_{k} - a_{k})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9028em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2547em;vertical-align:-0.8943em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8943em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p><p>经过 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 次迭代后，得到区间 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>a</mi><mi>n</mi></msub><mo separator="true">,</mo><msub><mi>b</mi><mi>n</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[a_n,b_n]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span>，且 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>b</mi><mi>n</mi></msub><mo>−</mo><msub><mi>a</mi><mi>n</mi></msub><mo>≤</mo><mi>δ</mi></mrow><annotation encoding="application/x-tex">b_n-a_n \le \delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03785em;">δ</span></span></span></span>，于是可得：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>b</mi><mi>n</mi></msub><mo>−</mo><msub><mi>a</mi><mi>n</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><msub><mi>F</mi><mn>1</mn></msub><msub><mi>F</mi><mn>2</mn></msub></mfrac><mo stretchy="false">(</mo><msub><mi>b</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><msub><mi>F</mi><mn>1</mn></msub><msub><mi>F</mi><mn>2</mn></msub></mfrac><mfrac><msub><mi>F</mi><mn>2</mn></msub><msub><mi>F</mi><mn>3</mn></msub></mfrac><mo stretchy="false">(</mo><msub><mi>b</mi><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>−</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>⋯</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><msub><mi>F</mi><mn>1</mn></msub><msub><mi>F</mi><mn>2</mn></msub></mfrac><mfrac><msub><mi>F</mi><mn>2</mn></msub><msub><mi>F</mi><mn>3</mn></msub></mfrac><mo>⋯</mo><mfrac><msub><mi>F</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>F</mi><mi>n</mi></msub></mfrac><mo stretchy="false">(</mo><msub><mi>b</mi><mn>1</mn></msub><mo>−</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><msub><mi>F</mi><mn>1</mn></msub><msub><mi>F</mi><mi>n</mi></msub></mfrac><mo stretchy="false">(</mo><msub><mi>b</mi><mn>1</mn></msub><mo>−</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>≤</mo><mi>δ</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}b_n-a_n &amp;= \frac{F_1}{F_2}(b_{n-1}-a_{n-1}) \\&amp;= \frac{F_1}{F_2} \frac{F_2}{F_3}(b_{n-2}-a_{n-2}) \\&amp;= \cdots \\&amp;= \frac{F_1}{F_2} \frac{F_2}{F_3} \cdots \frac{F_{n-1}}{F_n}(b_{1}-a_{1}) \\&amp;= \frac{F_1}{F_n}(b_1-a_1) \le \delta\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:11.4853em;vertical-align:-5.4927em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.9927em;"><span style="top:-7.9927em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-5.4963em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"></span></span><span style="top:-3.5203em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"></span></span><span style="top:-1.5em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"></span></span><span style="top:0.9963em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.4927em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.9927em;"><span style="top:-7.9927em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.836em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-5.4963em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.836em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.836em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-3.5203em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="minner">⋯</span></span></span><span style="top:-1.5em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.836em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.836em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.836em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:0.9963em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.836em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03785em;">δ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.4927em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>在已知上界 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>δ</mi></mrow><annotation encoding="application/x-tex">\delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03785em;">δ</span></span></span></span> 的情况下可以直接计算出 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 的值，于是迭代 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 次即可得到最终的步长值 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\lambda_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 或 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>μ</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\mu_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p><h4 id="3-2-3-二分法">3.2.3 二分法</h4><p>基于「一阶导数」进行选点。</p><p>单调性存在于导数上。极值点左边导数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">&lt;0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，极值点右边导数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">&gt;0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，于是可以进行二分搜索。</p><h4 id="3-2-4-插值法">3.2.4 插值法</h4><p>基于「函数值、一阶导数」进行选点。</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404170915756.png" alt="三点二次插值法"></p><p><strong>三点二次插值法</strong>。给定初始迭代区间 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>a</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>b</mi><mn>0</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[a_0,b_0]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span> 和初始迭代解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>t</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">t_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7651em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>。每次在目标函数上取三个点来拟合一个二次函数，通过拟合出来的二次函数的最小值，来更新三个点为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>b</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>t</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">a_1,b_1,t_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，直到区间小于上界长度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>δ</mi></mrow><annotation encoding="application/x-tex">\delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03785em;">δ</span></span></span></span>，终止迭代，极小值就是当前状态二次函数的最小值。</p><p>那么我们如何求解当前状态二次函数的最小值呢？假定三个已知点为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>&lt;</mo><msub><mi>x</mi><mn>2</mn></msub><mo>&lt;</mo><msub><mi>x</mi><mn>3</mn></msub></mrow><annotation encoding="application/x-tex">x_1&lt;x_2&lt;x_3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，且 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>&gt;</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>&lt;</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>3</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x_1)&gt;f(x_2)&lt;f(x_3)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>，我们需要知道二次函数的三个参数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo separator="true">,</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">a,b,c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">c</span></span></span></span>，刚好三个已知点可以得到三个方程，从而解出二次函数的三个未知数。</p><p><strong>两点二次插值法</strong>。只不过少取了一个点，多利用了一个点的导数值来确定二次函数的三个参数罢了，其余迭代过程与三点二次插值完全一致。</p><h3 id="3-3-不精确线性搜索算法">3.3 不精确线性搜索算法</h3><div class="note note-light">            <p>精确搜索有时会导致搜索时间过长，尤其是当最优解离当前点还很远时。接下来我们介绍不精确线性搜索方法，在确保每一步的函数值都有充分下降的必要条件下，确保收敛并提升计算效率。真的有这么厉害的算法吗？其实根本逻辑很简单，<strong>Armijo 准则</strong>限定了搜索方向的上界、<strong>Goldstein 准则</strong>在前者的基础上又限定了搜索方向的下界（防止过小导致收敛过慢）、<strong>Wolfe 准则</strong>在前者的基础上又完善了下界的约束（确保不会把可行解排除在搜索区间外）</p>          </div><p>符号定义：我们在已知上一步解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">x_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 和下次迭代的下降方向 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">d_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的基础上，需要寻找合适的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span>。于是唯一变量就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span>，我们直接定义关于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 的一元函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><mi>α</mi><msub><mi>d</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(\alpha) = f(x_k + \alpha d_k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">ϕ</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>g</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">g_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 表示梯度。</p><h4 id="3-3-1-Armijo-准则">3.3.1 Armijo 准则</h4><p>只限定搜索上界，给定初始点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>、系数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span></span></span></span>：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><mi>α</mi><msub><mi>d</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mo>≤</mo><mi>ρ</mi><mo>⋅</mo><msubsup><mi>g</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>d</mi><mi>k</mi></msub><mo>⋅</mo><mi>α</mi><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x_k + \alpha d_k) \le \rho \cdot g_k^Td_k \cdot \alpha + f(x_k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1383em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p><h4 id="3-3-2-Goldstein-准则">3.3.2 Goldstein 准则</h4><p>又限定了搜索下界，同样给定初始点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>、系数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span></span></span></span>：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><mi>α</mi><msub><mi>d</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mo>≤</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>ρ</mi><mo>⋅</mo><msubsup><mi>g</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>d</mi><mi>k</mi></msub><mo>⋅</mo><mi>α</mi><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><mi>α</mi><msub><mi>d</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mo>≥</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>⋅</mo><msubsup><mi>g</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>d</mi><mi>k</mi></msub><mo>⋅</mo><mi>α</mi><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}f(x_k + \alpha d_k) \le &amp;\rho \cdot g_k^Td_k \cdot \alpha + f(x_k)\\f(x_k + \alpha d_k) \ge &amp;(1-\rho) \cdot g_k^Td_k \cdot \alpha + f(x_k)\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.1027em;vertical-align:-1.3013em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8013em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span></span></span><span style="top:-2.3587em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3013em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8013em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal">ρ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-2.3587em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">ρ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3013em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><h4 id="3-3-3-Wolfe-准则">3.3.3 Wolfe 准则</h4><p>完善了搜索下界的约束，即保证新点的梯度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>g</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mi>T</mi></msubsup><msub><mi>d</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">g_{k+1}^Td_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1828em;vertical-align:-0.3414em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3414em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 不低于老点梯度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>g</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>d</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">g_{k}^Td_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1244em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> 倍。给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub><mo separator="true">,</mo><mi>ρ</mi><mo separator="true">,</mo><mi>σ</mi></mrow><annotation encoding="application/x-tex">x_0,\rho,\sigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">ρ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><mi>α</mi><msub><mi>d</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mo>≤</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>ρ</mi><mo>⋅</mo><msubsup><mi>g</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>d</mi><mi>k</mi></msub><mo>⋅</mo><mi>α</mi><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>g</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><msub><mi>α</mi><mi>k</mi></msub><msub><mi>d</mi><mi>k</mi></msub><msup><mo stretchy="false">)</mo><mi>T</mi></msup><msub><mi>d</mi><mi>k</mi></msub><mo>≥</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>σ</mi><msubsup><mi>g</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>d</mi><mi>k</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mn>0</mn><mo>&lt;</mo><mi>ρ</mi><mo>&lt;</mo><mi>σ</mi><mo>&lt;</mo><mn>1</mn></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}f(x_k + \alpha d_k) \le &amp;\rho \cdot g_k^Td_k \cdot \alpha + f(x_k)\\g(x_k + \alpha_kd_k)^Td_k \ge&amp; \sigma g_{k}^Td_k\\0 &lt; \rho &lt; \sigma &lt; 1\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.6027em;vertical-align:-2.0513em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5513em;"><span style="top:-4.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span></span></span><span style="top:-3.1087em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span></span></span><span style="top:-1.6087em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">ρ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.0513em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5513em;"><span style="top:-4.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal">ρ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-3.1087em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.5513em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><blockquote><p>参考：</p><ul><li><a href="https://www.bilibili.com/video/BV11N411m75b/">路人 - 视频 - 更通俗的解释</a></li><li><a href="https://blog.csdn.net/luzhanbo207/article/details/121559905">路人 - 博客 - 更通俗的解释</a></li></ul></blockquote><h2 id="第四章「无约束最优化」">第四章「无约束最优化」</h2><div class="note note-light">            <p>本章介绍无约束函数的最优化算法。其中：</p><ul><li>最速下降法基于「一阶梯度」。最基本的方法</li><li>牛顿法基于「二阶梯度」。最主要的方法</li><li>共轭梯度法基于「一阶梯度」。解大型最优化问题的首选</li><li>拟牛顿法基于「函数值和一阶梯度」。尤其是其中的 BFGS 是目前最成功的方法</li></ul>          </div>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-6f5deacf" role="button" aria-expanded="false" aria-controls="collapse-6f5deacf">        <div class="fold-arrow">▶</div>极小化 Rosenbrock 函数的 python 代码      </div>      <div class="fold-collapse collapse" id="collapse-6f5deacf">        <div class="fold-content">          <p>目标函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span>、一阶梯度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span></span></span></span>、二阶梯度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">G</span></span></span></span>、初始点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">def</span> <span class="hljs-title function_">f</span>(<span class="hljs-params">x: np.ndarray</span>) -&gt; <span class="hljs-built_in">float</span>:<br>    <span class="hljs-keyword">return</span> (<span class="hljs-number">1</span> - x[<span class="hljs-number">0</span>])**<span class="hljs-number">2</span> + <span class="hljs-number">100</span> * (x[<span class="hljs-number">1</span>] - x[<span class="hljs-number">0</span>]**<span class="hljs-number">2</span>)**<span class="hljs-number">2</span><br><br><br><span class="hljs-keyword">def</span> <span class="hljs-title function_">g</span>(<span class="hljs-params">x: np.ndarray</span>) -&gt; np.ndarray:<br>    grad_x = -<span class="hljs-number">2</span> * (<span class="hljs-number">1</span> - x[<span class="hljs-number">0</span>]) - <span class="hljs-number">400</span> * x[<span class="hljs-number">0</span>] * (x[<span class="hljs-number">1</span>] - x[<span class="hljs-number">0</span>]**<span class="hljs-number">2</span>)<br>    grad_y = <span class="hljs-number">200</span> * (x[<span class="hljs-number">1</span>] - x[<span class="hljs-number">0</span>]**<span class="hljs-number">2</span>)<br>    <span class="hljs-keyword">return</span> np.array([grad_x, grad_y])<br><br><br><span class="hljs-keyword">def</span> <span class="hljs-title function_">G</span>(<span class="hljs-params">x: np.ndarray</span>) -&gt; np.ndarray:<br>    grad_xx = <span class="hljs-number">2</span> - <span class="hljs-number">400</span> * x[<span class="hljs-number">1</span>] + <span class="hljs-number">1200</span> * x[<span class="hljs-number">0</span>]**<span class="hljs-number">2</span><br>    grad_xy = -<span class="hljs-number">400</span> * x[<span class="hljs-number">0</span>]<br>    grad_yx = -<span class="hljs-number">400</span> * x[<span class="hljs-number">0</span>]<br>    grad_yy = <span class="hljs-number">200</span><br>    <span class="hljs-keyword">return</span> np.array([<br>        [grad_xx, grad_xy],<br>        [grad_yx, grad_yy]<br>    ])<br><br><br>initial_point = [-<span class="hljs-number">1.2</span>, <span class="hljs-number">1</span>]<br></code></pre></td></tr></tbody></table></figure><p>已知最优点为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">x^*=(1, 1)^T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0913em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span></span></span>，最优解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msup><mi>x</mi><mo>∗</mo></msup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f(x^*)=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，以书中例题初始点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1.2</mn><mo separator="true">,</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">(-1.2,1)^T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0913em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1.2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span></span></span> 为例开始迭代。</p><p><strong>一、最速下降法</strong></p><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">def</span> <span class="hljs-title function_">gradient_descent</span>(<span class="hljs-params">initial_point, max_iter=<span class="hljs-number">5</span>, eps=<span class="hljs-number">1e-6</span></span>):<br>    x = np.array(initial_point)<br>    points = [x]<br>    gradients = [g(x)]<br>    alphas = []<br><br>    <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(max_iter):<br>        grad = gradients[-<span class="hljs-number">1</span>]<br><br>        <span class="hljs-comment"># 搜索方向</span><br>        direction = -grad<br><br>        <span class="hljs-comment"># 步长因子：无法确定准确的步长最小化函数，因此此处采用二分法搜索最优步长</span><br>        alpha = <span class="hljs-number">1</span><br>        <span class="hljs-keyword">while</span> f(x + alpha * direction) &gt; f(x):<br>            alpha /= <span class="hljs-number">2</span><br><br>        x = x + alpha * direction<br>        points.append(x)<br>        gradients.append(g(x))<br>        alphas.append(alpha)<br><br>        <span class="hljs-keyword">if</span> np.linalg.norm(grad) &lt; eps:<br>            <span class="hljs-keyword">break</span><br><br>    <span class="hljs-keyword">return</span> points, gradients, alphas<br><br><br>points, gradients, alphas = gradient_descent(initial_point, max_iter=<span class="hljs-number">100</span>, eps=<span class="hljs-number">1e-6</span>)<br><br><span class="hljs-keyword">for</span> i, (point, grad, alpha) <span class="hljs-keyword">in</span> <span class="hljs-built_in">enumerate</span>(<span class="hljs-built_in">zip</span>(points, gradients, [<span class="hljs-number">1</span>] + alphas)):<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"Iteration <span class="hljs-subst">{i}</span>:"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Point       = <span class="hljs-subst">{point}</span>"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Gradient    = <span class="hljs-subst">{grad}</span>"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Step Size   = <span class="hljs-subst">{alpha}</span>"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Direction   = <span class="hljs-subst">{-grad}</span>"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Function Val= <span class="hljs-subst">{f(point)}</span>\n"</span>)<br></code></pre></td></tr></tbody></table></figure><p>迭代100次后输出为：</p><figure class="highlight makefile"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br></pre></td><td class="code"><pre><code class="hljs makefile">Iteration 98:<br>  Point       = [0.93432802 0.87236513]<br>  Gradient    = [ 0.0942865  -0.12074477]<br>  Step Size   = 0.00390625<br>  Direction   = [-0.0942865   0.12074477]<br>  Function Val= 0.004349256784446673<br><br>Iteration 99:<br>  Point       = [0.93414387 0.87260096]<br>  Gradient    = [-0.12281587 -0.00476179]<br>  Step Size   = 0.001953125<br>  Direction   = [0.12281587 0.00476179]<br>  Function Val= 0.004337086557103718<br><br>Iteration 100:<br>  Point       = [0.93438374 0.87261026]<br>  Gradient    = [ 0.04171114 -0.09254423]<br>  Step Size   = 0.001953125<br>  Direction   = [-0.04171114  0.09254423]<br>  Function Val= 0.004326904052586884<br></code></pre></td></tr></tbody></table></figure><p><strong>二、牛顿法</strong></p><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">def</span> <span class="hljs-title function_">newton_method</span>(<span class="hljs-params">initial_point, max_iter=<span class="hljs-number">5</span>, eps=<span class="hljs-number">1e-6</span></span>):<br>    x = np.array(initial_point)<br>    points = [x]<br>    gradients = [g(x)]<br>    Hessians = [G(x)]<br><br>    <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(max_iter):<br>        grad = gradients[-<span class="hljs-number">1</span>]<br>        Hessian = Hessians[-<span class="hljs-number">1</span>]<br><br>        <span class="hljs-comment"># 搜索方向</span><br>        direction = np.linalg.inv(Hessian) @ grad<br><br>        <span class="hljs-comment"># 步长因子：假定使用固定步长的牛顿法</span><br>        alpha = <span class="hljs-number">1</span><br><br>        x = x - alpha * direction<br>        points.append(x)<br>        gradients.append(g(x))<br>        Hessians.append(G(x))<br><br>        <span class="hljs-keyword">if</span> np.linalg.norm(grad) &lt; eps:<br>            <span class="hljs-keyword">break</span><br><br>    <span class="hljs-keyword">return</span> points, gradients, Hessians<br><br><br>points, gradients, Hessians = newton_method(initial_point, max_iter=<span class="hljs-number">50</span>, eps=<span class="hljs-number">1e-6</span>)<br><br><span class="hljs-keyword">for</span> i, (point, grad, Hessian) <span class="hljs-keyword">in</span> <span class="hljs-built_in">enumerate</span>(<span class="hljs-built_in">zip</span>(points, gradients, Hessians)):<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"Iteration <span class="hljs-subst">{i}</span>:"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Point       = <span class="hljs-subst">{point}</span>"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Gradient    = <span class="hljs-subst">{grad}</span>"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Hessian     = <span class="hljs-subst">{Hessian}</span>"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Function Val= <span class="hljs-subst">{f(point)}</span>\n"</span>)<br></code></pre></td></tr></tbody></table></figure><p>迭代7次即收敛：</p><figure class="highlight makefile"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br></pre></td><td class="code"><pre><code class="hljs makefile">Iteration 5:<br>  Point       = [0.9999957  0.99999139]<br>  Gradient    = [-8.60863343e-06 -2.95985458e-11]<br>  Hessian     = [[ 801.99311306 -399.99827826]<br> [-399.99827826  200.        ]]<br>  Function Val= 1.8527397192178054e-11<br><br>Iteration 6:<br>  Point       = [1. 1.]<br>  Gradient    = [ 7.41096051e-09 -3.70548037e-09]<br>  Hessian     = [[ 802.00000001 -400.        ]<br> [-400.          200.        ]]<br>  Function Val= 3.4326461875363225e-20<br><br>Iteration 7:<br>  Point       = [1. 1.]<br>  Gradient    = [-0.  0.]<br>  Hessian     = [[ 802. -400.]<br> [-400.  200.]]<br>  Function Val= 0.0<br></code></pre></td></tr></tbody></table></figure><p><strong>三、共轭梯度法</strong></p><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">def</span> <span class="hljs-title function_">conjugate_gradient</span>(<span class="hljs-params">initial_point, max_iter=<span class="hljs-number">5</span>, eps=<span class="hljs-number">1e-6</span></span>):<br>    x = np.array(initial_point)<br>    points = [x]<br>    gradients = [g(x)]<br>    directions = [-g(x)]<br>    alphas = []<br><br>    <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(max_iter):<br>        grad = gradients[-<span class="hljs-number">1</span>]<br><br>        <span class="hljs-comment"># 搜索方向：FR公式</span><br>        <span class="hljs-keyword">if</span> i == <span class="hljs-number">0</span>:<br>            direction = -grad<br>        <span class="hljs-keyword">else</span>:<br>            beta = np.dot(g(x), g(x)) / np.dot(g(points[-<span class="hljs-number">2</span>]), g(points[-<span class="hljs-number">2</span>]))<br>            direction = -g(x) + beta * direction<br><br>        <span class="hljs-comment"># 步长因子：精确线搜索直接得到闭式解</span><br>        alpha = -np.dot(grad, direction) / np.dot(direction, G(x) @ direction)<br>        <br><br>        x = x + alpha * direction<br><br>        directions.append(direction)<br>        alphas.append(alpha)<br>        points.append(x)<br>        gradients.append(g(x))<br><br>        <span class="hljs-keyword">if</span> np.linalg.norm(grad) &lt; eps:<br>            <span class="hljs-keyword">break</span><br><br>    <span class="hljs-keyword">return</span> points, gradients, alphas<br><br><br>points, gradients, alphas = conjugate_gradient(initial_point, max_iter=<span class="hljs-number">1000</span>, eps=<span class="hljs-number">1e-6</span>)<br><br><span class="hljs-keyword">for</span> i, (point, grad, alpha) <span class="hljs-keyword">in</span> <span class="hljs-built_in">enumerate</span>(<span class="hljs-built_in">zip</span>(points, gradients, alphas)):<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"Iteration <span class="hljs-subst">{i}</span>:"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Point       = <span class="hljs-subst">{point}</span>"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Gradient    = <span class="hljs-subst">{grad}</span>"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Step Size   = <span class="hljs-subst">{alpha}</span>"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Direction   = <span class="hljs-subst">{-grad}</span>"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Function Val= <span class="hljs-subst">{f(point)}</span>\n"</span>)<br></code></pre></td></tr></tbody></table></figure><p>迭代1000次后输出为：</p><figure class="highlight makefile"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br></pre></td><td class="code"><pre><code class="hljs makefile">Iteration 997:<br>  Point       = [0.9999994  0.99999875]<br>  Gradient    = [ 1.90794906e-05 -1.01414007e-05]<br>  Step Size   = 0.0005161468619784313<br>  Direction   = [-1.90794906e-05  1.01414007e-05]<br>  Function Val= 6.191018745155016e-13<br><br>Iteration 998:<br>  Point       = [0.99999931 0.99999861]<br>  Gradient    = [ 5.43686111e-06 -3.40374227e-06]<br>  Step Size   = 0.0005078748917694624<br>  Direction   = [-5.43686111e-06  3.40374227e-06]<br>  Function Val= 4.986125950068217e-13<br><br>Iteration 999:<br>  Point       = [0.9999993 0.9999986]<br>  Gradient    = [ 1.34784246e-06 -1.36924747e-06]<br>  Step Size   = 0.0005356250138048412<br>  Direction   = [-1.34784246e-06  1.36924747e-06]<br>  Function Val= 4.881643528976312e-13<br></code></pre></td></tr></tbody></table></figure><p><strong>四、拟牛顿法</strong></p><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">def</span> <span class="hljs-title function_">bfgs</span>(<span class="hljs-params">initial_point, max_iter=<span class="hljs-number">5</span>, eps=<span class="hljs-number">1e-6</span></span>):<br>    x = np.array(initial_point)<br>    points = [x]<br>    gradients = [g(x)]<br>    B = G(x)<br><br>    <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(max_iter):<br>        grad = gradients[-<span class="hljs-number">1</span>]<br><br>        <span class="hljs-comment"># 迭代公式</span><br>        x = x - np.linalg.inv(B) @ grad<br><br>        <span class="hljs-comment"># 更新 B 矩阵</span><br>        s = x - points[-<span class="hljs-number">1</span>]<br>        y = g(x) - gradients[-<span class="hljs-number">1</span>]<br>        B = B + np.outer(y, y) / (s @ y) - (B @ np.outer(s, s) @ B) / (s @ B @ s)<br><br>        points.append(x)<br>        gradients.append(g(x))<br><br>        <span class="hljs-keyword">if</span> np.linalg.norm(grad) &lt; eps:<br>            <span class="hljs-keyword">break</span><br><br>    <span class="hljs-keyword">return</span> points, gradients<br><br><br>points, gradients = bfgs(initial_point, max_iter=<span class="hljs-number">1000</span>, eps=<span class="hljs-number">1e-6</span>)<br><br><span class="hljs-keyword">for</span> i, (point, grad) <span class="hljs-keyword">in</span> <span class="hljs-built_in">enumerate</span>(<span class="hljs-built_in">zip</span>(points, gradients)):<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"Iteration <span class="hljs-subst">{i}</span>:"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Point       = <span class="hljs-subst">{point}</span>"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Gradient    = <span class="hljs-subst">{grad}</span>"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Function Val= <span class="hljs-subst">{f(point)}</span>\n"</span>)<br></code></pre></td></tr></tbody></table></figure><p>迭代 78 次收敛：</p><figure class="highlight makefile"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br></pre></td><td class="code"><pre><code class="hljs makefile">Iteration 76:<br>  Point       = [1.00000075 0.99999921]<br>  Gradient    = [ 0.000918   -0.00045825]<br>  Function Val= 5.255482679080297e-10<br><br>Iteration 77:<br>  Point       = [1.00000006 1.00000012]<br>  Gradient    = [ 6.46055099e-07 -2.63592925e-07]<br>  Function Val= 3.7061757712619374e-15<br><br>Iteration 78:<br>  Point       = [1. 1.]<br>  Gradient    = [6.75185684e-10 4.68913797e-10]<br>  Function Val= 6.510026595267928e-19<br></code></pre></td></tr></tbody></table></figure>        </div>      </div>    </div><h3 id="4-1-最速下降法">4.1 最速下降法</h3><p>放一张生动的图：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406132016327.png" alt="最速下降法 - 迭代示意图"></p><p>迭代公式：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>x</mi><mi>k</mi></msub><mo>−</mo><msub><mi>α</mi><mi>k</mi></msub><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x_{k+1} = x_k - \alpha_k \nabla f(x_k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p><p>每次迭代时，<strong>下降方向</strong> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">d_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 采用当前解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">x_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 处的负梯度方向 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">- \nabla f(x_k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>，<strong>步长因子</strong> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>α</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 采用精确线性搜索算法的计算结果。</p><p>易证相邻迭代解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>k</mi></msub><mo separator="true">,</mo><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">x_k,x_{k+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span> 的方向 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mi>k</mi></msub><mo separator="true">,</mo><msub><mi>d</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">d_k,d_{k+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9028em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span> 是正交的：由于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><mi>α</mi><msub><mi>d</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(\alpha) = f(x_k + \alpha d_k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">ϕ</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>，在采用线搜索找最优步长时，步长的搜索结果 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>α</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 即为使得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>ϕ</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\phi'(\alpha)=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">ϕ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 的解，于是可得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>=</mo><msup><mi>ϕ</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>ϕ</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><msub><mi>α</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><msub><mi>α</mi><mi>k</mi></msub><msub><mi>d</mi><mi>k</mi></msub><mo stretchy="false">)</mo><msub><mi>d</mi><mi>k</mi></msub><mo>=</mo><mo>−</mo><msubsup><mi>d</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mi>T</mi></msubsup><mo>⋅</mo><msub><mi>d</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">0=\phi'(\alpha) = \phi'(\alpha_k) = \nabla f(x_k+\alpha_k d_k)d_k = -d_{k+1}^T \cdot d_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">ϕ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">ϕ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1828em;vertical-align:-0.3414em;"></span><span class="mord">−</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3414em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>d</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mi>T</mi></msubsup><mo>⋅</mo><msub><mi>d</mi><mi>k</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d_{k+1}^T \cdot d_k = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1828em;vertical-align:-0.3414em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3414em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>。如图：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406132046068.png" alt="相邻迭代解的方向正交"></p><p>也正因为搜索方向正交的特性导致最速下降法的收敛速度往往不尽如人意。优点在于程序设计简单并且计算和存储量都不大，以及对初始点的要求不高。</p><h3 id="4-2-牛顿法">4.2 牛顿法</h3><p>放一张生动的图：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406140902324.gif" alt="牛顿法 - 迭代示意图"></p><div class="note note-warning">            <p>注：本图演示的是求解一维函数零点的迭代过程，需要使得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f(x)=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，因此比值式为原函数除以导函数。后续介绍的是极小化函数的过程，需要使得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f'(x)=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，因此比值式为一阶导数除以二阶导数，高维就是二阶梯度的逆乘以一阶梯度。</p>          </div><p>迭代公式：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>x</mi><mi>k</mi></msub><mo>−</mo><msup><mi mathvariant="normal">∇</mi><mn>2</mn></msup><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><msup><mo stretchy="false">)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x_{k+1} = x_k - \nabla^2f(x_k)^{-1}\nabla f(x_k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord">∇</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p><p>牛顿法相较于最速下降法有了更快的收敛速度，但是由于需要计算和存储海塞矩阵导致计算量增加并且有些目标函数可能根本求不出二阶梯度。同时牛顿法对于初始迭代点的选择比最速下降法要苛刻的多。</p><h3 id="4-3-共轭梯度法">4.3 共轭梯度法</h3><p>我们利用共轭梯度法解决「正定二次函数」的极小化问题。由于最速下降法中相邻迭代点的方向是正交的导致搜索效率下降，牛顿法又由于需要计算和存储海塞矩阵导致存储开销过大，共轭梯度法的核心思想是<strong>相邻两个迭代点的搜索方向是关于正定二次型的正定阵正交的</strong>。这样既保证了迭代收敛的速度也避免了计算存储海塞矩阵的开销。美中不足的是当共轭梯度法解决其他问题是往往会出现对线搜索的过度依赖，一旦线搜索变差会导致整个迭代过程精度变差。</p><p>概念补充：</p><ol><li>共轭：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 与 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> 共轭当且仅当 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mi>T</mi></msup><mi>G</mi><mi>y</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x^TGy=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0358em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，其中 G 为对称正定阵</li><li>正定二次：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mi>T</mi></msup><mi>G</mi><mi>x</mi><mo>−</mo><msup><mi>b</mi><mi>T</mi></msup><mi>x</mi><mo>+</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">f(x) =\frac{1}{2} x^TGx - b^Tx + c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.9247em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span></span></span></span></li></ol><p>公式推导：</p><ol><li><p>首先给定初始迭代点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，收敛阈值 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ϵ</span></span></span></span>，迭代公式是不变的：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><msub><mi>α</mi><mi>k</mi></msub><msub><mi>d</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">x_{k+1} = x_k + \alpha_k d_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，关键在于计算每一次迭代过程中的步长因子 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>α</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>​ 和搜索方向 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">d_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>​</p></li><li><p>确定步长因子 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>α</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>α</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mrow><mi>min</mi><mo>⁡</mo></mrow><mi>α</mi></munder><mi>ϕ</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mrow><mi>min</mi><mo>⁡</mo></mrow><mi>α</mi></munder><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><mi>α</mi><msub><mi>d</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mrow><mi>min</mi><mo>⁡</mo></mrow><mi>α</mi></munder><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><mi>α</mi><msub><mi>d</mi><mi>k</mi></msub><msup><mo stretchy="false">)</mo><mi>T</mi></msup><mi>G</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><mi>α</mi><msub><mi>d</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mo>−</mo><msup><mi>b</mi><mi>T</mi></msup><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><mi>α</mi><msub><mi>d</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mo>+</mo><mi>c</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mrow><mi>min</mi><mo>⁡</mo></mrow><mi>α</mi></munder><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><msubsup><mi>x</mi><mi>k</mi><mi>T</mi></msubsup><mi>G</mi><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><mn>2</mn><mi>α</mi><msubsup><mi>x</mi><mi>k</mi><mi>T</mi></msubsup><mi>G</mi><msub><mi>d</mi><mi>k</mi></msub><mo>+</mo><msup><mi>α</mi><mn>2</mn></msup><msubsup><mi>d</mi><mi>K</mi><mi>T</mi></msubsup><msub><mi>d</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mo>−</mo><msup><mi>b</mi><mi>T</mi></msup><msub><mi>x</mi><mi>k</mi></msub><mo>−</mo><msup><mi>b</mi><mi>T</mi></msup><mi>α</mi><msub><mi>d</mi><mi>k</mi></msub><mo>+</mo><mi>c</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mrow><mi>min</mi><mo>⁡</mo></mrow><mi>α</mi></munder><mfrac><mn>1</mn><mn>2</mn></mfrac><msubsup><mi>x</mi><mi>k</mi><mi>T</mi></msubsup><mi>G</mi><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><mi>α</mi><msubsup><mi>x</mi><mi>k</mi><mi>T</mi></msubsup><mi>G</mi><msub><mi>d</mi><mi>k</mi></msub><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>α</mi><mn>2</mn></msup><msubsup><mi>d</mi><mi>k</mi><mi>T</mi></msubsup><mi>G</mi><msub><mi>d</mi><mi>k</mi></msub><mo>−</mo><msup><mi>b</mi><mi>T</mi></msup><mi>α</mi><msub><mi>d</mi><mi>k</mi></msub><mo>+</mo><mi>c</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\alpha &amp;= \min_{\alpha} \phi(\alpha) \\&amp;= \min_{\alpha} f(x_k+\alpha d_k) \\&amp;= \min_{\alpha} \frac{1}{2}(x_k+\alpha d_k)^TG(x_k+\alpha d_k) - b^T(x_k+\alpha d_k) + c \\&amp;= \min_{\alpha} \frac{1}{2} (x_k^TGx_k + 2\alpha x_k^T G d_k + \alpha^2d_K^Td_k) - b^Tx_k - b^T \alpha d_k + c\\&amp;= \min_{\alpha} \frac{1}{2} x_k^TGx_k + \alpha x_k^T G d_k + \frac{1}{2}\alpha^2d_k^T G d_k - b^T \alpha d_k + c\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:10.6443em;vertical-align:-5.0722em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.5722em;"><span style="top:-8.0536em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span><span style="top:-6.2136em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"></span></span><span style="top:-3.8922em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"></span></span><span style="top:-1.5707em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"></span></span><span style="top:0.7507em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.0722em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.5722em;"><span style="top:-8.0536em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">min</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">ϕ</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mclose">)</span></span></span><span style="top:-6.2136em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">min</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-3.8922em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">min</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">c</span></span></span><span style="top:-1.5707em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">min</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">K</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">c</span></span></span><span style="top:0.7507em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">min</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.0722em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>由于目标函数是正定二次型，显然可以直接求出步长因子的闭式解：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi>d</mi><mi>ϕ</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><mi>α</mi></mrow></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msubsup><mi>x</mi><mi>k</mi><mi>T</mi></msubsup><mi>G</mi><msub><mi>d</mi><mi>k</mi></msub><mo>+</mo><mi>α</mi><msubsup><mi>d</mi><mi>k</mi><mi>T</mi></msubsup><mi>G</mi><msub><mi>d</mi><mi>k</mi></msub><mo>−</mo><msup><mi>b</mi><mi>T</mi></msup><msub><mi>d</mi><mi>k</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>0</mn></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\frac{d \phi(\alpha)}{d\alpha} &amp;= x_k^T G d_k + \alpha d_k^T G d_k - b^Td_k \\&amp;=0\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.913em;vertical-align:-1.7065em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.2065em;"><span style="top:-4.2065em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">ϕ</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.3805em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.7065em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.2065em;"><span style="top:-4.2065em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.3805em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.7065em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>于是可以导出当前的步长因子 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>α</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的闭式解为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi>α</mi><mi>k</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><msup><mi>b</mi><mi>T</mi></msup><mo>−</mo><msup><mi>x</mi><mi>T</mi></msup><mi>G</mi><mo stretchy="false">)</mo><msub><mi>d</mi><mi>k</mi></msub></mrow><mrow><msubsup><mi>d</mi><mi>k</mi><mi>T</mi></msubsup><mi>G</mi><msub><mi>d</mi><mi>k</mi></msub></mrow></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>−</mo><mfrac><mrow><msubsup><mi>g</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>d</mi><mi>k</mi></msub></mrow><mrow><msubsup><mi>d</mi><mi>k</mi><mi>T</mi></msubsup><mi>G</mi><msub><mi>d</mi><mi>k</mi></msub></mrow></mfrac></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\alpha_k &amp;= \frac{(b^T - x^TG)d_k}{d_k^TGd_k} \\&amp;= -\frac{g_k^T d_k}{d_k^TGd_k}\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.6655em;vertical-align:-2.5828em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.0828em;"><span style="top:-5.0828em;"><span class="pstrut" style="height:3.5183em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.5183em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.5828em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.0828em;"><span style="top:-5.0828em;"><span class="pstrut" style="height:3.5183em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5183em;"><span style="top:-2.2869em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8231em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0144em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.5183em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5183em;"><span style="top:-2.2869em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8231em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0144em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.5828em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li><li><p>确定搜索方向 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">d_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>d</mi><mi>k</mi></msub><mo>=</mo><mo>−</mo><msub><mi>g</mi><mi>k</mi></msub><mo>+</mo><mi>β</mi><msub><mi>d</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">d_k = -g_k + \beta d_{k-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord">−</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.9028em;vertical-align:-0.2083em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span></span></p><p>可见只需要确定组合系数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span>。由于共轭梯度法遵循相邻迭代点的搜索方向共轭，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>d</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mi>T</mi></msubsup><mi>G</mi><msub><mi>d</mi><mi>k</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d_{k-1}^TGd_k=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1828em;vertical-align:-0.3414em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3414em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，因此对上式两侧同时左乘 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>d</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mi>T</mi></msubsup><mi>G</mi></mrow><annotation encoding="application/x-tex">d_{k-1}^TG</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1828em;vertical-align:-0.3414em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3414em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span></span></span></span>，有：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msubsup><mi>d</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mi>T</mi></msubsup><mi>G</mi><msub><mi>d</mi><mi>k</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>−</mo><msubsup><mi>d</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mi>T</mi></msubsup><mi>G</mi><msub><mi>g</mi><mi>k</mi></msub><mo>+</mo><mi>β</mi><msubsup><mi>d</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mi>T</mi></msubsup><mi>G</mi><msub><mi>d</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>0</mn></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}d_{k-1}^TGd_k &amp;= -d_{k-1}^TGg_k + \beta d_{k-1}^TGd_{k-1} \\&amp;= 0\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0513em;vertical-align:-1.2757em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7757em;"><span style="top:-3.8843em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.3843em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2757em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7757em;"><span style="top:-3.8843em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">−</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.3843em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2757em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>于是可得当前的组合系数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span> 为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>β</mi><mo>=</mo><mfrac><mrow><msubsup><mi>d</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mi>T</mi></msubsup><mi>G</mi><msub><mi>g</mi><mi>k</mi></msub></mrow><mrow><msubsup><mi>d</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mi>T</mi></msubsup><mi>G</mi><msub><mi>d</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">\beta = \frac{d_{k-1}^TGg_k}{d_{k-1}^TGd_{k-1}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.6455em;vertical-align:-1.0728em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5728em;"><span style="top:-2.2869em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8231em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3596em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.7314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3414em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0728em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p> <div class="note note-warning">            <p>上述组合系数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span> 的结果是共轭梯度最原始的表达式，后人又进行了变形，<s>没证出来，难崩，直接背吧</s>，给出 FR 的组合系数表达式：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>β</mi><mo>=</mo><mfrac><mrow><msubsup><mi>g</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>g</mi><mi>k</mi></msub></mrow><mrow><msubsup><mi>g</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mi>T</mi></msubsup><msub><mi>g</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mfrac></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\beta = \frac{g_k^Tg_k}{g_{k-1}^T g_{k-1}}\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.8911em;vertical-align:-1.1956em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6956em;"><span style="top:-3.6956em;"><span class="pstrut" style="height:3.5183em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5183em;"><span style="top:-2.2869em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8231em;"><span style="top:-2.3987em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3596em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0728em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.1956em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>          </div><p>当然了由于初始迭代时没有前一个搜索方向，因此直接用初始点的梯度作为搜索方向，即：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>d</mi><mn>0</mn></msub><mo>=</mo><mo>−</mo><msub><mi>g</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">d_0=-g_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord">−</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p><p>于是可以导出当前的搜索方向 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">d_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的闭式解为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>d</mi><mi>k</mi></msub><mo>=</mo><mo>−</mo><msub><mi>g</mi><mi>k</mi></msub><mo>+</mo><mfrac><mrow><msubsup><mi>g</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>g</mi><mi>k</mi></msub></mrow><mrow><msubsup><mi>g</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mi>T</mi></msubsup><msub><mi>g</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mfrac><msub><mi>d</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">d_k = -g_k + \frac{g_k^Tg_k}{g_{k-1}^T g_{k-1}} d_{k-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord">−</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.5911em;vertical-align:-1.0728em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5183em;"><span style="top:-2.2869em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8231em;"><span style="top:-2.3987em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3596em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0728em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span></span></p></li></ol><p>迭代公式：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><msub><mi>α</mi><mi>k</mi></msub><msub><mi>d</mi><mi>k</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mo lspace="0em" rspace="0em">=</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><mo stretchy="false">(</mo><mo>−</mo><mfrac><mrow><msubsup><mi>g</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>d</mi><mi>k</mi></msub></mrow><mrow><msubsup><mi>d</mi><mi>k</mi><mi>T</mi></msubsup><mi>G</mi><msub><mi>d</mi><mi>k</mi></msub></mrow></mfrac><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo>−</mo><msub><mi>g</mi><mi>k</mi></msub><mo>+</mo><mfrac><mrow><msubsup><mi>g</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>g</mi><mi>k</mi></msub></mrow><mrow><msubsup><mi>g</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mi>T</mi></msubsup><msub><mi>g</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mfrac><msub><mi>d</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}x_{k+1} =&amp; x_k + \alpha_k d_k\\=&amp; x_k + (-\frac{g_k^T d_k}{d_k^TGd_k}) (-g_k + \frac{g_k^Tg_k}{g_{k-1}^T g_{k-1}} d_{k-1})\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.3911em;vertical-align:-1.9456em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.4456em;"><span style="top:-5.1239em;"><span class="pstrut" style="height:3.5183em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span><span style="top:-2.9456em;"><span class="pstrut" style="height:3.5183em;"></span><span class="mord"><span class="mrel">=</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.9456em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.4456em;"><span style="top:-5.1239em;"><span class="pstrut" style="height:3.5183em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.9456em;"><span class="pstrut" style="height:3.5183em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5183em;"><span style="top:-2.2869em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8231em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0144em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span><span class="mopen">(</span><span class="mord">−</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5183em;"><span style="top:-2.2869em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8231em;"><span style="top:-2.3987em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3596em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0728em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.9456em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><h3 id="4-4-拟牛顿法">4.4 拟牛顿法</h3><p>4.1 和 4.2 介绍的基于一阶梯度和二阶梯度的下降法都可以统一成下面的表达式：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>x</mi><mi>k</mi></msub><mo>−</mo><msub><mi>α</mi><mi>k</mi></msub><msub><mi>B</mi><mi>k</mi></msub><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x_{k+1} = x_k - \alpha_k B_k \nabla f(x_k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p><ul><li>4.1 的最速下降法的步长因子通过精确线搜索获得，海塞矩阵的逆 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>B</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">B_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 不存在，可以看做为单位阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span></span></span></span></li><li>4.2 的牛顿法的步长因子同样可以通过精确线搜索获得，当然也可以设置为定值，海塞矩阵的逆 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>B</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">B_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 对应二阶梯度的逆 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msup><mi mathvariant="normal">∇</mi><mn>2</mn></msup><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">(\nabla^2 f(x_k))^{-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord">∇</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span></li></ul><p>前者收敛速度差、后者计算量和存储量大，我们尝试构造一个对称正定阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>B</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">B_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 来近似代替二阶梯度的逆，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>B</mi><mi>k</mi></msub><mo>≈</mo><mo stretchy="false">(</mo><msup><mi mathvariant="normal">∇</mi><mn>2</mn></msup><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">B_k \approx (\nabla^2 f(x_k))^{-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord">∇</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span>，使得该法具备较快的收敛速度与较少的内存开销。</p><p>介绍最著名的 <strong>BFGS</strong> 拟牛顿法，它的核心思想是每次迭代过程中对其进行快速校正，从而在确保收敛速度的情况下提升计算效率。迭代公式如下：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>x</mi><mi>k</mi></msub><mo>−</mo><msubsup><mi>B</mi><mi>k</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><msub><mi>g</mi><mi>k</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mtext>记:&nbsp;</mtext><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>s</mi><mi>k</mi></msub><mo>=</mo><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>x</mi><mi>k</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>y</mi><mi>k</mi></msub><mo>=</mo><msub><mi>g</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>g</mi><mi>k</mi></msub></mrow></mstyle></mtd></mtr></mtable></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mtext>则:&nbsp;</mtext><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>B</mi><mn>0</mn></msub><mo>=</mo><msup><mi mathvariant="normal">∇</mi><mn>2</mn></msup><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><msub><mi>B</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>B</mi><mi>k</mi></msub><mo>+</mo><mfrac><mrow><msub><mi>y</mi><mi>k</mi></msub><msubsup><mi>y</mi><mi>k</mi><mi>T</mi></msubsup></mrow><mrow><msubsup><mi>s</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>y</mi><mi>k</mi></msub></mrow></mfrac><mo>−</mo><mfrac><mrow><msub><mi>B</mi><mi>k</mi></msub><msub><mi>s</mi><mi>k</mi></msub><msubsup><mi>s</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>B</mi><mi>k</mi></msub></mrow><mrow><msubsup><mi>s</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>B</mi><mi>k</mi></msub><msub><mi>s</mi><mi>k</mi></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr></mtable></mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}&amp;x_{k+1} = x_k - B_k^{-1}g_k \\&amp;\text{记: }\begin{cases}s_k= x_{k+1} - x_k\\y_k=g_{k+1} - g_k\end{cases}\\&amp;\text{则: }\begin{cases}B_0 = \nabla^2f(x_0)\\\displaystyle B_{k+1} = B_k + \frac{y_ky_k^T}{s_k^Ty_k} - \frac{B_ks_ks_k^TB_k}{s_k^TB_ks_k}\end{cases}\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:9.0969em;vertical-align:-4.2985em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.7985em;"><span style="top:-8.1707em;"><span class="pstrut" style="height:4.2364em;"></span><span class="mord"></span></span><span style="top:-5.7607em;"><span class="pstrut" style="height:4.2364em;"></span><span class="mord"></span></span><span style="top:-1.9743em;"><span class="pstrut" style="height:4.2364em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.2985em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.7985em;"><span style="top:-8.1707em;"><span class="pstrut" style="height:4.2364em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.4086em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2914em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-5.7607em;"><span class="pstrut" style="height:4.2364em;"></span><span class="mord"><span class="mord"></span><span class="mord text"><span class="mord cjk_fallback">记</span><span class="mord">:&nbsp;</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-1.9743em;"><span class="pstrut" style="height:4.2364em;"></span><span class="mord"><span class="mord"></span><span class="mord text"><span class="mord cjk_fallback">则</span><span class="mord">:&nbsp;</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-2.5em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.492em;"><span class="pstrut" style="height:3.15em;"></span><span style="height:0.016em;width:0.8889em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.8889em" height="0.016em" style="width:0.8889em" viewBox="0 0 888.89 16" preserveAspectRatio="xMinYMin"><path d="M384 0 H504 V16 H384z M384 0 H504 V16 H384z"></path></svg></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.292em;"><span class="pstrut" style="height:3.15em;"></span><span style="height:0.016em;width:0.8889em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.8889em" height="0.016em" style="width:0.8889em" viewBox="0 0 888.89 16" preserveAspectRatio="xMinYMin"><path d="M384 0 H504 V16 H384z M384 0 H504 V16 H384z"></path></svg></span></span><span style="top:-4.3em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.2364em;"><span style="top:-4.7467em;"><span class="pstrut" style="height:3.5183em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord">∇</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-2.7964em;"><span class="pstrut" style="height:3.5183em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5183em;"><span style="top:-2.2869em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8231em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0144em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5183em;"><span style="top:-2.2869em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8231em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0144em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.7364em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.2985em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><blockquote><p>参考：</p><ul><li><a href="https://max.book118.com/html/2017/0606/111956214.shtm">西北工业大学 - 课件 - 含例题与解析</a></li><li><a href="https://www.math.pku.edu.cn/teachers/lidf/docs/statcomp/html/_statcompbook/opt-uncons.html#opt-uncons">北京大学 - 教材 - 含例题与解析</a></li><li><a href="https://blog.csdn.net/Serendipity_zyx/article/details/120515338">路人 - 博客 - 更通俗的解释</a></li><li><a href="https://zh.wikipedia.org/wiki/%E7%89%9B%E9%A1%BF%E6%B3%95">牛顿法 - wiki - 更权威的解释</a></li></ul></blockquote><h2 id="第五章「无约束最优化」最小二乘">第五章「无约束最优化」最小二乘</h2><div class="note note-light">            <p>本章继续介绍无约束函数的最优化算法。不过和第四章的区别在于现在的目标函数是二次函数，称为「最小二乘问题」。</p><p>所谓的无约束最小二乘问题，本质上是第四章介绍的无约束问题的一个子集，只不过因为使用场景很多所以单独拿出来进行讨论。也正因为使用场景多，学者们针对此类问题设计出了更加高效的最优化算法。</p><p>无约束最小二乘问题的形式定义为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><munder><mrow><mi>min</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>∈</mo><msup><mi>R</mi><mi>n</mi></msup></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><mo stretchy="false">[</mo><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">]</mo><mn>2</mn></msup><mo separator="true">,</mo><mspace width="1em"></mspace><mi>m</mi><mo>≥</mo><mi>n</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\min_{x\in R^n}f(x)=\frac{1}{2}\sum_{i=1}^m[r_{i}(x)]^2,\quad m\ge n\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.2291em;vertical-align:-1.3645em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8645em;"><span style="top:-3.8645em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-2.3557em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5935em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">min</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7717em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3645em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r_i(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 称为「残量函数」。本质上最小二乘问题就是寻找一个函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><msub><mi>α</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x,\alpha_i),(i=1,2,\cdots,m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span> 来拟合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span>，于是问题就转化为了</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>min</mi><mo>⁡</mo><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><mo stretchy="false">[</mo><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">]</mo><mn>2</mn></msup></mrow><mo>=</mo><mi>min</mi><mo>⁡</mo><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><msub><mi>α</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>−</mo><msub><mi>b</mi><mi>i</mi></msub><msup><mo stretchy="false">]</mo><mn>2</mn></msup></mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\min { \sum_{i=1}^m [r_i(x)]^2 }=\min { \sum_{i=1}^m [f(x,\alpha_i) - b_i]^2 }\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.2291em;vertical-align:-1.3645em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8645em;"><span style="top:-3.8645em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mop">min</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop">min</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3645em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>当 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r_i(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 为线性函数时，当前问题为线性最小二乘问题；当 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r_i(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 为非线性函数时，当前问题为非线性最小二乘问题。本章将分别讨论这两种最小二乘问题的优化求解策略。</p>          </div><h3 id="5-1-线性最小二乘">5.1 线性最小二乘</h3><p>此时可以直接将目标函数写成：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>min</mi><mo>⁡</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mi>A</mi><mi>x</mi><mo>−</mo><mi>b</mi><mi mathvariant="normal">∣</mi><msup><mi mathvariant="normal">∣</mi><mn>2</mn></msup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mi>T</mi></msup><msup><mi>A</mi><mi>T</mi></msup><mi>A</mi><mi>x</mi><mo>−</mo><msup><mi>b</mi><mi>T</mi></msup><mi>A</mi><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>b</mi><mi>T</mi></msup><mi>b</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\min f(x)&amp;=\frac{1}{2}|| Ax-b ||^2\\&amp;= \frac{1}{2}x^TA^TAx-b^TAx+\frac{1}{2}b^Tb\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.6149em;vertical-align:-2.0574em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5574em;"><span style="top:-4.5574em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mop">min</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.0574em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5574em;"><span style="top:-4.5574em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord">∣∣</span><span class="mord mathnormal">A</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">b</span><span class="mord">∣</span><span class="mord"><span class="mord">∣</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">A</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">A</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.0574em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>利用一阶必要条件可得：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msup><mi>A</mi><mi>T</mi></msup><mi>A</mi><mi>x</mi><mo>−</mo><msup><mi>A</mi><mi>T</mi></msup><mi>b</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>0</mn></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\nabla f(x)&amp;=A^TAx - A^Tb\\&amp;=0\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0513em;vertical-align:-1.2757em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7757em;"><span style="top:-3.8843em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span><span style="top:-2.3843em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2757em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7757em;"><span style="top:-3.8843em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">A</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">b</span></span></span><span style="top:-2.3843em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2757em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>于是可得最优闭式解：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup><mo>=</mo><mo stretchy="false">(</mo><msup><mi>A</mi><mi>T</mi></msup><mi>A</mi><msup><mo stretchy="false">)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mi>A</mi><mi>T</mi></msup><mi>b</mi></mrow><annotation encoding="application/x-tex">x^*=(A^T A)^{-1}A^Tb</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7387em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7387em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1413em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">A</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">b</span></span></span></span></span></p><p>当然 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>A</mi><mi>T</mi></msup><mi>A</mi></mrow><annotation encoding="application/x-tex">A^TA</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8413em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">A</span></span></span></span> 并不都是可逆的，并且在数据量足够大时，即使可逆也会让求逆操作即为耗时。针对此问题，提出了线性最小二乘的 QR 正交分解算法。</p><h3 id="5-2-非线性最小二乘">5.2 非线性最小二乘</h3><p>同样可以采用第四章学到的各种下降迭代算法，这里引入高斯牛顿法，推导的解的迭代公式为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>=</mo><msup><mi>x</mi><mi>k</mi></msup><mo>−</mo><mo stretchy="false">(</mo><msubsup><mi>A</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>A</mi><mi>k</mi></msub><msup><mo stretchy="false">)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><msubsup><mi>A</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>r</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">x^{k+1}=x^k - (A_k^TA_k)^{-1}A_k^Tr_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8991em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9824em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1413em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p><p>其中：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>A</mi><mi>k</mi></msub><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">∇</mi><msub><mi>r</mi><mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">∇</mi><msub><mi>r</mi><mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">∇</mi><msub><mi>r</mi><mi>m</mi></msub><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo separator="true">,</mo><mspace width="1em"></mspace><msub><mi>r</mi><mi>k</mi></msub><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>r</mi><mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>r</mi><mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>r</mi><mi>m</mi></msub><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}A_k =  \begin{bmatrix}\nabla r_1(x_k)\\\nabla r_2(x_k)\\\vdots \\\nabla r_m(x_k)\end{bmatrix},\quadr_k = \begin{bmatrix}r_1(x_k)\\r_2(x_k)\\\vdots \\r_m(x_k)\end{bmatrix}\\\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.76em;vertical-align:-2.63em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.13em;"><span style="top:-5.13em;"><span class="pstrut" style="height:4.98em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.95em;"><span style="top:-4.95em;"><span class="pstrut" style="height:7.4em;"></span><span style="width:0.667em;height:5.400em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="5.400em" viewBox="0 0 667 5400"><path d="M403 1759 V84 H666 V0 H319 V1759 v1800 v1759 h347 v-84H403z M403 1759 V0 H319 V1759 v1800 v1759 h84z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.45em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.98em;"><span style="top:-5.8275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">∇</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-4.6275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">∇</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-2.7675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-1.5675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">∇</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.48em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.95em;"><span style="top:-4.95em;"><span class="pstrut" style="height:7.4em;"></span><span style="width:0.667em;height:5.400em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="5.400em" viewBox="0 0 667 5400"><path d="M347 1759 V0 H0 V84 H263 V1759 v1800 v1759 H0 v84 H347zM347 1759 V0 H263 V1759 v1800 v1759 h84z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.45em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.95em;"><span style="top:-4.95em;"><span class="pstrut" style="height:7.4em;"></span><span style="width:0.667em;height:5.400em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="5.400em" viewBox="0 0 667 5400"><path d="M403 1759 V84 H666 V0 H319 V1759 v1800 v1759 h347 v-84H403z M403 1759 V0 H319 V1759 v1800 v1759 h84z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.45em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.98em;"><span style="top:-5.8275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-4.6275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-2.7675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-1.5675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.48em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.95em;"><span style="top:-4.95em;"><span class="pstrut" style="height:7.4em;"></span><span style="width:0.667em;height:5.400em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="5.400em" viewBox="0 0 667 5400"><path d="M347 1759 V0 H0 V84 H263 V1759 v1800 v1759 H0 v84 H347zM347 1759 V0 H263 V1759 v1800 v1759 h84z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.45em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.63em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><blockquote><p>参考：</p><ul><li><a href="http://faculty.bicmr.pku.edu.cn/~wenzw/optbook/lect/14-lsp-new-zxx.pdf">北京大学 - 课件 - 非线性最小二乘</a></li><li><a href="https://zhuanlan.zhihu.com/p/124934931">路人 - 博客 - 通俗解释非线性最小二乘</a></li></ul></blockquote><h2 id="第六章「约束最优化」二次规划">第六章「约束最优化」二次规划</h2><div class="note note-light">            <p>目标函数是二次函数，约束函数是线性函数。一般形式为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>min</mi><mo>⁡</mo><mspace width="1em"></mspace><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mi>T</mi></msup><mi>G</mi><mi>x</mi><mo>+</mo><msup><mi>g</mi><mi>T</mi></msup><mi>x</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>s</mi><mi mathvariant="normal">.</mi><mi>t</mi><mi mathvariant="normal">.</mi><mspace width="1em"></mspace><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msubsup><mi>a</mi><mi>i</mi><mi>T</mi></msubsup><mi>x</mi><mo>=</mo><msub><mi>b</mi><mi>i</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo separator="true">,</mo><mi>i</mi><mo>∈</mo><mi>S</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msubsup><mi>a</mi><mi>i</mi><mi>T</mi></msubsup><mi>x</mi><mo>≥</mo><msub><mi>b</mi><mi>i</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo separator="true">,</mo><mi>i</mi><mo>∈</mo><mi>M</mi></mrow></mstyle></mtd></mtr></mtable></mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}&amp;\min \quad q(x) = \frac{1}{2}x^TGx + g^Tx \\&amp;s.t.\quad \begin{cases}a_i^Tx = b_i&amp; ,i\in S\\a_i^Tx \ge b_i&amp; ,i \in M\\\end{cases}\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.6075em;vertical-align:-2.5537em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.0537em;"><span style="top:-5.4823em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"></span></span><span style="top:-2.7463em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.5537em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.0537em;"><span style="top:-5.4823em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">min</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">x</span></span></span><span style="top:-2.7463em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal">s</span><span class="mord">.</span><span class="mord mathnormal">t</span><span class="mord">.</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.5537em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>本章将分别讨论约束函数「含有等式」和「含有不等式」两类二次规划问题。</p>          </div><h3 id="6-1-等式约束二次规划">6.1 等式约束二次规划</h3><p>我们引入<strong>拉格朗日方法 (Lagrange Method, 简称 LM)</strong>。此时可以用矩阵表示问题和约束条件，并且不加证明的给出最优解和对应乘子就是满足 KKT 条件下的解。</p><p>拉格朗日函数：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>λ</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mi>T</mi></msup><mi>G</mi><mi>x</mi><mo>+</mo><msup><mi>g</mi><mi>T</mi></msup><mi>x</mi><mo>−</mo><msup><mi>λ</mi><mi>T</mi></msup><mo stretchy="false">(</mo><msup><mi>A</mi><mi>T</mi></msup><mi>x</mi><mo>−</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(x,\lambda) = \frac{1}{2}x^TGx+g^Tx -\lambda^T(A^Tx - b)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">λ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0858em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1413em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">b</span><span class="mclose">)</span></span></span></span></span></p><p>一阶必要条件：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>L</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>G</mi><mi>x</mi><mo>+</mo><mi>g</mi><mo>−</mo><mi>A</mi><mi>λ</mi><mo>=</mo><mn>0</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>L</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">∂</mi><mi>λ</mi></mrow></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msup><mi>A</mi><mi>x</mi></msup><mo>−</mo><mi>b</mi><mo>=</mo><mn>0</mn></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\frac{\partial L(x, \lambda) }{\partial x} &amp;= Gx+g-A\lambda = 0 \\\frac{\partial L(x, \lambda) }{\partial \lambda} &amp;= A^x - b = 0\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.826em;vertical-align:-2.163em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.663em;"><span style="top:-4.663em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">λ</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">λ</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">λ</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.163em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.663em;"><span style="top:-4.663em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">G</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal">λ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.163em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>最优解的矩阵形式：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>G</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mi>A</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msup><mi>A</mi><mi>T</mi></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msup><mi>x</mi><mo>∗</mo></msup></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msup><mi>λ</mi><mo>∗</mo></msup></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mo>−</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>g</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\begin{bmatrix}G &amp; -A \\-A^T &amp; 0\end{bmatrix}\begin{bmatrix}x^* \\\lambda^*\end{bmatrix}=-\begin{bmatrix}g \\b\end{bmatrix}\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.7013em;vertical-align:-1.1007em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6007em;"><span style="top:-3.6007em;"><span class="pstrut" style="height:3.4507em;"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4507em;"><span style="top:-3.6107em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">G</span></span></span><span style="top:-2.4093em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9507em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4507em;"><span style="top:-3.6107em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord mathnormal">A</span></span></span><span style="top:-2.4093em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9507em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.1007em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><h3 id="6-2-不等式约束二次规划">6.2 不等式约束二次规划</h3><p>我们引入<strong>有效集方法 (Active Set Method, 简称 ASM)</strong>。首先显然的最优解一定成立于等式约束，或成立于不等式取等。我们可以直接枚举每一种约束条件的组合（所有等式+枚举的不等式，并将不等式看做等式约束），然后判定当前的解是否满足没有选择的不等式约束。这种方法是可行的但是计算量极大。于是有效集方法应运而生。</p><p>算法流程：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406191553692.png" alt="Primal ASM 算法"></p><p>算法解析：</p><ul><li><strong>计算初始可行点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">x^{(0)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.888em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span></strong>，可用线性规划中的大 M 法计算获得</li><li><strong>计算前进方向 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">p_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></strong>，通过求解等式约束二次规划子问题获得</li><li>若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>k</mi></msub><mo>=</mo><mn mathvariant="bold">0</mn></mrow><annotation encoding="application/x-tex">p_k = \bf{0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">0</span></span></span></span></span></span>​，则<strong>计算工作集约束 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>W</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">W_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 对应的所有拉格朗日乘子 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\lambda_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></strong><ul><li>若所有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mi>i</mi></msub><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\lambda_i \ge 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，则停止迭代，得到最优解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup><mo>=</mo><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">x^*=x^{(k)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.888em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span></li><li>若存在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mi>i</mi></msub><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\lambda_i &lt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，则在工作集中剔除 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\lambda_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 最小时对应的约束 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">a_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>，得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>W</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>W</mi><mi>k</mi></msub><mo>∖</mo><mo stretchy="false">{</mo><mtext> </mtext><msub><mi>a</mi><mi>j</mi></msub><mtext> </mtext><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">W_{k+1}=W_k \setminus \set{a_j}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∖</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mopen">{</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">}</span></span></span></span></li></ul></li><li>若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>k</mi></msub><mo mathvariant="normal">≠</mo><mn mathvariant="bold">0</mn></mrow><annotation encoding="application/x-tex">p_k \ne \bf{0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">0</span></span></span></span></span></span>，则<strong>计算步长因子 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>α</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></strong>，并置 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mo>=</mo><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup><mo>+</mo><msub><mi>α</mi><mi>k</mi></msub><msub><mi>p</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">x^{(k+1)} = x^{(k)} + \alpha_kp_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.888em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9713em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><ul><li>若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>α</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\alpha_k=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，则工作集不变，得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>W</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>W</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">W_{k+1}=W_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></li><li>若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>α</mi><mi>k</mi></msub><mo>&lt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\alpha_k &lt; 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，则在工作集中添加 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>α</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 最小时对应的约束 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">a_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>，得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>W</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>W</mi><mi>k</mi></msub><mo>⋃</mo><mo stretchy="false">{</mo><mtext> </mtext><msub><mi>a</mi><mi>j</mi></msub><mtext> </mtext><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">W_{k+1}=W_k \bigcup \set {a_j}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">⋃</span><span class="mopen">{</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">}</span></span></span></span></li></ul></li></ul><blockquote><p>参考：</p><ul><li><a href="https://blog.csdn.net/dymodi/article/details/50358278">路人 - 博客 - 引入+原理+示例</a></li></ul></blockquote><h2 id="第七章「约束最优化」">第七章「约束最优化」</h2><div class="note note-light">            <p>本章我们简单讨论<strong>约束最优化问题</strong>中，针对<strong>等式约束</strong>的「二次罚函数」算法，以及拉格朗日乘子法。</p>          </div><h3 id="二次罚函数法">二次罚函数法</h3><p>对于等式约束问题：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>min</mi><mo>⁡</mo><mspace width="1em"></mspace><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>s</mi><mi mathvariant="normal">.</mi><mi>t</mi><mi mathvariant="normal">.</mi><mspace width="1em"></mspace><msub><mi>a</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mtext>&nbsp;</mtext><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><mi>p</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}&amp;\min\quad f(x) \\&amp;s.t.\quad a_i(x) = 0,\ i=1,2,...,p\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.75em;"><span class="pstrut" style="height:2.84em;"></span><span class="mord"></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:2.84em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">min</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal">s</span><span class="mord">.</span><span class="mord mathnormal">t</span><span class="mord">.</span><span class="mspace" style="margin-right:1em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace">&nbsp;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">...</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>我们定义二次罚函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>E</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P_E(x, \sigma)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mclose">)</span></span></span></span>，其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> 为惩罚因子：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>P</mi><mi>E</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>σ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>σ</mi><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>p</mi></munderover><msubsup><mi>a</mi><mi>i</mi><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P_E(x, \sigma) = f(x) + \frac{1}{2} \sigma\sum_{i=1}^p a_i^2(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.9762em;vertical-align:-1.2777em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6985em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3471em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></p><p>不加证明的给出结论：当惩罚因子 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">\to \infty</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord">∞</span></span></span></span> 时，目标函数趋于最小值。下面给出算法流程：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406120820032.png" alt="等式约束的二次罚函数法"></p><h3 id="拉格朗日乘子法">拉格朗日乘子法</h3><p>目标函数：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406281100567.png" alt="目标函数"></p><p>拉格朗日函数：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406281102611.png" alt="拉格朗日函数"></p><p>KKT 条件：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406281102383.png" alt="KKT 条件"></p><p>在实际求解时，我们只需要罗列上述 KKT 条件中的 (1) 和 (5)，另外三个显然不需要再罗列了。我们只需要对 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 个不等式约束条件对应的乘子进行是否为零的讨论即可，显然需要讨论 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>m</mi></msup></mrow><annotation encoding="application/x-tex">2^m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6644em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span></span></span></span></span></span></span></span> 次。</p><h2 id="后记">后记</h2><p>下面罗列一下考试题型（考前就透露完了🤣）：</p><h3 id="一、选择-6-2"><strong>一、选择 <code>6 * 2'</code></strong></h3><ul><li><p>严格凸函数的定义（ch1.2）（判断<a href="https://www.cnblogs.com/bigmonkey/p/11913079.html">海塞矩阵是否正定</a>即可）</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-a7ec1960" role="button" aria-expanded="false" aria-controls="collapse-a7ec1960">        <div class="fold-arrow">▶</div>例题      </div>      <div class="fold-collapse collapse" id="collapse-a7ec1960">        <div class="fold-content">          <p>考虑函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">f(x, y) = x^2 + y^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>。我们计算它的 Hessian 矩阵：</p><ol><li><p>计算一阶导数：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>f</mi><mi>x</mi></msub><mo>=</mo><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mn>2</mn><mi>x</mi></mrow><annotation encoding="application/x-tex">f_x = \frac{\partial f}{\partial x} = 2x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span><span class="mord mathnormal">x</span></span></span></span></span></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>f</mi><mi>y</mi></msub><mo>=</mo><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mn>2</mn><mi>y</mi></mrow><annotation encoding="application/x-tex">f_y = \frac{\partial f}{\partial y} = 2y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span></span></p></li><li><p>计算二阶导数：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>f</mi><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>=</mo><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">f_{xx} = \frac{\partial^2 f}{\partial x^2} = 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">xx</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span></span></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>f</mi><mrow><mi>y</mi><mi>y</mi></mrow></msub><mo>=</mo><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">f_{yy} = \frac{\partial^2 f}{\partial y^2} = 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">yy</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3715em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span></span></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>f</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>=</mo><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f_{xy} = \frac{\partial^2 f}{\partial x \partial y} = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3715em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">x</span><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>f</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f_{yx} = \frac{\partial^2 f}{\partial y \partial x} = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3715em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p></li><li><p>构造 Hessian 矩阵：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>H</mi><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>f</mi><mrow><mi>x</mi><mi>x</mi></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>f</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>f</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>f</mi><mrow><mi>y</mi><mi>y</mi></mrow></msub></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">H = \begin{pmatrix}f_{xx} &amp; f_{xy} \\f_{yx} &amp; f_{yy}\end{pmatrix} = \begin{pmatrix}2 &amp; 0 \\0 &amp; 2\end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">xx</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">yy</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p></li><li><p>检查 Hessian 矩阵是否为正定矩阵。对于一个 2x2 的对称矩阵：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>H</mi><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">H = \begin{pmatrix}a &amp; b \\b &amp; c\end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p><p>要判断它是否正定，可以使用以下条件：</p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">a &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></li><li>矩阵的行列式 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mi>c</mi><mo>−</mo><msup><mi>b</mi><mn>2</mn></msup><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">ac - b^2 &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">a</span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8532em;vertical-align:-0.0391em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></li></ul></li></ol><p>对于我们的 Hessian 矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>a</mi><mo>=</mo><mn>2</mn><mo separator="true">,</mo><mspace width="1em"></mspace><mi>b</mi><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mspace width="1em"></mspace><mi>c</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">   a = 2, \quad b = 0, \quad c = 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span></span></p><p>显然：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>2</mn><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">   2 &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6835em;vertical-align:-0.0391em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>行列式</mtext><mo>=</mo><mn>2</mn><mo>⋅</mo><mn>2</mn><mo>−</mo><msup><mn>0</mn><mn>2</mn></msup><mo>=</mo><mn>4</mn><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">   \text{行列式} = 2 \cdot 2 - 0^2 = 4 &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord cjk_fallback">行列式</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6835em;vertical-align:-0.0391em;"></span><span class="mord">4</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p><p>因此，Hessian 矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> 是正定的，所以函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">f(x, y) = x^2 + y^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 是严格凸函数。</p>        </div>      </div>    </div></li><li><p>海塞矩阵正定负定和极值的关系（ch1.3）（显然的）</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-6b88c869" role="button" aria-expanded="false" aria-controls="collapse-6b88c869">        <div class="fold-arrow">▶</div>讲解      </div>      <div class="fold-collapse collapse" id="collapse-6b88c869">        <div class="fold-content">          <p>驻点的性质与海塞矩阵的行列式（determinant，det）和特征值（eigenvalues）的符号密切相关。以下是具体的关系：</p><ol><li><p><strong>正定矩阵</strong>：如果海塞矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> 是正定矩阵（即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>det</mtext><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\text{det}(H) &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">det</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 且 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f_{xx} &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">xx</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>），则驻点是局部极小值点。这是因为此时所有的特征值都是正的，函数在该点附近是向上的碗形结构。</p></li><li><p><strong>负定矩阵</strong>：如果海塞矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> 是负定矩阵（即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>det</mtext><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\text{det}(H) &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">det</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 且 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f_{xx} &lt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">xx</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>），则驻点是局部极大值点。这是因为此时所有的特征值都是负的，函数在该点附近是向下的碗形结构。</p></li><li><p><strong>不定矩阵</strong>：如果海塞矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> 的行列式小于零（即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>det</mtext><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\text{det}(H) &lt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">det</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>），则驻点是鞍点。这是因为此时特征值的符号不同，函数在该点附近的形状像马鞍。</p></li><li><p><strong>退化矩阵</strong>：如果海塞矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> 的行列式等于零（即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>det</mtext><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\text{det}(H) = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">det</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>），则无法通过二阶导数判断驻点的性质，需要进一步分析。这是因为此时海塞矩阵是奇异的，不能提供足够的信息来确定驻点的性质。</p></li></ol><p>总结以上内容可以得出如下的关系：</p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>det</mtext><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\text{det}(H) &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">det</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 且 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f_{xx} &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">xx</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>：局部极小值点。</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>det</mtext><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\text{det}(H) &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">det</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 且 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f_{xx} &lt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">xx</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>：局部极大值点。</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>det</mtext><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\text{det}(H) &lt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">det</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>：鞍点。</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>det</mtext><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\text{det}(H) = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">det</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>：需要进一步分析。</li></ul>        </div>      </div>    </div></li><li><p>求二元函数的极值（ch1.3）（先利用一阶必要条件求出所有驻点，然后利用海塞矩阵的正定性判定极值点。答案：一个极大一个极小，另两个什么都不是）</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-cbf53915" role="button" aria-expanded="false" aria-controls="collapse-cbf53915">        <div class="fold-arrow">▶</div>例题      </div>      <div class="fold-collapse collapse" id="collapse-cbf53915">        <div class="fold-content">          <p><strong>问题：</strong></p><p>给定函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>3</mn><mi>x</mi><mo>+</mo><msup><mi>y</mi><mn>3</mn></msup><mo>−</mo><mn>3</mn><mi>y</mi></mrow><annotation encoding="application/x-tex">f(x, y) = x^3 - 3x + y^3 - 3y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">3</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">3</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>，求其所有极值点。</p><p><strong>解题步骤：</strong></p><ol><li><p><strong>计算一阶梯度以求驻点</strong>：</p><p>计算函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x, y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> 对 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> 的一阶偏导数：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">\frac{\partial f}{\partial x} = 3x^2 - 3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9474em;vertical-align:-0.0833em;"></span><span class="mord">3</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span></span></span></span></span></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">\frac{\partial f}{\partial y} = 3y^2 - 3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0585em;vertical-align:-0.1944em;"></span><span class="mord">3</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span></span></span></span></span></p><p>解方程组 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\frac{\partial f}{\partial x} = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2772em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.10764em;">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\frac{\partial f}{\partial y} = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4133em;vertical-align:-0.4811em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.10764em;">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">3x^2 - 3 = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9474em;vertical-align:-0.0833em;"></span><span class="mord">3</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">3y^2 - 3 = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0585em;vertical-align:-0.1944em;"></span><span class="mord">3</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p><p>化简方程组得到：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><mi>x</mi><mo>=</mo><mo>±</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x^2 = 1 \implies x = \pm 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6684em;vertical-align:-0.024em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">±</span><span class="mord">1</span></span></span></span></span></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><mi>y</mi><mo>=</mo><mo>±</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">y^2 = 1 \implies y = \pm 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0585em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6684em;vertical-align:-0.024em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">±</span><span class="mord">1</span></span></span></span></span></p><p>所以，驻点为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1, 1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1, -1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-1, 1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-1, -1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span></span></span></span>。</p></li><li><p><strong>计算二阶梯度以检验驻点的性质</strong>：</p><p>计算二阶偏导数：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>6</mn><mi>x</mi></mrow><annotation encoding="application/x-tex">\frac{\partial^2 f}{\partial x^2} = 6x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">6</span><span class="mord mathnormal">x</span></span></span></span></span></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>6</mn><mi>y</mi></mrow><annotation encoding="application/x-tex">\frac{\partial^2 f}{\partial y^2} = 6y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.3715em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">6</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span></span></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\frac{\partial^2 f}{\partial x \partial y} = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.3715em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">x</span><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p><p>对于每个驻点，计算 Hessian 矩阵：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>H</mi><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>6</mn><mi>x</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>6</mn><mi>y</mi></mrow></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">H = \begin{pmatrix}6x &amp; 0 \\0 &amp; 6y\end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span><span class="mord mathnormal">x</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p><p><strong>驻点 (1, 1)：</strong></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>6</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>6</mn></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">H(1,1) = \begin{pmatrix}6 &amp; 0 \\0 &amp; 6\end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p><p>行列式为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>det</mtext><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mn>6</mn><mo>×</mo><mn>6</mn><mo>−</mo><mn>0</mn><mo>×</mo><mn>0</mn><mo>=</mo><mn>36</mn><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\text{det}(H(1,1)) = 6 \times 6 - 0 \times 0 = 36 &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">det</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">6</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">6</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6835em;vertical-align:-0.0391em;"></span><span class="mord">36</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p><p>主对角线元素 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>6</mn><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">6 &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6835em;vertical-align:-0.0391em;"></span><span class="mord">6</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，所以 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1, 1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 是<strong>局部极小值点</strong>。</p><p><strong>驻点 (-1, -1)：</strong></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>6</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>6</mn></mrow></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">H(-1,-1) = \begin{pmatrix}-6 &amp; 0 \\0 &amp; -6\end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">6</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">6</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p><p>行列式为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>det</mtext><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>6</mn><mo>×</mo><mo>−</mo><mn>6</mn><mo>−</mo><mn>0</mn><mo>×</mo><mn>0</mn><mo>=</mo><mn>36</mn><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\text{det}(H(-1,-1)) = -6 \times -6 - 0 \times 0 = 36 &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">det</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">6</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">6</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6835em;vertical-align:-0.0391em;"></span><span class="mord">36</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p><p>主对角线元素 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>6</mn><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">-6 &lt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">6</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，所以 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-1, -1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 是<strong>局部极大值点</strong>。</p><p><strong>驻点 (1, -1)：</strong></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>6</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>6</mn></mrow></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">H(1,-1) = \begin{pmatrix}6 &amp; 0 \\0 &amp; -6\end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">6</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p><p>行列式为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>det</mtext><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mn>6</mn><mo>×</mo><mo>−</mo><mn>6</mn><mo>−</mo><mn>0</mn><mo>×</mo><mn>0</mn><mo>=</mo><mo>−</mo><mn>36</mn><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\text{det}(H(1,-1)) = 6 \times -6 - 0 \times 0 = -36 &lt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">det</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">6</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">6</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">36</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p><p>由于行列式为负，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1, -1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 是<strong>鞍点</strong>。</p><p><strong>驻点 (-1, 1)：</strong></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>6</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>6</mn></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">H(-1,1) = \begin{pmatrix}-6 &amp; 0 \\0 &amp; 6\end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">6</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p><p>行列式为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>det</mtext><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>6</mn><mo>×</mo><mn>6</mn><mo>−</mo><mn>0</mn><mo>×</mo><mn>0</mn><mo>=</mo><mo>−</mo><mn>36</mn><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\text{det}(H(-1,1)) = -6 \times 6 - 0 \times 0 = -36 &lt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">det</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">6</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">6</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">36</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p><p>由于行列式为负，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-1, 1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 是<strong>鞍点</strong>。</p></li></ol><p><strong>结论：</strong></p><p>函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>3</mn><mi>x</mi><mo>+</mo><msup><mi>y</mi><mn>3</mn></msup><mo>−</mo><mn>3</mn><mi>y</mi></mrow><annotation encoding="application/x-tex">f(x, y) = x^3 - 3x + y^3 - 3y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">3</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">3</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> 在点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1, 1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 处有一个局部极小值点，在点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-1, -1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 处有一个局部极大值点，而在点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1, -1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-1, 1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 处为鞍点。</p>        </div>      </div>    </div></li><li><p>拉格朗日函数中等式乘子的性质（ch1.3）（等式约束的乘子 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\lambda_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是任意实数，不等式约束的乘子 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>μ</mi><mi>i</mi></msub><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\mu_i\ge 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>）</p></li></ul><h3 id="二、填空-5-2"><strong>二、填空 <code>5 * 2'</code></strong></h3><ul><li><p>求解给定点的下降方向（ch1.3）</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-49bb6dd1" role="button" aria-expanded="false" aria-controls="collapse-49bb6dd1">        <div class="fold-arrow">▶</div>例题      </div>      <div class="fold-collapse collapse" id="collapse-49bb6dd1">        <div class="fold-content">          <p>求解一个函数的下降方向（Descent Direction），可以使用其梯度（Gradient）。梯度的反方向是函数值下降最快的方向。具体来说：</p><ol><li><p><strong>一元函数：</strong></p><ul><li>对于一元函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>，其梯度是导数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f'(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>。</li><li>下降方向是梯度的负方向，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">-f'(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>。</li></ul><p><strong>例子：</strong><br>设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">f(x) = x^2 + 2x + 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>。</p><ul><li>首先计算导数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">f'(x) = 2x + 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span>。</li><li>在某点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 处，下降方向是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mo stretchy="false">(</mo><mn>2</mn><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">-f'(x_0) = -(2x_0 + 2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mopen">(</span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mclose">)</span></span></span></span>。</li></ul></li><li><p><strong>二元函数：</strong></p><ul><li>对于二元函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x, y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span>，其梯度是偏导数的向量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">(</mo><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac><mo separator="true">,</mo><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\nabla f(x, y) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.8em;vertical-align:-0.65em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.10764em;">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.10764em;">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">)</span></span></span></span></span></span>。</li><li>下降方向是梯度的负方向，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">(</mo><mo>−</mo><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac><mo separator="true">,</mo><mo>−</mo><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">-\nabla f(x, y) = \left( -\frac{\partial f}{\partial x}, -\frac{\partial f}{\partial y} \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.8em;vertical-align:-0.65em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size2">(</span></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.10764em;">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.10764em;">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">)</span></span></span></span></span></span>。</li></ul><p><strong>例子：</strong><br>设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi></mrow><annotation encoding="application/x-tex">f(x, y) = x^2 + y^2 + 2x + 2y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>。</p><ul><li>首先计算偏导数：<p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\frac{\partial f}{\partial x} = 2x + 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span></span></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\frac{\partial f}{\partial y} = 2y + 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span></span></p></li><li>在某点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x_0, y_0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 处，梯度是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>2</mn><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><mn>2</mn><mo separator="true">,</mo><mn>2</mn><msub><mi>y</mi><mn>0</mn></msub><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\nabla f(x_0, y_0) = (2x_0 + 2, 2y_0 + 2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mclose">)</span></span></span></span>。</li><li>下降方向是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mo>−</mo><mo stretchy="false">(</mo><mn>2</mn><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo>−</mo><mo stretchy="false">(</mo><mn>2</mn><msub><mi>y</mi><mn>0</mn></msub><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">-\nabla f(x_0, y_0) = (-(2x_0 + 2), -(2y_0 + 2))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mopen">(</span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mopen">(</span><span class="mord">2</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mclose">))</span></span></span></span>。</li></ul></li></ol>        </div>      </div>    </div></li><li><p>线性规划的基本概念（ch2.1）</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-9f553363" role="button" aria-expanded="false" aria-controls="collapse-9f553363">        <div class="fold-arrow">▶</div>讲解      </div>      <div class="fold-collapse collapse" id="collapse-9f553363">        <div class="fold-content">          <p>这个不知道要怎么考，大致罗列一下所有的基本概念。</p><ul><li>定义：目标函数和约束条件都是线性的</li><li>图解法：对于二元函数，可以使用平面图法进行求解。那么共有 4 种可能的结果，分别为：有唯一解、有无穷多解、有无界解、无可行解</li><li>性质：分别从可行域和最优解两个角度展开：<ul><li>对于可行域，如果可行域是非空则可行域一定是凸集，这个很显然，用 ch1.2 的可行域凸集定理证明即可</li><li>对于最优解，首先最优解如果存在则一定存在于所有约束条件中取等时，其次最优解如果存在一定是在可行域的顶点上。</li></ul></li></ul>        </div>      </div>    </div></li><li><p>拟牛顿法的近似海塞矩阵公式（ch4.4）（见书 4.4.9和 4.4.10）</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-4c49716f" role="button" aria-expanded="false" aria-controls="collapse-4c49716f">        <div class="fold-arrow">▶</div>讲解      </div>      <div class="fold-collapse collapse" id="collapse-4c49716f">        <div class="fold-content">          <p>对于原函数，有：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>G</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mo>≈</mo><msub><mi>g</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>g</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">G_{k+1}(x_{k+1}-x_k) \approx g_{k+1} - g_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">G</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p><p>我们希望构造出的对称矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>B</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">B_{k+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span> 满足上式中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>G</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">G_{k+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">G</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span> 的条件，于是就有下面两种拟牛顿条件，其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>H</mi><mi>k</mi></msub><mo>=</mo><msubsup><mi>B</mi><mi>k</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">H_k=B_k^{-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1555em;vertical-align:-0.3013em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8542em;"><span style="top:-2.3987em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.1031em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span></span></span></span>：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>B</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>g</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>g</mi><mi>k</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>H</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>g</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>g</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>x</mi><mi>k</mi></msub></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}B_{k+1}(x_{k+1}-x_k)= g_{k+1} - g_k\\H_{k+1}(g_{k+1} - g_k) = x_{k+1}-x_k\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>我们记 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>s</mi><mi>k</mi></msub><mo>=</mo><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>x</mi><mi>k</mi></msub><mo separator="true">,</mo><msub><mi>y</mi><mi>k</mi></msub><mo>=</mo><msub><mi>g</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>g</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">s_k=x_{k+1}-x_k,y_k=g_{k+1} - g_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>。则关于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>B</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">B_{k+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span> 的 BFGS 校正迭代公式如下：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>B</mi><mn>0</mn></msub><mo>=</mo><msup><mi mathvariant="normal">∇</mi><mn>2</mn></msup><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><msub><mi>B</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>B</mi><mi>k</mi></msub><mo>+</mo><mfrac><mrow><msub><mi>y</mi><mi>k</mi></msub><msubsup><mi>y</mi><mi>k</mi><mi>T</mi></msubsup></mrow><mrow><msubsup><mi>s</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>y</mi><mi>k</mi></msub></mrow></mfrac><mo>−</mo><mfrac><mrow><msub><mi>B</mi><mi>k</mi></msub><msub><mi>s</mi><mi>k</mi></msub><msubsup><mi>s</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>B</mi><mi>k</mi></msub></mrow><mrow><msubsup><mi>s</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>B</mi><mi>k</mi></msub><msub><mi>s</mi><mi>k</mi></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases}B_0 = \nabla^2f(x_0)\\\displaystyle B_{k+1} = B_k + \frac{y_ky_k^T}{s_k^Ty_k} - \frac{B_ks_ks_k^TB_k}{s_k^TB_ks_k}\end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.9728em;vertical-align:-1.7364em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-2.5em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.492em;"><span class="pstrut" style="height:3.15em;"></span><span style="height:0.016em;width:0.8889em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.8889em" height="0.016em" style="width:0.8889em" viewBox="0 0 888.89 16" preserveAspectRatio="xMinYMin"><path d="M384 0 H504 V16 H384z M384 0 H504 V16 H384z"></path></svg></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.292em;"><span class="pstrut" style="height:3.15em;"></span><span style="height:0.016em;width:0.8889em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.8889em" height="0.016em" style="width:0.8889em" viewBox="0 0 888.89 16" preserveAspectRatio="xMinYMin"><path d="M384 0 H504 V16 H384z M384 0 H504 V16 H384z"></path></svg></span></span><span style="top:-4.3em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.2364em;"><span style="top:-4.7467em;"><span class="pstrut" style="height:3.5183em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord">∇</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-2.7964em;"><span class="pstrut" style="height:3.5183em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5183em;"><span style="top:-2.2869em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8231em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0144em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5183em;"><span style="top:-2.2869em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8231em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0144em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.7364em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>        </div>      </div>    </div></li></ul><h3 id="三、证明-1-10"><strong>三、证明 <code>1 * 10'</code></strong></h3><ul><li><p>证明高维凸规划问题（ch1.2）</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-0f7ac154" role="button" aria-expanded="false" aria-controls="collapse-0f7ac154">        <div class="fold-arrow">▶</div>例题      </div>      <div class="fold-collapse collapse" id="collapse-0f7ac154">        <div class="fold-content">          <p>关键在于证明可行域是凸集，目标函数是凸函数。证明可行域是凸集比较简单，书 ch1.2 中的「定理1.2.18」给出了详细的可行域凸性判定定理。证明目标函数是凸函数有三种方法，书 ch1.2 中「定理1.2.19-1.2.21」分别从函数值、一阶导数、二阶导数三个角度进行了凸函数判定定理的介绍，下面仅从二阶导数的角度给出凸函数判定的示例。</p><p><strong>二元凸函数</strong></p><p>选取函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>4</mn><msup><mi>y</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">f(x, y) = 3x^2 + 2xy + 4y^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord">3</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord">4</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>，计算 Hessian 矩阵：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>6</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>8</mn></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">H(f) = \begin{pmatrix}\frac{\partial^2 f}{\partial x^2} &amp; \frac{\partial^2 f}{\partial x \partial y} \\\frac{\partial^2 f}{\partial y \partial x} &amp; \frac{\partial^2 f}{\partial y^2}\end{pmatrix} = \begin{pmatrix}6 &amp; 2 \\2 &amp; 8\end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.1023em;vertical-align:-1.3011em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8011em;"><span style="top:-3.8011em;"><span class="pstrut" style="height:3.07em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.07em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.10764em;">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.07em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.07em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.10764em;">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3011em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8011em;"><span style="top:-3.8011em;"><span class="pstrut" style="height:3.07em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.07em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight">x</span><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.10764em;">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.07em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.07em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.10764em;">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3011em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size4">)</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">8</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p><p>Hessian 矩阵的特征值均为正数，因此海塞矩阵是正定的，因此 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>4</mn><msup><mi>y</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">f(x, y) = 3x^2 + 2xy + 4y^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord">3</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord">4</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 是凸函数并且是严格凸的。</p><p><strong>三元凸函数</strong></p><p>选取函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mi>z</mi><mo>+</mo><msup><mi>z</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">g(x, y, z) = 2x^2 + 2xy + 3y^2 + 2xz + z^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord">3</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>，计算 Hessian 矩阵：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>g</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>g</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>g</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi><mi mathvariant="normal">∂</mi><mi>z</mi></mrow></mfrac></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>g</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>g</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>g</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi><mi mathvariant="normal">∂</mi><mi>z</mi></mrow></mfrac></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>g</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>z</mi><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>g</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>z</mi><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>g</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>z</mi><mn>2</mn></msup></mrow></mfrac></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>4</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>6</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">H(g) = \begin{pmatrix}\frac{\partial^2 g}{\partial x^2} &amp; \frac{\partial^2 g}{\partial x \partial y} &amp; \frac{\partial^2 g}{\partial x \partial z} \\\frac{\partial^2 g}{\partial y \partial x} &amp; \frac{\partial^2 g}{\partial y^2} &amp; \frac{\partial^2 g}{\partial y \partial z} \\\frac{\partial^2 g}{\partial z \partial x} &amp; \frac{\partial^2 g}{\partial z \partial y} &amp; \frac{\partial^2 g}{\partial z^2}\end{pmatrix} = \begin{pmatrix}4 &amp; 2 &amp; 2 \\2 &amp; 6 &amp; 0 \\2 &amp; 0 &amp; 2\end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:4.6534em;vertical-align:-2.0767em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35em;"><span style="top:-4.35em;"><span class="pstrut" style="height:6.2em;"></span><span style="width:0.875em;height:4.200em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.875em" height="4.200em" viewBox="0 0 875 4200"><path d="M863,9c0,-2,-2,-5,-6,-9c0,0,-17,0,-17,0c-12.7,0,-19.3,0.3,-20,1c-5.3,5.3,-10.3,11,-15,17c-242.7,294.7,-395.3,682,-458,1162c-21.3,163.3,-33.3,349,-36,557 l0,684c0.2,6,0,26,0,60c2,159.3,10,310.7,24,454c53.3,528,210,949.7,470,1265c4.7,6,9.7,11.7,15,17c0.7,0.7,7,1,19,1c0,0,18,0,18,0c4,-4,6,-7,6,-9c0,-2.7,-3.3,-8.7,-10,-18c-135.3,-192.7,-235.5,-414.3,-300.5,-665c-65,-250.7,-102.5,-544.7,-112.5,-882c-2,-104,-3,-167,-3,-189l0,-692c0,-162.7,5.7,-314,17,-454c20.7,-272,63.7,-513,129,-723c65.3,-210,155.3,-396.3,270,-559c6.7,-9.3,10,-15.3,10,-18z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.85em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5767em;"><span style="top:-4.5767em;"><span class="pstrut" style="height:3.07em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.07em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.0256em;"><span class="pstrut" style="height:3.07em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.07em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-1.4744em;"><span class="pstrut" style="height:3.07em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.07em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.0767em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5767em;"><span style="top:-4.5767em;"><span class="pstrut" style="height:3.07em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.07em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight">x</span><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.0256em;"><span class="pstrut" style="height:3.07em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.07em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-1.4744em;"><span class="pstrut" style="height:3.07em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.07em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.0767em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5767em;"><span style="top:-4.5767em;"><span class="pstrut" style="height:3.07em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.07em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight">x</span><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.0256em;"><span class="pstrut" style="height:3.07em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.07em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-1.4744em;"><span class="pstrut" style="height:3.07em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.07em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.0767em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35em;"><span style="top:-4.35em;"><span class="pstrut" style="height:6.2em;"></span><span style="width:0.875em;height:4.200em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.875em" height="4.200em" viewBox="0 0 875 4200"><path d="M76,0c-16.7,0,-25,3,-25,9c0,2,2,6.3,6,13c21.3,28.7,42.3,60.3,63,95c96.7,156.7,172.8,332.5,228.5,527.5c55.7,195,92.8,416.5,111.5,664.5c11.3,139.3,17,290.7,17,454c0,28,1.7,43,3.3,45l0,609c-3,4,-3.3,16.7,-3.3,38c0,162,-5.7,313.7,-17,455c-18.7,248,-55.8,469.3,-111.5,664c-55.7,194.7,-131.8,370.3,-228.5,527c-20.7,34.7,-41.7,66.3,-63,95c-2,3.3,-4,7,-6,11c0,7.3,5.7,11,17,11c0,0,11,0,11,0c9.3,0,14.3,-0.3,15,-1c5.3,-5.3,10.3,-11,15,-17c242.7,-294.7,395.3,-681.7,458,-1161c21.3,-164.7,33.3,-350.7,36,-558l0,-744c-2,-159.3,-10,-310.7,-24,-454c-53.3,-528,-210,-949.7,-470,-1265c-4.7,-6,-9.7,-11.7,-15,-17c-0.7,-0.7,-6.7,-1,-18,-1z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.85em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.6em;vertical-align:-1.55em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.875em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.875em" height="3.600em" viewBox="0 0 875 3600"><path d="M863,9c0,-2,-2,-5,-6,-9c0,0,-17,0,-17,0c-12.7,0,-19.3,0.3,-20,1c-5.3,5.3,-10.3,11,-15,17c-242.7,294.7,-395.3,682,-458,1162c-21.3,163.3,-33.3,349,-36,557 l0,84c0.2,6,0,26,0,60c2,159.3,10,310.7,24,454c53.3,528,210,949.7,470,1265c4.7,6,9.7,11.7,15,17c0.7,0.7,7,1,19,1c0,0,18,0,18,0c4,-4,6,-7,6,-9c0,-2.7,-3.3,-8.7,-10,-18c-135.3,-192.7,-235.5,-414.3,-300.5,-665c-65,-250.7,-102.5,-544.7,-112.5,-882c-2,-104,-3,-167,-3,-189l0,-92c0,-162.7,5.7,-314,17,-454c20.7,-272,63.7,-513,129,-723c65.3,-210,155.3,-396.3,270,-559c6.7,-9.3,10,-15.3,10,-18z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.875em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.875em" height="3.600em" viewBox="0 0 875 3600"><path d="M76,0c-16.7,0,-25,3,-25,9c0,2,2,6.3,6,13c21.3,28.7,42.3,60.3,63,95c96.7,156.7,172.8,332.5,228.5,527.5c55.7,195,92.8,416.5,111.5,664.5c11.3,139.3,17,290.7,17,454c0,28,1.7,43,3.3,45l0,9c-3,4,-3.3,16.7,-3.3,38c0,162,-5.7,313.7,-17,455c-18.7,248,-55.8,469.3,-111.5,664c-55.7,194.7,-131.8,370.3,-228.5,527c-20.7,34.7,-41.7,66.3,-63,95c-2,3.3,-4,7,-6,11c0,7.3,5.7,11,17,11c0,0,11,0,11,0c9.3,0,14.3,-0.3,15,-1c5.3,-5.3,10.3,-11,15,-17c242.7,-294.7,395.3,-681.7,458,-1161c21.3,-164.7,33.3,-350.7,36,-558l0,-144c-2,-159.3,-10,-310.7,-24,-454c-53.3,-528,-210,-949.7,-470,-1265c-4.7,-6,-9.7,-11.7,-15,-17c-0.7,-0.7,-6.7,-1,-18,-1z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>Hessian 矩阵的特征值均为正数，因此海塞矩阵是正定的，因此 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mi>z</mi><mo>+</mo><msup><mi>z</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">g(x, y, z) = 2x^2 + 2xy + 3y^2 + 2xz + z^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord">3</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 是凸函数并且是严格凸的。</p><p>当然我们也可以通过计算主子矩阵的行列式来代替计算矩阵的特征值，上述二元同样也可以。例如对于下述对称矩阵：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>A</mi><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>1</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>1</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>1</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>1</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">A = \begin{pmatrix} 2 &amp; -1 &amp; 0 \\ -1 &amp; 2 &amp; -1 \\ 0 &amp; -1 &amp; 2 \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.6em;vertical-align:-1.55em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.875em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.875em" height="3.600em" viewBox="0 0 875 3600"><path d="M863,9c0,-2,-2,-5,-6,-9c0,0,-17,0,-17,0c-12.7,0,-19.3,0.3,-20,1c-5.3,5.3,-10.3,11,-15,17c-242.7,294.7,-395.3,682,-458,1162c-21.3,163.3,-33.3,349,-36,557 l0,84c0.2,6,0,26,0,60c2,159.3,10,310.7,24,454c53.3,528,210,949.7,470,1265c4.7,6,9.7,11.7,15,17c0.7,0.7,7,1,19,1c0,0,18,0,18,0c4,-4,6,-7,6,-9c0,-2.7,-3.3,-8.7,-10,-18c-135.3,-192.7,-235.5,-414.3,-300.5,-665c-65,-250.7,-102.5,-544.7,-112.5,-882c-2,-104,-3,-167,-3,-189l0,-92c0,-162.7,5.7,-314,17,-454c20.7,-272,63.7,-513,129,-723c65.3,-210,155.3,-396.3,270,-559c6.7,-9.3,10,-15.3,10,-18z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.875em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.875em" height="3.600em" viewBox="0 0 875 3600"><path d="M76,0c-16.7,0,-25,3,-25,9c0,2,2,6.3,6,13c21.3,28.7,42.3,60.3,63,95c96.7,156.7,172.8,332.5,228.5,527.5c55.7,195,92.8,416.5,111.5,664.5c11.3,139.3,17,290.7,17,454c0,28,1.7,43,3.3,45l0,9c-3,4,-3.3,16.7,-3.3,38c0,162,-5.7,313.7,-17,455c-18.7,248,-55.8,469.3,-111.5,664c-55.7,194.7,-131.8,370.3,-228.5,527c-20.7,34.7,-41.7,66.3,-63,95c-2,3.3,-4,7,-6,11c0,7.3,5.7,11,17,11c0,0,11,0,11,0c9.3,0,14.3,-0.3,15,-1c5.3,-5.3,10.3,-11,15,-17c242.7,-294.7,395.3,-681.7,458,-1161c21.3,-164.7,33.3,-350.7,36,-558l0,-144c-2,-159.3,-10,-310.7,-24,-454c-53.3,-528,-210,-949.7,-470,-1265c-4.7,-6,-9.7,-11.7,-15,-17c-0.7,-0.7,-6.7,-1,-18,-1z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>我们计算其所有主子矩阵的行列式：</p><ul><li>一阶主子矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\begin{pmatrix} 2 \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.85em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span> 的行列式为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span>。</li><li>二阶主子矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>1</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>1</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\begin{pmatrix} 2 &amp; -1 \\ -1 &amp; 2 \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span> 的行列式为：<p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>det</mi><mo>⁡</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>1</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>1</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><mo>=</mo><mn>2</mn><mo>⋅</mo><mn>2</mn><mo>−</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>4</mn><mo>−</mo><mn>1</mn><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">\det \begin{pmatrix} 2 &amp; -1 \\ -1 &amp; 2 \end{pmatrix} = 2 \cdot 2 - (-1) \cdot (-1) = 4 - 1 = 3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mop">det</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">4</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span></span></span></span></span></p></li><li>三阶主子矩阵即矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 本身的行列式：<p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">∣</mo><mtable rowspacing="0.16em" columnalign="center center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>1</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>1</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>1</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>1</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd></mtr></mtable><mo fence="true">∣</mo></mrow><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">\det(A) = \begin{vmatrix} 2 &amp; -1 &amp; 0 \\ -1 &amp; 2 &amp; -1 \\ 0 &amp; -1 &amp; 2 \end{vmatrix}=4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.6em;vertical-align:-1.55em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.333em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.333em" height="3.600em" viewBox="0 0 333 3600"><path d="M145 15 v585 v2400 v585 c2.667,10,9.667,15,21,15c10,0,16.667,-5,20,-15 v-585 v-2400 v-585 c-2.667,-10,-9.667,-15,-21,-15c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v2400 v585 h43z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.333em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.333em" height="3.600em" viewBox="0 0 333 3600"><path d="M145 15 v585 v2400 v585 c2.667,10,9.667,15,21,15c10,0,16.667,-5,20,-15 v-585 v-2400 v-585 c-2.667,-10,-9.667,-15,-21,-15c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v2400 v585 h43z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">4</span></span></span></span></span></p></li></ul><p>由于所有主子矩阵的行列式都大于零，矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 是正定的。假如这是一个函数的海塞矩阵，则该函数是凸函数并且是严格凸的。</p>        </div>      </div>    </div></li></ul><h3 id="四、计算-68"><strong>四、计算 <code>68'</code></strong></h3><ul><li><p>计算点到超平面的距离，转化为等式约束的最优化问题（ch1.3）</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-4236d39e" role="button" aria-expanded="false" aria-controls="collapse-4236d39e">        <div class="fold-arrow">▶</div>讲解      </div>      <div class="fold-collapse collapse" id="collapse-4236d39e">        <div class="fold-content">          <p>这个应该是很显然的一道题，我们定义目标函数为点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> 到超平面上点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 的距离，约束条件为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 在超平面上。求解方法就是构造一个拉格朗日函数，然后利用一阶必要条件求解即可。</p>        </div>      </div>    </div></li><li><p>线性规划问题中求对偶问题表达式（ch2.3.1）</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-e2dc11d3" role="button" aria-expanded="false" aria-controls="collapse-e2dc11d3">        <div class="fold-arrow">▶</div>例题      </div>      <div class="fold-collapse collapse" id="collapse-e2dc11d3">        <div class="fold-content">          <p>直接看 2.3.1 的实战演练即可。</p>        </div>      </div>    </div></li><li><p>线性规划问题中已知原问题最优解，求解对偶问题最优解（ch2.3.2）（利用互补松弛定理、原问题和对偶问题最优解相等）</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-17bf9d6f" role="button" aria-expanded="false" aria-controls="collapse-17bf9d6f">        <div class="fold-arrow">▶</div>讲解      </div>      <div class="fold-collapse collapse" id="collapse-17bf9d6f">        <div class="fold-content">          <p>见 ch2.3.2 有详细求解步骤。注意可能试卷中给的是原问题的最优解，需要求解对偶问题的最优解，原理是一样的，把原问题看成对偶问题，对偶问题看成原问题，就和 ch2.3.2 的求解逻辑完全一致了。</p>        </div>      </div>    </div></li><li><p>0.618 法求精确步长（ch3.2）</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-ee1d904a" role="button" aria-expanded="false" aria-controls="collapse-ee1d904a">        <div class="fold-arrow">▶</div>讲解      </div>      <div class="fold-collapse collapse" id="collapse-ee1d904a">        <div class="fold-content">          <p>每次缩函数值大的点即可。</p>        </div>      </div>    </div></li><li><p>最速下降法、牛顿法求解方法及其优缺点，共轭梯度法的优缺点（ch4）（直接给步长）</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-774ef8ba" role="button" aria-expanded="false" aria-controls="collapse-774ef8ba">        <div class="fold-arrow">▶</div>讲解      </div>      <div class="fold-collapse collapse" id="collapse-774ef8ba">        <div class="fold-content">          <p>同样很显然的一道题，透露说只要迭代一步？总之就那两个迭代公式，况且步长都给了，记一下方向的迭代即可，最速下降法就是负梯度方向，牛顿法就是二阶梯度的逆乘负梯度作为新的方向。至于优缺点，简单记忆一下即可。最速下降法无非程序简单但因为搜索方向是正交的导致收敛速度差，牛顿法虽然收敛速度快了但是存储的内容太多导致计算量变大，开销增加。</p><p>至于共轭梯度法的优缺点，优点就是该法是最速下降法和牛顿法的结合，每次的搜索方向是共轭的，这样就不用存储海塞矩阵并且收敛速度往往比最速下降法更快。主要用于解决正定二次函数的极小化问题。但在解决其余问题时可能会对搜索步长有极高的依赖，一旦搜索步长不够精准会导致整体的精度下降，收敛速度也会下降。</p>        </div>      </div>    </div></li><li><p><s>有效集方法解不等式约束的二次规划问题（ch6）（小概率考到，考到就gg，因为不会）</s></p></li><li><p>等式约束下的二次罚函数法（ch7）（一个约束函数，一个等式约束条件，共 4 问）</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-fa06789a" role="button" aria-expanded="false" aria-controls="collapse-fa06789a">        <div class="fold-arrow">▶</div>例题      </div>      <div class="fold-collapse collapse" id="collapse-fa06789a">        <div class="fold-content">          <ol><li><p>用拉格朗日乘子法求出最优解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">x^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span> 和拉格朗日乘子 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>λ</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">\lambda^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span></p></li><li><p>写出二次罚函数表达式 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>E</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P_E(x,\sigma)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mclose">)</span></span></span></span>，讨论罚因子 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> 在什么取值范围下可以使海塞矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="normal">∇</mi><mn>2</mn></msup><msub><mi>P</mi><mi>E</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\nabla^2P_E(x,\sigma)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord">∇</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mclose">)</span></span></span></span> 正定</p></li><li><p>求最优解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">x^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span>（直接计算一阶梯度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∇</mi></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">∇</span></span></span></span> 并用 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> 表示 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">x^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span> 即可）</p></li><li><p>当 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>σ</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">\sigma \to \infty</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord">∞</span></span></span></span>​ 时求最优解，并判断是否和第一问计算结果一致</p></li></ol><p>示例：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202407081507550.jpg" alt="等式约束的二次罚函数法"></p>        </div>      </div>    </div></li><li><p>约束最优化问题，约束条件中只有不等式（ch7）（用拉格朗日乘子法，已知有 3 个不等式，且 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">2^3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span> 个答案中只有一个是合法结果）</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-f639b72c" role="button" aria-expanded="false" aria-controls="collapse-f639b72c">        <div class="fold-arrow">▶</div>例题      </div>      <div class="fold-collapse collapse" id="collapse-f639b72c">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202407081608997.jpg" alt="拉格朗日乘子法解一般约束优化问题"></p>        </div>      </div>    </div></li></ul><p><s>很遗憾将这门课学成了面向已知考试题型的过拟合形式。我并不觉得我掌握了多少优化理论的知识，最多称得上知道了优化问题的大致分类和一些基本的优化应用。从我的笔记就可以看出，自第三章开始就没怎么涉及到理论的证明，确实是证不明白🤡。但这门课自下届开始就被取消了hh。。。。你我皆是局内人，祝好。</s></p>]]></content>
     
     
     <categories>
@@ -989,11 +989,11 @@
   
   
   <entry>
-    <title>ProbAndStat</title>
-    <link href="/GPA/4th-term/ProbAndStat/"/>
-    <url>/GPA/4th-term/ProbAndStat/</url>
+    <title>PyAlgo</title>
+    <link href="/GPA/4th-term/PyAlgo/"/>
+    <url>/GPA/4th-term/PyAlgo/</url>
     
-    <content type="html"><![CDATA[<h2 id="概率论与数理统计">概率论与数理统计</h2><h2 id="前言">前言</h2><p>学科情况：</p><table><thead><tr><th style="text-align:center">主讲教师</th><th style="text-align:center">学分配额</th><th style="text-align:center">学科类别</th></tr></thead><tbody><tr><td style="text-align:center">周效尧</td><td style="text-align:center">4</td><td style="text-align:center">学科基础课</td></tr></tbody></table><p>成绩组成：</p><table><thead><tr><th style="text-align:center">平时</th><th style="text-align:center">测验（×2）</th><th style="text-align:center">期末</th></tr></thead><tbody><tr><td style="text-align:center">10%</td><td style="text-align:center">30%</td><td style="text-align:center">60%</td></tr></tbody></table><p>教材情况：</p><table><thead><tr><th style="text-align:center">课程名称</th><th style="text-align:center">选用教材</th><th style="text-align:center">版次</th><th style="text-align:center">作者</th><th style="text-align:center">出版社</th><th style="text-align:center">ISBN号</th></tr></thead><tbody><tr><td style="text-align:center">概率论与数理统计Ⅰ</td><td style="text-align:center">《概率论与数理统计》</td><td style="text-align:center">第一版</td><td style="text-align:center">刘国祥王晓谦     等主编</td><td style="text-align:center">科学出版社</td><td style="text-align:center">978-7-03-038317-4</td></tr></tbody></table><p>学习资源：</p><ul><li><span class="github-emoji"><span>📺</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/1f4fa.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span> 视频资源：<a href="https://www.bilibili.com/video/BV1ot411y7mU">《概率论与数理统计》教学视频全集（宋浩）</a></li><li><span class="github-emoji"><span>📖</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/1f4d6.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span>  教材答案：<ul><li>本地路径（自用）：[概率论答案.pdf](D:\华为云盘\2. Score\4. 概率论与数理统计\概率论答案.pdf)</li><li>网盘资源（共享）：<a href="https://pan.baidu.com/s/1yeC0rxatHaLeNHQaW85Kpw?pwd=448w">提取码：448w</a></li></ul></li></ul><h2 id="第1章-事件与概率">第1章 事件与概率</h2><h3 id="1-1-随机事件与样本空间">1.1 随机事件与样本空间</h3><h4 id="1-1-1-样本空间">1.1.1 样本空间</h4><p>随机事件发生的总集合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Ω</span></span></span></span></p><h4 id="1-1-2-随机事件">1.1.2 随机事件</h4><p>事件是否发生取决于观察结果的事件</p><h4 id="1-1-3-事件之间的关系与运算">1.1.3 事件之间的关系与运算</h4><ol><li>包含：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>⊂</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \subset B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> or <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mo>⊂</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">B \subset A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span></li><li>相等：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>=</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A=B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></li><li>并（和）：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>∪</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \cup B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></li><li>交（积）：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>∩</mo><mi>B</mi><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi>A</mi><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \cap B \quad (AB)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:1em;"></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span></li><li><strong>互斥</strong>（互不相容）：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>B</mi><mo>=</mo><mi mathvariant="normal">Φ</mi></mrow><annotation encoding="application/x-tex">AB=\Phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Φ</span></span></span></span></li><li><strong>对立事件</strong>（余事件）：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>∩</mo><mi>B</mi><mo>=</mo><mi mathvariant="normal">Φ</mi><mo>∧</mo><mi>A</mi><mo>∪</mo><mi>B</mi><mo>=</mo><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">A \cap B=\Phi \land A \cup B=\Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Φ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Ω</span></span></span></span></li><li>差：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>−</mo><mi>B</mi><mo>=</mo><mi>A</mi><mo>∩</mo><mover accent="true"><mi>B</mi><mo stretchy="true">‾</mo></mover><mo>=</mo><mi>A</mi><mover accent="true"><mi>B</mi><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">A-B=A \cap \overline{B} = A \overline B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8833em;"></span><span class="mord mathnormal">A</span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span></span></li><li>德摩根律</li></ol><p>将事件发生的概率论转化为<strong>集合论</strong>进行计算与分析</p><h3 id="1-2-概率的三种定义及其性质">1.2 概率的三种定义及其性质</h3><h4 id="1-2-1-概率的统计定义">1.2.1 概率的统计定义</h4><p>从频率出发得到。</p><h4 id="1-2-2-概率的古典定义">1.2.2 概率的古典定义</h4><p>特征：</p><ul><li>样本空间是有限集</li><li>等可能性（试验中每个基本事件发生的概率是等可能的）</li></ul><p>内容：</p><ol><li>模型与计算公式</li><li>基本组合分析公式<ul><li>乘法、加法原理</li><li>排列公式</li><li>组合公式</li></ul></li><li>实例<ul><li>超几何概率</li><li>分房问题</li><li>生日问题</li></ul></li><li>古典概率的基本性质</li></ol><h4 id="1-2-3-概率的几何定义">1.2.3 概率的几何定义</h4><p>特征：</p><ul><li>样本空间不可列</li><li>等可能性</li></ul><p>内容：</p><ol><li>模型与计算公式</li><li>实例<ul><li>一维几何图形：公交车乘车问题</li><li>二维几何图形：会面问题、蒲丰（Buffon）投针问题</li></ul></li><li>几何概率的基本性质</li></ol><h4 id="1-2-4-概率的性质">1.2.4 概率的性质</h4><p>pass</p><h3 id="1-3-常用概型公式">1.3 常用概型公式</h3><h4 id="1-3-1-条件概率计算公式">1.3.1 条件概率计算公式</h4><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mi mathvariant="normal">∣</mi><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mi>B</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">P(B|A) = \frac{P(AB)}{P(A)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord">∣</span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><h4 id="1-3-2-乘法原理计算公式">1.3.2 乘法原理计算公式</h4><ul><li><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>基本</mtext><mo>:</mo><mtext>前提</mtext><mo>:</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\text{基本}:\text{前提}:P(A)&gt;0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord cjk_fallback">基本</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord cjk_fallback">前提</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mi mathvariant="normal">∣</mi><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(AB) = P(A)P(B|A)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord">∣</span><span class="mord mathnormal">A</span><span class="mclose">)</span></span></span></span></span></p></li><li><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>推广</mtext><mo>:</mo><mtext>前提</mtext><mo>:</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mn>1</mn></msub><msub><mi>A</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>A</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\text{推广}:\text{前提}:P(A_1A_2,...,A_n)&gt;0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord cjk_fallback">推广</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord cjk_fallback">前提</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">...</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mn>1</mn></msub><msub><mi>A</mi><mn>2</mn></msub><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><msub><mi>A</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mn>2</mn></msub><mi mathvariant="normal">∣</mi><msub><mi>A</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mn>3</mn></msub><mi mathvariant="normal">∣</mi><msub><mi>A</mi><mn>1</mn></msub><msub><mi>A</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>⋯</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>n</mi></msub><mi mathvariant="normal">∣</mi><msub><mi>A</mi><mn>1</mn></msub><msub><mi>A</mi><mn>2</mn></msub><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><msub><mi>A</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(A_1A_2...A_n) = P(A_1)P(A_2|A_1)P(A_3|A_1A_2) \cdots P(A_n|A_1A_2...A_{n-1})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">...</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">...</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p></li></ul><h4 id="1-3-3-全概公式">1.3.3 全概公式</h4><p>我们将样本空间 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Ω</span></span></span></span> 完全划分为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个互斥的区域，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>A</mi><mi>i</mi></msub></mstyle></mrow><annotation encoding="application/x-tex">\Omega = \displaystyle \sum_{i=1}^{n} A_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Ω</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> ，则在样本空间中事件 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> 发生的概率 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(B)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span> 就是在各子样本空间中概率之和，经过上述乘法公式变形，计算公式如下：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.16em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mi mathvariant="normal">Ω</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><msub><mi>A</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><msub><mi>A</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mo>⋯</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><msub><mi>A</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mi mathvariant="normal">∣</mi><msub><mi>A</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mi mathvariant="normal">∣</mi><msub><mi>A</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mo>⋯</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mi mathvariant="normal">∣</mi><msub><mi>A</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mi mathvariant="normal">∣</mi><msub><mi>A</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{equation*}\begin{aligned}    P(B) &amp;= P(B \Omega) \\    &amp;= P(BA_1) + P(BA_2) + \cdots + P(BA_n) \\    &amp;= P(A_1)P(B|A_1) + P(A_2)P(B|A_2) + \cdots + P(A_n)P(B|A_n) \\    &amp;= \sum_{i=1}^n P(A_i)P(B|A_i)\end{aligned}\end{equation*}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:7.7291em;vertical-align:-3.6145em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.1145em;"><span style="top:-6.1145em;"><span class="pstrut" style="height:6.1145em;"></span><span class="mord"><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.1145em;"><span style="top:-6.9259em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span><span style="top:-5.4259em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"></span></span><span style="top:-3.9259em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"></span></span><span style="top:-1.6145em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.6145em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.1145em;"><span style="top:-6.9259em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord">Ω</span><span class="mclose">)</span></span></span><span style="top:-5.4259em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-3.9259em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-1.6145em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.6145em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.6145em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><h4 id="1-3-4-贝叶斯公式">1.3.4 贝叶斯公式</h4><p>在上述全概公式的背景之下，现在希望求解事件 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> 在第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> 个子样本空间 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">A_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 中发生的概率，或者说第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> 个子样本空间对于事件 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> 的发生贡献了多少概率，记作 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>j</mi></msub><mi mathvariant="normal">∣</mi><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(A_j|B)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span> ，计算公式如下：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>j</mi></msub><mi mathvariant="normal">∣</mi><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mi mathvariant="normal">∣</mi><msub><mi>A</mi><mi>j</mi></msub><mo stretchy="false">)</mo></mrow><mstyle scriptlevel="0" displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mi mathvariant="normal">∣</mi><msub><mi>A</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mstyle></mfrac></mrow><annotation encoding="application/x-tex">P(A_j|B) = \frac{P(A_j)P(B|A_j)}{\displaystyle \sum_{i=1}^n P(A_i)P(B|A_i)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:4.2461em;vertical-align:-2.8191em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.11em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-3.8814em;"><span class="pstrut" style="height:3.6514em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-4.3284em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.8191em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><div class="note note-light">            <p>可以发现全概公式是计算事件发生的所有子样本空间的概率贡献，而贝叶斯公式是计算事件发生的总概率中某些子样本空间的概率贡献，前者是正向思维，后者是逆向思维</p>          </div><h3 id="1-4-事件的独立性及伯努利概型">1.4 事件的独立性及伯努利概型</h3><h4 id="1-4-1-独立性">1.4.1 独立性</h4><p><strong>定义</strong></p><ul><li><p>基本：若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo separator="true">,</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A,B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">A</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> 相互独立，则满足：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(AB)=P(A)P(B)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span></span></p></li><li><p>推广：若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>A</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>A</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">A_1,A_2,...,A_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">...</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 相互独立，则满足：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="normal">∀</mi><mspace width="1em"></mspace><mn>1</mn><mo>≤</mo><msub><mi>i</mi><mn>1</mn></msub><mo>&lt;</mo><msub><mi>i</mi><mn>2</mn></msub><mo>&lt;</mo><mo>⋯</mo><mo>&lt;</mo><msub><mi>i</mi><mi>k</mi></msub><mo>≤</mo><mi>n</mi><mtext>&nbsp;</mtext><mo stretchy="false">(</mo><mi>k</mi><mo>=</mo><mn>2</mn><mo separator="true">,</mo><mn>3</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>s</mi><mi mathvariant="normal">.</mi><mi>t</mi><mi mathvariant="normal">.</mi><mspace width="1em"></mspace><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><msub><mi>i</mi><mn>1</mn></msub></msub><msub><mi>A</mi><msub><mi>i</mi><mn>2</mn></msub></msub><mo>⋯</mo><msub><mi>A</mi><msub><mi>i</mi><mi>k</mi></msub></msub><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><msub><mi>i</mi><mn>1</mn></msub></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><msub><mi>i</mi><mn>2</mn></msub></msub><mo stretchy="false">)</mo><mo>⋯</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><msub><mi>i</mi><mi>k</mi></msub></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\forall \quad 1 \le i_1&lt;i_2&lt;\cdots&lt;i_k \le n\ (k=2,3,\cdots,n) \\s.t. \quad P(A_{i_1}A_{i_2}\cdots A_{i_k}) = P(A_{i_1})P(A_{i_2})\cdots P(A_{i_k})\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∀</span><span class="mspace" style="margin-right:1em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">n</span><span class="mspace">&nbsp;</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="mord">.</span><span class="mord mathnormal">t</span><span class="mord">.</span><span class="mspace" style="margin-right:1em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1512em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2559em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1512em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2559em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ul><p><strong>定理</strong></p><ul><li><p>基本：若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo separator="true">,</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A,B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">A</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> 相互独立，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo separator="true">,</mo><mover accent="true"><mi>B</mi><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">A,\overline{B}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">A</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span></span> 相互独立；<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>A</mi><mo stretchy="true">‾</mo></mover><mo separator="true">,</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\overline{A},B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0778em;vertical-align:-0.1944em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> 相互独立；<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>A</mi><mo stretchy="true">‾</mo></mover><mo separator="true">,</mo><mover accent="true"><mi>B</mi><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\overline{A},\overline{B}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0778em;vertical-align:-0.1944em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span></span> 相互独立</p></li><li><p>推广：若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>A</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>A</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">A_1,A_2,...,A_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">...</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 相互独立，则其中任意 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo stretchy="false">(</mo><mn>2</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">k(2 \le k \le n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mopen">(</span><span class="mord">2</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 个也相互独立，且满足：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>P</mi><mo stretchy="false">(</mo><mover accent="true"><msub><mi>A</mi><msub><mi>i</mi><mn>1</mn></msub></msub><mo>^</mo></mover><mover accent="true"><msub><mi>A</mi><msub><mi>i</mi><mn>2</mn></msub></msub><mo>^</mo></mover><mo>⋯</mo><mover accent="true"><msub><mi>A</mi><msub><mi>i</mi><mi>k</mi></msub></msub><mo>^</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mover accent="true"><msub><mi>A</mi><msub><mi>i</mi><mn>1</mn></msub></msub><mo>^</mo></mover><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mover accent="true"><msub><mi>A</mi><msub><mi>i</mi><mn>2</mn></msub></msub><mo>^</mo></mover><mo stretchy="false">)</mo><mo>⋯</mo><mi>P</mi><mo stretchy="false">(</mo><mover accent="true"><msub><mi>A</mi><msub><mi>i</mi><mi>k</mi></msub></msub><mo>^</mo></mover><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>s</mi><mi mathvariant="normal">.</mi><mi>t</mi><mi mathvariant="normal">.</mi><mspace width="1em"></mspace><mover accent="true"><msub><mi>A</mi><msub><mi>i</mi><mi>j</mi></msub></msub><mo>^</mo></mover><mo>=</mo><mi>A</mi><mtext>&nbsp;</mtext><mi>o</mi><mi>r</mi><mtext>&nbsp;</mtext><mover accent="true"><mi>A</mi><mo stretchy="true">‾</mo></mover><mtext>&nbsp;</mtext><mo stretchy="false">(</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}P(\hat{A_{i_1}}\hat{A_{i_2}}\cdots \hat{A_{i_k}}) = P(\hat{A_{i_1}})P(\hat{A_{i_2}})\cdots P(\hat{A_{i_k}}) \\s.t. \quad \hat{A_{i_j}} = A \ or \ \overline{A}\ (j=1,2,\cdots,k)\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.2135em;vertical-align:-1.3568em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8568em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span></span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">^</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span></span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">^</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1512em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2559em;"><span></span></span></span></span></span></span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">^</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2559em;"><span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span></span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">^</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span></span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">^</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1512em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2559em;"><span></span></span></span></span></span></span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">^</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2559em;"><span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-2.3032em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="mord">.</span><span class="mord mathnormal">t</span><span class="mord">.</span><span class="mspace" style="margin-right:1em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2819em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3473em;"><span></span></span></span></span></span></span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">^</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3473em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">A</span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right:0.02778em;">or</span><span class="mspace">&nbsp;</span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace">&nbsp;</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3568em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ul><p><strong>概念辨析</strong></p><ul><li><p>两两独立：对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个事件，两两独立，而不考虑三个及以上的关系。</p></li><li><p>相互独立：对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个事件，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mo>→</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">2 \to n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个事件的独立关系都需要考虑。</p></li><li><p>总结：对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个事件，满足两两独立需要 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>C</mi><mi>n</mi><mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">C_n^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0611em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> 个等式关系，对于相互独立需要 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>n</mi></msup><mo>−</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">2^n-(n+1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7477em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 个等式关系，因此：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>两两独立</mtext><mo>⊂</mo><mtext>相互独立</mtext></mrow><annotation encoding="application/x-tex">\text{两两独立} \subset \text{相互独立}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord text"><span class="mord cjk_fallback">两两独立</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord cjk_fallback">相互独立</span></span></span></span></span></span></p></li></ul><h4 id="1-4-2-伯努利概型">1.4.2 伯努利概型</h4><p>定义：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 重伯努利概型</p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 重：发生 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 次独立试验</li><li>伯努利概型：每次试验只有两种可能的结果</li></ul><p>模型：</p><ul><li><p>二项概率公式：n 次独立重复试验发生 k 次的概率：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mi>C</mi><mi>n</mi><mi>k</mi></msubsup><msup><mi>p</mi><mi>k</mi></msup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><msup><mo stretchy="false">)</mo><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup></mrow><annotation encoding="application/x-tex">C_n^k p^k (1-p)^{n-k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1491em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1491em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span></span></span></span></span></p></li><li><p>几何概率公式：在第 n 次试验首次成功的概率：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><msup><mo stretchy="false">)</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>p</mi></mrow><annotation encoding="application/x-tex">(1-p)^{n-1}p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord mathnormal">p</span></span></span></span></span></p></li></ul><h2 id="第2章-随机事件及其分布">第2章 随机事件及其分布</h2><div class="note note-light">            <p>我们知道，解决事件发生概率的问题，除了事件表示以外，我们还关心每一个事件发生的概率 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(X=k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span>，以及某些事件发生的概率 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mo stretchy="false">[</mo><mi>r</mi><mi>a</mi><mi>n</mi><mi>g</mi><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(X=[range))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">an</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord mathnormal">e</span><span class="mclose">))</span></span></span></span>。接下来我们将：</p><ul><li>首先介绍<strong>随机变量的概念</strong>以及<strong>分布函数的概念</strong></li><li>接着介绍随机变量对应的概率发生情况组成的集合。离散型的叫<strong>分布列</strong>，连续型的叫<strong>概率密度函数</strong>，并在其中贯穿分布函数的应用</li><li>最后介绍<strong>分布函数的复合</strong>。从离散型和连续型随机变量两个方向展开</li></ul>          </div><h3 id="2-1-随机变量及其概率分布">2.1 随机变量及其概率分布</h3><h4 id="2-1-1-随机变量的概念">2.1.1 随机变量的概念</h4><p>总的来说，随机变量就是一个样本空间与实数集的映射。我们定义样本空间 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi><mo>=</mo><mo stretchy="false">{</mo><mi>ω</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\Omega=\{ \omega \}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Ω</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="mclose">}</span></span></span></span>，其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span></span></span></span> 表示所有可能的事件，实数集 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span></span></span></span>，随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span>，则随机变量满足以下映射关系</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>X</mi><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>=</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">X(\omega)=R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span></span></span></span></span></p><h4 id="2-1-2-随机变量的分布函数">2.1.2 随机变量的分布函数</h4><ol><li>分布函数的定义：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(x)=P(X \le x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></li><li>分布函数的性质：<ul><li>非负有界性：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≤</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">0 \le F(x) \le 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></li><li>单调不减性：若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>&lt;</mo><msub><mi>x</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_1 &lt; x_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>≤</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(x_1) \le F(x_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mi>F</mi><mo stretchy="false">(</mo><mo>−</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><mo>−</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle F(-\infty) = \lim_{x \to -\infty} F(x) = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord">−</span><span class="mord">∞</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.5083em;vertical-align:-0.7583em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mrel mtight">→</span><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7583em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mi>F</mi><mo stretchy="false">(</mo><mo>+</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle F(+\infty) = \lim_{x \to +\infty} F(x) = 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord">+</span><span class="mord">∞</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.5083em;vertical-align:-0.7583em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mrel mtight">→</span><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7583em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></li><li>右连续性：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><msubsup><mi>x</mi><mn>0</mn><mo>+</mo></msubsup></mrow></munder><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mspace width="1em"></mspace><mo stretchy="false">(</mo><mo>−</mo><mi mathvariant="normal">∞</mi><mo>&lt;</mo><msub><mi>x</mi><mn>0</mn></msub><mo>&lt;</mo><mo>+</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \lim_{x\to x_0^+}F(x) = F(x_0)\quad(-\infty &lt; x_0 &lt; +\infty)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.8087em;vertical-align:-1.0587em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.2593em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mrel mtight">→</span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.821em;"><span style="top:-2.1885em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span><span style="top:-2.9043em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mbin mtight">+</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3115em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0587em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:1em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">∞</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">+</span><span class="mord">∞</span><span class="mclose">)</span></span></span></span></li></ul></li></ol><h3 id="2-2-离散型随机变量及其分布列">2.2 离散型随机变量及其分布列</h3><h4 id="2-2-1-离散性随机变量的分布列">2.2.1 离散性随机变量的分布列</h4><p>随机变量的取值都是整数，有以下三种表示方法</p><ol><li><p>公式法</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>p</mi><mi>k</mi></msub><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mo separator="true">,</mo><mspace width="1em"></mspace><mi>k</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo></mrow><annotation encoding="application/x-tex">p_k = P(X=x_k),\quad k = 1,2,\cdots,</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span></span></span></span></span></p></li><li><p>服从法</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>X</mi><mo>∼</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>1</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>2</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>3</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>p</mi><mn>1</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>p</mi><mn>2</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>p</mi><mn>3</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">X \sim \begin{pmatrix}x_1 &amp; x_2 &amp; x_3 &amp; \cdots \\p_1 &amp; p_2 &amp; p_3 &amp; \cdots\end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="minner">⋯</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p></li><li><p>表格法</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.16em" columnalign="center center center center center" columnlines="solid none none none" columnspacing="1em" rowlines="solid"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>X</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>1</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>2</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>3</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>P</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>p</mi><mn>1</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>p</mi><mn>2</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>p</mi><mn>3</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{array}{c|cccc}X &amp; x_1 &amp; x_2 &amp; x_3 &amp; \cdots \\\hlineP &amp; p_1 &amp; p_2 &amp; p_3 &amp; \cdots\end{array}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.45em;"><span class="pstrut" style="height:3.45em;"></span><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="vertical-separator" style="height:2.4em;border-right-width:0.04em;border-right-style:solid;margin:0 -0.02em;vertical-align:-0.95em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="minner">⋯</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span style="top:-3.7em;"><span class="pstrut" style="height:3.45em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span></span></span></p></li></ol><h4 id="2-2-2-常用离散性随机变量及其分布列">2.2.2 常用离散性随机变量及其分布列</h4><ul><li><p>0-1分布：即一个事件只有两面性，我们称这样的随机变量服从0-1分布或者两点分布，记作</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>X</mi><mo>∼</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><mo>−</mo><mi>p</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>p</mi></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">X \sim \begin{pmatrix}0 &amp; 1 \\1-p &amp; p\end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p></li><li><p>二项分布：其实就是 n 重伯努利试验，我们称这样的随机变量服从二项分布，分布列为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>C</mi><mi>n</mi><mi>k</mi></msubsup><msup><mi>p</mi><mi>k</mi></msup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><msup><mo stretchy="false">)</mo><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup></mrow><annotation encoding="application/x-tex">P(X=k) = C_n^kp^k(1-p)^{n-k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0991em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0991em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span></span></span></span>，记作</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>B</mi><mo stretchy="false">(</mo><mi>n</mi><mo separator="true">,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sim B(n,p)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mclose">)</span></span></span></span></span></p></li><li><p>几何分布：同样是伯努利事件，现在需要求解第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 次事件首次发生的概率，此时分布列为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><msup><mo stretchy="false">)</mo><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>p</mi></mrow><annotation encoding="application/x-tex">P(X=k)=(1-p)^{k-1}p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0991em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord mathnormal">p</span></span></span></span>，记作</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>G</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sim G(p)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">G</span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mclose">)</span></span></span></span></span></p></li><li><p>超几何分布：就是在 N 件含有 M 件次品的样品中无放回的抽取 n 件，问其中含有次品数量的分布列，为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msubsup><mi>C</mi><mi>M</mi><mi>k</mi></msubsup><msubsup><mi>C</mi><mrow><mi>N</mi><mo>−</mo><mi>M</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msubsup></mrow><msubsup><mi>C</mi><mi>N</mi><mi>n</mi></msubsup></mfrac><mo separator="true">,</mo><mspace width="1em"></mspace><mi>k</mi><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mi>min</mi><mo>⁡</mo><mrow><mo stretchy="false">(</mo><mi>n</mi><mo separator="true">,</mo><mi>M</mi><mo stretchy="false">)</mo></mrow></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle P(X=k)=\frac{C_M^k C_{N-M}^{n-k}}{C_N^n}, \quad k=0,1,2,\cdots,\min{(n, M)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.6106em;vertical-align:-0.9795em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6311em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6462em;"><span style="top:-2.4065em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2935em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.7419em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-2.4247em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2753em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8892em;"><span style="top:-2.4065em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span></span></span></span><span style="top:-3.1031em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3519em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9795em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">min</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mclose">)</span></span></span></span></span>，记作</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>X</mi><mo>∼</mo><mtext>超几何分布</mtext><mo stretchy="false">(</mo><mi>n</mi><mo separator="true">,</mo><mi>N</mi><mo separator="true">,</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sim \text{超几何分布}(n,N,M)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord cjk_fallback">超几何分布</span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mclose">)</span></span></span></span></span></p></li><li><p>泊松分布：当二项分布中，试验次数很大或者概率很小时，可以近似为泊松分布，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>C</mi><mi>n</mi><mi>k</mi></msubsup><msup><mi>p</mi><mi>k</mi></msup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><msup><mo stretchy="false">)</mo><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup><mo>→</mo><mfrac><msup><mi>λ</mi><mi>k</mi></msup><mrow><mi>k</mi><mo stretchy="false">!</mo></mrow></mfrac><msup><mi>e</mi><mrow><mo>−</mo><mi>λ</mi></mrow></msup></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle P(X=k)=C_n^k p^k(1-p)^{n-k} \to \frac{\lambda^k}{k!}e^{-\lambda}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1491em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1491em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2121em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5261em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">!</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">λ</span></span></span></span></span></span></span></span></span></span></span></span>，其中常数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\lambda &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">λ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，记作</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>P</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sim P(\lambda)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal">λ</span><span class="mclose">)</span></span></span></span></span></p><p>显然，泊松分布含有下面两个性质</p><ol><li><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>&gt;</mo><mn>0</mn><mo separator="true">,</mo><mi>k</mi><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo separator="true">,</mo><mo>⋯</mo></mrow><annotation encoding="application/x-tex">P(X=k) &gt; 0,k=0,1,\cdots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span></span></span></span></p></li><li><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \sum_{k=0}^\infty P(X=k)=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.9535em;vertical-align:-1.3021em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8479em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">∞</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3021em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></p>     <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-60042435" role="button" aria-expanded="false" aria-controls="collapse-60042435">        <div class="fold-arrow">▶</div>泊松分布正规性证明      </div>      <div class="fold-collapse collapse" id="collapse-60042435">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403151012817.png" alt="泊松分布正规性证明"></p>        </div>      </div>    </div></li></ol></li></ul><h3 id="2-3-连续型随机变量及其概率密度函数">2.3 连续型随机变量及其概率密度函数</h3><p>说白了其实就是离散性随机变量的积分加强版。现在随着事件发生的不同取值 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>，随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 发生的概率 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(X=x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 变成了连续的取值了（学名概率密度函数），于是分布函数（离散的叫分布列）的取值就没那么容易求了（其实一重定积分就可以）。接下来就从定义、性质、应用三个角度出发介绍概率密度函数以及相应的随机变量的分布函数。</p><h4 id="2-3-1-连续型随机变量的密度函数">2.3.1 连续型随机变量的密度函数</h4><p>概率密度函数，简称：密度函数 or 概率密度</p><ul><li><p>定义：设随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的分布函数为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>，如果存在非负可积函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>，使下式成立，则称 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 为连续型随机变量，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的概率密度函数</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">∀</mi><mi>x</mi><mo>∈</mo><mi>R</mi><mo separator="true">,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi>x</mi></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi>d</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">\forall x \in R,F(x) = \int_{-\infty}^{x} p(t)dt</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord">∀</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3846em;vertical-align:-0.9703em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span></span></span></p></li><li><p>性质：</p><ol><li><p>非负性：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">p(x) \ge 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></p></li><li><p>正规性：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\int_{-\infty}^{+\infty} p(x)dx = 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3804em;vertical-align:-0.4142em;"></span><span class="mop"><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9662em;"><span style="top:-2.3442em;margin-left:-0.1945em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.2579em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4142em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></p></li><li><p>可积性：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∀</mi><msub><mi>x</mi><mn>1</mn></msub><mo>≤</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo>≤</mo><mi>X</mi><mo>≤</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>−</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><msub><mi>x</mi><mn>1</mn></msub><msub><mi>x</mi><mn>2</mn></msub></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\forall x_1 \le x_2,P(x_1 \le X \le x_2) = F(x_2) - F(x_1) = \int_{x_1}^{x_2}p(x)dx</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord">∀</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.3152em;vertical-align:-0.4559em;"></span><span class="mop"><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8593em;"><span style="top:-2.3442em;margin-left:-0.1945em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2579em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4559em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span></span></p></li><li><p>分布函数可导性：若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 处连续，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>F</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F'(x) = p(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></p></li><li><p>已知事件但无意义性：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∀</mi><mi>x</mi><mo>∈</mo><mi>R</mi><mo separator="true">,</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\forall x \in R, P(X=x) = F(x) - F(x) = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord">∀</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></p><ul><li>离散型变量可以通过列举随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的取值来计算概率，但连续型随机变量这么做是无意义的</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">P(A) = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 不能推出 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 是不可能事件，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">P(A)=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 不能推出 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 是必然事件</li><li>对于连续型随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 有：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo>&lt;</mo><mi>X</mi><mo>&lt;</mo><msub><mi>X</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo>&lt;</mo><mi>X</mi><mo>≤</mo><msub><mi>X</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo>≤</mo><mi>X</mi><mo>&lt;</mo><msub><mi>X</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo>≤</mo><mi>X</mi><mo>≤</mo><msub><mi>X</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(x_1 &lt; X &lt; X_2)=P(x_1 &lt; X \le X_2)=P(x_1 \le X &lt; X_2)=P(x_1 \le X \le X_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></li></ul></li><li><p>实际描述性：密度函数的数值反映了随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 取 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 的临近值的概率的大小，因为</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi mathvariant="normal">Δ</mi><mi>x</mi><mo>≈</mo><msubsup><mo>∫</mo><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi>d</mi><mi>t</mi><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo>≤</mo><mi>X</mi><mo>≤</mo><mi>x</mi><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(x)\Delta x \approx \int_{x}^{x+\Delta x} p(t)dt = F(x+\Delta x) - F(x) = P(x \le X \le x+\Delta x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord">Δ</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.5032em;vertical-align:-0.9119em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5912em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mbin mtight">+</span><span class="mord mtight">Δ</span><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Δ</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Δ</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></p></li></ol></li></ul><h4 id="2-3-2-常用连续型随机变量及其密度函数">2.3.2 常用连续型随机变量及其密度函数</h4><table><thead><tr><th style="text-align:center"></th><th style="text-align:center">分布定义式</th><th style="text-align:center">概率密度函数</th><th style="text-align:center">分布函数</th></tr></thead><tbody><tr><td style="text-align:center">均匀分布</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>U</mi><mo stretchy="false">[</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">X \sim U[a,b]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mopen">[</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mclose">]</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mfrac><mn>1</mn><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow></mfrac><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>a</mi><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>b</mi><mo separator="true">,</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>0</mn><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>其他</mtext></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">p(x) = \begin{cases} \frac{1}{b-a}, &amp; a \le x \le b, \\ 0, &amp; \text{其他} \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight">a</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4033em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">0</span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">b</span><span class="mpunct">,</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">其他</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>0</mn><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>x</mi><mo>&lt;</mo><mi>a</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mfrac><mrow><mi>x</mi><mo>−</mo><mi>a</mi></mrow><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow></mfrac><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>a</mi><mo>≤</mo><mi>x</mi><mo>&lt;</mo><mi>b</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>x</mi><mo>≥</mo><mi>b</mi></mrow></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">F(x) = \begin{cases} 0, &amp; x &lt; a \\ \frac{x - a}{b - a}, &amp; a \le x &lt; b \\ 1, &amp; x \ge b \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:4.32em;vertical-align:-1.91em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35em;"><span style="top:-2.2em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.192em;"><span class="pstrut" style="height:3.15em;"></span><span style="height:0.316em;width:0.8889em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.8889em" height="0.316em" style="width:0.8889em" viewBox="0 0 888.89 316" preserveAspectRatio="xMinYMin"><path d="M384 0 H504 V316 H384z M384 0 H504 V316 H384z"></path></svg></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.292em;"><span class="pstrut" style="height:3.15em;"></span><span style="height:0.316em;width:0.8889em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.8889em" height="0.316em" style="width:0.8889em" viewBox="0 0 888.89 316" preserveAspectRatio="xMinYMin"><path d="M384 0 H504 V316 H384z M384 0 H504 V316 H384z"></path></svg></span></span><span style="top:-4.6em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.85em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em;"><span style="top:-4.41em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">0</span><span class="mpunct">,</span></span></span><span style="top:-2.97em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8023em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight">a</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4033em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span></span></span><span style="top:-1.53em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">1</span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.91em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em;"><span style="top:-4.41em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">a</span></span></span><span style="top:-2.97em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">b</span></span></span><span style="top:-1.53em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.91em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td></tr><tr><td style="text-align:center">指数分布</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>e</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sim e (\lambda)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">e</span><span class="mopen">(</span><span class="mord mathnormal">λ</span><span class="mclose">)</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>0</mn><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>x</mi><mo>&lt;</mo><mn>0</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>λ</mi><msup><mi>e</mi><mrow><mo>−</mo><mi>λ</mi><mi>x</mi></mrow></msup><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>x</mi><mo>≥</mo><mn>0</mn></mrow></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">p(x) = \begin{cases} 0, &amp; x &lt; 0 \\ \lambda e^{-\lambda x} , &amp; x \ge 0 \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">0</span><span class="mpunct">,</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">λ</span><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>0</mn><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>x</mi><mo>&lt;</mo><mn>0</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>λ</mi><mi>x</mi></mrow></msup><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>x</mi><mo>≥</mo><mn>0</mn></mrow></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">F(x) = \begin{cases} 0, &amp; x &lt; 0 \\ 1- e^{-\lambda x}, &amp; x \ge 0 \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">0</span><span class="mpunct">,</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">λ</span><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td></tr><tr><td style="text-align:center">正态分布</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><mi>μ</mi><mo separator="true">,</mo><msup><mi>σ</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sim N(\mu,\sigma^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mopen">(</span><span class="mord mathnormal">μ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><msqrt><mrow><mn>2</mn><mi>π</mi></mrow></msqrt><mi>σ</mi></mrow></mfrac><msup><mi>e</mi><mrow><mo>−</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>μ</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mrow><mn>2</mn><msup><mi>σ</mi><mn>2</mn></msup></mrow></mfrac></mrow></msup><mo separator="true">,</mo><mspace width="1em"></mspace><mo>−</mo><mi mathvariant="normal">∞</mi><mo>&lt;</mo><mi>x</mi><mo>&lt;</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">p(x) = \frac{1}{\sqrt{2 \pi} \sigma } e^{- \frac{(x - \mu)^2}{2 \sigma ^2}} , \quad -\infty &lt; x &lt; + \infty</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.8671em;vertical-align:-0.538em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.551em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord sqrt mtight"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9128em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight" style="padding-left:0.833em;"><span class="mord mtight">2</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">π</span></span></span><span style="top:-2.8728em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail mtight" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1272em;"><span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.538em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.3291em;"><span style="top:-3.4534em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.251em;"><span style="top:-2.5062em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9384em;"><span style="top:-2.9384em;margin-right:0.1em;"><span class="pstrut" style="height:2.6444em;"></span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.5021em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight">x</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight">μ</span><span class="mclose mtight"><span class="mclose mtight">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0484em;"><span style="top:-3.0484em;margin-right:0.1em;"><span class="pstrut" style="height:2.6444em;"></span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4938em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord">∞</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">+</span><span class="mord">∞</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><msqrt><mrow><mn>2</mn><mi>π</mi></mrow></msqrt><mi>σ</mi></mrow></mfrac><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi>x</mi></msubsup><msup><mi>e</mi><mrow><mo>−</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>y</mi><mo>−</mo><mi>μ</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mrow><mn>2</mn><msup><mi>σ</mi><mn>2</mn></msup></mrow></mfrac></mrow></msup><mi>d</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">F(x) = \frac{1}{\sqrt{2 \pi} \sigma } \int_{- \infty}^x e^{- \frac{(y - \mu)^2}{2 \sigma ^2}} dy</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.8671em;vertical-align:-0.538em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.551em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord sqrt mtight"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9128em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight" style="padding-left:0.833em;"><span class="mord mtight">2</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">π</span></span></span><span style="top:-2.8728em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail mtight" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1272em;"><span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.538em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8593em;"><span style="top:-2.3442em;margin-left:-0.1945em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.2579em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4142em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.3291em;"><span style="top:-3.4534em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.251em;"><span style="top:-2.5062em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9384em;"><span style="top:-2.9384em;margin-right:0.1em;"><span class="pstrut" style="height:2.6444em;"></span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.5021em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight">μ</span><span class="mclose mtight"><span class="mclose mtight">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0484em;"><span style="top:-3.0484em;margin-right:0.1em;"><span class="pstrut" style="height:2.6444em;"></span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4938em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span></td></tr></tbody></table><p>补充说明：</p><ul><li><p>指数分布：其中参数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\lambda &gt;0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">λ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></p></li><li><p>正态分布：一般正态函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> <strong>转化为标准正态函数</strong> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Φ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Phi(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Φ</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 公式：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">Φ</mi><mo stretchy="false">(</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mi>μ</mi></mrow><mi>σ</mi></mfrac><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(x) = \Phi(\frac{x - \mu}{\sigma})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.9463em;vertical-align:-0.686em;"></span><span class="mord">Φ</span><span class="mopen">(</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span></span></span></span></span></p><p>于是对于计算一般正态函数的函数值，就可以通过下式将其转化为标准正态函数，最后查表即可：</p></li></ul><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">Φ</mi><mo stretchy="false">(</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mi>μ</mi></mrow><mi>σ</mi></mfrac><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(X \le x) = F(x) = \Phi (\frac{x - \mu}{\sigma})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.9463em;vertical-align:-0.686em;"></span><span class="mord">Φ</span><span class="mopen">(</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span></span></span></span></span></p><h3 id="2-4-随机变量函数的分布">2.4 随机变量函数的分布</h3><div class="note note-light">            <p>本目主要介绍给定一个随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的分布情况，通过一个关系式 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y=g(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 来求解随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span>​ 的分布情况</p>          </div><h4 id="2-4-1-离散型随机变量函数的分布">2.4.1 离散型随机变量函数的分布</h4><p>通过关系式 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y=g(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>​ 将所有的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 的取值全部枚举出来，然后一一统计即可。</p><h4 id="2-4-2-连续型随机变量函数的分布">2.4.2 连续型随机变量函数的分布</h4><p>给定随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的概率密度函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p_X(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>，以及关系式 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y=g(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>，求解随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 的分布函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F_Y(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span>、概率密度函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p_Y(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span></p><ul><li><p><strong>方法一</strong>：先求解随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 的分布函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F_Y(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span>，再通过对其求导得到概率密度函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p_Y(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span>​</p><p>即先 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>P</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>≤</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>P</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>P</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F_Y(y) = P_Y(Y \le y) = P_Y(g(X) \le y) = P_X(X \le f(y)) = F_X(f(y))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">))</span></span></span></span> 得到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 的分布函数</p><p>再对 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F_Y(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> 求导得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><msub><mi>p</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>y</mi></mrow></mfrac><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>y</mi></mrow></mfrac><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>F</mi><mi>X</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋅</mo><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>p</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋅</mo><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle p_Y(y) = \frac{d}{dy} F_Y(y) = \frac{d}{dy} F_X(f(y)) = F_X'(f(y)) \cdot f'(y) = p_X(f(y)) \cdot f'(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0519em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-2.453em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0519em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0519em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span></p></li><li><p><strong>方法二</strong>：如果关系式 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y=g(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 单调且反函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mi>h</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x=h(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> 连续可导，则可以直接得出随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 的概率密度函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p_Y(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> 为下式。其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span> 为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Y=g(X)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span> 的取值范围（<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 应该怎么取值，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> 就应该怎么取值，从而计算出 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> 的取值范围）</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>p</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>p</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋅</mo><mi mathvariant="normal">∣</mi><msup><mi>h</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mi mathvariant="normal">∣</mi><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>α</mi><mo>&lt;</mo><mi>y</mi><mo>&lt;</mo><mi>β</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>0</mn><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>其他</mtext></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">p_Y(y) = \begin{cases}p_X(h(y)) \cdot |h'(y)|, &amp; \alpha &lt; y &lt; \beta \\0, &amp; \text{其他}\end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">h</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mord">∣</span><span class="mpunct">,</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">0</span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">其他</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p></li></ul><h2 id="第3章-随机向量及其分布">第3章 随机向量及其分布</h2><div class="note note-light">            <p>实际生活中，只采用一个随机变量描述事件往往是不够的。本章引入多维的随机变量概念，构成随机向量，从二维开始，推广到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>​ 维。</p>          </div><h3 id="3-1-二维随机向量的联合分布">3.1 二维随机向量的联合分布</h3><div class="note note-light">            <p>现在我们讨论二维随机向量的联合分布。所谓的联合分布，其实就是一个曲面的概率密度（离散型就是点集），而分布函数就是对其积分得到的三维几何体的体积（散点和）而已。</p>          </div><h4 id="3-1-1-联合分布函数">3.1.1 联合分布函数</h4><p>定义：我们定义满足下式的二元函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> 为二维随机向量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span> 的联合分布函数</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mi>x</mi><mo stretchy="false">)</mo><mo>∩</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>≤</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mi>x</mi><mo separator="true">,</mo><mi>Y</mi><mo>≤</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(x,y) = P((X \le x) \cap (Y \le y)) = P(X \le x, Y \le y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">((</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span></span></p>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-7ae072b2" role="button" aria-expanded="false" aria-controls="collapse-7ae072b2">        <div class="fold-arrow">▶</div>几何意义：F(x,y) 即左下方无界矩形的面积      </div>      <div class="fold-collapse collapse" id="collapse-7ae072b2">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403271717479.png" alt="联合分布函数的几何意义"></p>        </div>      </div>    </div><p>性质：其实配合几何意义理解就会很容易了</p><ol><li>固定某一维度，另一维度是单调不减的</li><li>对于每个维度都是右连续的</li><li>固定某一维度，另一维度趋近于负无穷对应的函数值为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></li><li>二维前缀和性质，右上角的矩阵面积 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\ge 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></li></ol><h4 id="3-1-2-联合分布列">3.1.2 联合分布列</h4><p>定义：若二维随机向量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span> 的所有可能取值是至多可列的，则称 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span> 为二维离散型随机向量</p><p>表示：有两种表示二维随机向量分布列的方法，如下</p>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-819aef24" role="button" aria-expanded="false" aria-controls="collapse-819aef24">        <div class="fold-arrow">▶</div>二维随机向量分布列的表示方法      </div>      <div class="fold-collapse collapse" id="collapse-819aef24">        <div class="fold-content">          <ol><li><p>公式法</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>p</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><msub><mi>x</mi><mi>i</mi></msub><mo separator="true">,</mo><mi>Y</mi><mo>=</mo><msub><mi>y</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo separator="true">,</mo><mspace width="1em"></mspace><mi>i</mi><mo separator="true">,</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo></mrow><annotation encoding="application/x-tex">p_{ij} = P(X=x_i,Y = y_i), \quad i,j=1,2,\cdots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">i</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span></span></span></span></span></p></li><li><p>表格法：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403271728248.png" alt="二维联合分布列"></p></li></ol>        </div>      </div>    </div><p>性质：</p><ol><li>非负性：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>≥</mo><mn>0</mn><mo separator="true">,</mo><mspace width="1em"></mspace><mi>i</mi><mo separator="true">,</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo></mrow><annotation encoding="application/x-tex">p_{ij} \ge 0, \quad i,j=1,2,\cdots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9221em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">i</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span></span></span></span></li><li>正规性：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><munder><mo>∑</mo><mi>i</mi></munder><munder><mo>∑</mo><mi>j</mi></munder><msub><mi>p</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mn>1</mn></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \sum_{i} \sum_{j} p_{ij} = 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4638em;vertical-align:-1.4138em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4138em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></li></ol><h4 id="3-1-3-联合密度函数">3.1.3 联合密度函数</h4><p>定义：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi>x</mi></msubsup><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi>y</mi></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo stretchy="false">)</mo><mi>d</mi><mi>u</mi><mi>d</mi><mi>v</mi></mrow><annotation encoding="application/x-tex">F(x,y) = \int_{-\infty}^x \int_{-\infty}^y p(u,v)dudv</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3846em;vertical-align:-0.9703em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">u</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span></span></span></p><p>性质：</p><ol><li>非负性：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∀</mi><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>∈</mo><mi>R</mi><mo separator="true">,</mo><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\forall x,y \in R,p(x,y) \ge 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">∀</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></li><li>正规性：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mi>d</mi><mi>y</mi><mo>=</mo><mn>1</mn></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} p(x,y)dxdy = 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4915em;vertical-align:-0.9703em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></li></ol><p>结论：</p><ol><li><strong>联合分布函数</strong>相比于一元分布函数，其实就是从概率密度函数与 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 轴围成的面积转变为了概率密度曲面与 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mi>O</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">xOy</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>​ 平面围成的<strong>体积</strong></li><li>若概率密度曲面在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mi>O</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">xOy</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> 平面的投影为点集或线集，则对应的概率显然为零</li></ol><p>常见的连续型二维分布：</p><ol><li><p>二维均匀分布：假设该曲面与 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mi>O</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">xOy</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> 面的投影面积为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span>，则分布函数其实就是一个高为定值 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mi>S</mi></mfrac></mrow><annotation encoding="application/x-tex">\frac{1}{S}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">S</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 的柱体，密度函数为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mfrac><mn>1</mn><mi>S</mi></mfrac><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>G</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>0</mn><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>其他</mtext></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">p(x,y) = \begin{cases}\frac{1}{S}, &amp;(x,y) \in G \\0, &amp;\text{其他}\end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">S</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">0</span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">G</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">其他</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p></li><li><p>二元正态分布：不要求掌握密度函数，可以感受一下密度函数的图像：</p>     <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-e058054e" role="button" aria-expanded="false" aria-controls="collapse-e058054e">        <div class="fold-arrow">▶</div>二元正态分布 - 密度函数的图像      </div>      <div class="fold-collapse collapse" id="collapse-e058054e">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403290948792.png" alt="二元正态分布 - 密度函数的图像"></p>        </div>      </div>    </div></li></ol><p>计算题：往往给出一个二元密度函数，然后让我们求解（1）密度函数中的参数、（2）分布函数、（3）联合事件某个区域下的概率</p><p>（1）我们利用二元密度函数的正规性，直接积分值为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 即可</p><p>（2）划分区间后进行曲面积分即可，在曲面积分时往往结合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 型和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 型的二重积分进行</p><p>（3）画出概率密度曲面在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mi>O</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">xOy</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> 面的投影，然后积分即可</p><h3 id="3-2-二维随机向量的边缘分布">3.2 二维随机向量的边缘分布</h3><div class="note note-light">            <p>对于二元分布函数，我们也可以研究其中任意一个随机变量的分布情况，而不需要考虑另一个随机变量的取值情况。举一个实例就是，假如当前的随机向量是身高和体重，所谓的只研究其中一个随机变量，即边缘分布函数的情形就是，我们不考虑身高只考虑体重的分布情况；或者我们不考虑体重，只考虑身高的分布情况。接下来，我们将从边缘分布函数入手，逐渐学习离散型的分布列与连续型的分布函数。</p>          </div><h4 id="3-2-1-边缘分布函数">3.2.1 边缘分布函数</h4><p>我们称 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">F_X(x),F_Y(y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>) 分别为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span> 关于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo separator="true">,</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X,Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 的边缘分布函数，定义式为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mi>x</mi><mo separator="true">,</mo><mi>Y</mi><mo>&lt;</mo><mo>+</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>y</mi><mo>→</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mo>+</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>≤</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>&lt;</mo><mo>+</mo><mi mathvariant="normal">∞</mi><mo separator="true">,</mo><mi>Y</mi><mo>≤</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><mo>+</mo><mi mathvariant="normal">∞</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}F_X(x) = P(X \le x) = P(X \le x,Y &lt; +\infty) = \lim_{y \to +\infty} F(x,y) = F(x,+\infty) \\F_Y(y) = P(Y \le y) = P(X &lt; +\infty, Y \le y) = \lim_{x \to +\infty} F(x,y) = F(+\infty,y)\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.8744em;vertical-align:-1.6872em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.1872em;"><span style="top:-4.3472em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">+</span><span class="mord">∞</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="mrel mtight">→</span><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8361em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">+</span><span class="mord">∞</span><span class="mclose">)</span></span></span><span style="top:-2.3711em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">+</span><span class="mord">∞</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mrel mtight">→</span><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7583em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord">+</span><span class="mord">∞</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.6872em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><h4 id="3-2-2-边缘分布列">3.2.2 边缘分布列</h4><p>所谓的边缘分布列，就是固定一个随机变量，另外的随机变量取遍，组成的分布列。即：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>p</mi><mrow><mi>i</mi><mo>⋅</mo></mrow></msub><mo>=</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munderover><msub><mi>p</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo separator="true">,</mo><mspace width="1em"></mspace><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><msub><mi>y</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>p</mi><mrow><mo>⋅</mo><mi>j</mi></mrow></msub><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munderover><msub><mi>p</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo separator="true">,</mo><mspace width="1em"></mspace><mi>j</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}P(X=x_i) = p_{i\cdot}=\sum_{j=1}^{+\infty} p_{ij}, \quad i=1,2,\cdots \\P(Y=y_j) = p_{\cdot j}=\sum_{i=1}^{+\infty} p_{ij}, \quad j=1,2,\cdots\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:6.8081em;vertical-align:-3.1541em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.6541em;"><span style="top:-5.6541em;"><span class="pstrut" style="height:3.7583em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mord mtight">⋅</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7583em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4138em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span></span></span><span style="top:-2.1819em;"><span class="pstrut" style="height:3.7583em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">⋅</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7583em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.1541em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>我们称：</p><ul><li><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(X=x_i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 为随机向量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span> 关于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的边缘分布列</p></li><li><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><msub><mi>y</mi><mi>j</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(Y=y_j)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 为随机向量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span> 关于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 的边缘分布列</p></li></ul><h4 id="3-2-3-边缘密度函数">3.2.3 边缘密度函数</h4><p>所谓的的边缘密度函数，可以与边缘分布列进行类比，也就是固定一个随机变量，另外的随机变量取遍。只不过连续型的取遍就是无数个点，而离散型的取遍是可列个点，仅此而已。即：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi>p</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mo>+</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi>x</mi></msubsup><mrow><mo fence="true">[</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo stretchy="false">)</mo><mi>d</mi><mi>v</mi><mo fence="true">]</mo></mrow><mi>d</mi><mi>u</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mi>d</mi><mi>y</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}P(X=x) &amp;= p_X(x) \\&amp;= \frac{d}{dx} F_X(x) \\&amp;= \frac{d}{dx} F(x,+\infty) \\&amp;= \frac{d}{dx} \int_{-\infty}^{x} \left [ \int_{-\infty}^{+\infty} p(u,v) dv \right ] du \\&amp;= \int_{-\infty}^{+\infty} p(x,y) dy \\\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:11.7979em;vertical-align:-5.649em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.149em;"><span style="top:-8.8302em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span><span style="top:-6.7987em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"></span></span><span style="top:-4.4413em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"></span></span><span style="top:-1.9341em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"></span></span><span style="top:0.8574em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.649em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.149em;"><span style="top:-8.8302em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span><span style="top:-6.7987em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span><span style="top:-4.4413em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">+</span><span class="mord">∞</span><span class="mclose">)</span></span></span><span style="top:-1.9341em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">u</span></span></span><span style="top:0.8574em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.649em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi>p</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>y</mi></mrow></mfrac><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>y</mi></mrow></mfrac><mi>F</mi><mo stretchy="false">(</mo><mo>+</mo><mi mathvariant="normal">∞</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>y</mi></mrow></mfrac><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mo fence="true">[</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi>y</mi></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo stretchy="false">)</mo><mi>d</mi><mi>v</mi><mo fence="true">]</mo></mrow><mi>d</mi><mi>u</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>y</mi></mrow></mfrac><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi>y</mi></msubsup><mrow><mo fence="true">[</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo stretchy="false">)</mo><mi>d</mi><mi>u</mi><mo fence="true">]</mo></mrow><mi>d</mi><mi>v</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}P(Y=y) &amp;= p_Y(y) \\&amp;= \frac{d}{dy} F_Y(y) \\&amp;= \frac{d}{dy} F(+\infty,y) \\&amp;= \frac{d}{dy} \int_{-\infty}^{+\infty} \left [ \int_{-\infty}^{y} p(u,v) dv \right ] du \\&amp;= \frac{d}{dy} \int_{-\infty}^{y} \left [ \int_{-\infty}^{+\infty} p(u,v) du \right ] dv \\&amp;= \int_{-\infty}^{+\infty} p(x,y) dx \\\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:14.9783em;vertical-align:-7.2391em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:7.7391em;"><span style="top:-10.4204em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span><span style="top:-8.3889em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"></span></span><span style="top:-5.8371em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"></span></span><span style="top:-3.1354em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"></span></span><span style="top:-0.3439em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"></span></span><span style="top:2.4476em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:7.2391em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:7.7391em;"><span style="top:-10.4204em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span><span style="top:-8.3889em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span><span style="top:-5.8371em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord">+</span><span class="mord">∞</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span><span style="top:-3.1354em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">u</span></span></span><span style="top:-0.3439em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">u</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span><span style="top:2.4476em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:7.2391em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>我们称：</p><ul><li><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(X=x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 为随机向量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span> 关于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的边缘密度函数</p></li><li><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(Y=y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> 为随机向量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span> 关于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 的边缘密度函数</p></li></ul><h3 id="3-3-随机向量的条件分布">3.3 随机向量的条件分布</h3><div class="note note-light">            <p>本目主要介绍的是条件分布。所谓的条件分布，其实就是在约束一个随机变量为定值的情况下，另外一个随机变量的取值情况。与上述联合分布、边缘分布的区别在于：</p><ul><li>联合分布、边缘分布的分布函数是一个体积（散点和），概率密度（分布列）是一个曲面（点集）</li><li>条件分布的分布函数是一个面积（散点和），概率密度（分布列）是一个曲线（点集）</li></ul>          </div><h4 id="3-3-1-离散型随机向量的条件分布列和条件分布函数">3.3.1 离散型随机向量的条件分布列和条件分布函数</h4><p>条件分布列，即散点情况：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>p</mi><mrow><mi>i</mi><mi mathvariant="normal">∣</mi><mi>j</mi></mrow></msub><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><msub><mi>x</mi><mi>i</mi></msub><mtext>&nbsp;</mtext><mi mathvariant="normal">∣</mi><mtext>&nbsp;</mtext><mi>Y</mi><mo>=</mo><msub><mi>y</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><msub><mi>x</mi><mi>i</mi></msub><mo separator="true">,</mo><mi>Y</mi><mo>=</mo><msub><mi>y</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><msub><mi>y</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><msub><mi>p</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>p</mi><mrow><mo>⋅</mo><mi>j</mi></mrow></msub></mfrac><mo separator="true">,</mo><mspace width="1em"></mspace><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>p</mi><mrow><mi>j</mi><mi mathvariant="normal">∣</mi><mi>i</mi></mrow></msub><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><msub><mi>y</mi><mi>j</mi></msub><mtext>&nbsp;</mtext><mi mathvariant="normal">∣</mi><mtext>&nbsp;</mtext><mi>X</mi><mo>=</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><msub><mi>x</mi><mi>i</mi></msub><mo separator="true">,</mo><mi>Y</mi><mo>=</mo><msub><mi>y</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><msub><mi>p</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>p</mi><mrow><mi>i</mi><mo>⋅</mo></mrow></msub></mfrac><mo separator="true">,</mo><mspace width="1em"></mspace><mi>j</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}p_{i|j} = P(X=x_i\ |\ Y=y_j) = \frac{P(X=x_i,Y=y_i)}{P(Y=y_i)} = \frac{p_{ij}}{p_{\cdot j}}, \quad i=1,2,\cdots \\p_{j|i} = P(Y=y_j\ |\ X=x_i) = \frac{P(X=x_i,Y=y_i)}{P(X=x_i)} = \frac{p_{ij}}{p_{i\cdot }}, \quad j=1,2,\cdots\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.3621em;vertical-align:-2.4311em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9311em;"><span style="top:-4.9311em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mord mtight">∣</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace">&nbsp;</span><span class="mord">∣</span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">⋅</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9721em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span></span></span><span style="top:-2.2319em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mord mtight">∣</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace">&nbsp;</span><span class="mord">∣</span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mord mtight">⋅</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.4311em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>我们称：</p><ul><li><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mrow><mi>i</mi><mi mathvariant="normal">∣</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">p_{i|j}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7858em;vertical-align:-0.3552em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mord mtight">∣</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span></span></span></span> 为在给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><msub><mi>y</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">Y=y_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 的条件下 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的条件分布列</p></li><li><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mrow><mi>j</mi><mi mathvariant="normal">∣</mi><mi>i</mi></mrow></msub></mrow><annotation encoding="application/x-tex">p_{j|i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7858em;vertical-align:-0.3552em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mord mtight">∣</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span></span></span></span> 为在给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>=</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">X=x_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的条件下 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 的条件分布列</p></li></ul><p>条件分布函数，即点集情况：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><msub><mi>y</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mi>x</mi><mtext>&nbsp;</mtext><mi mathvariant="normal">∣</mi><mtext>&nbsp;</mtext><mi>Y</mi><mo>=</mo><msub><mi>y</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mo>=</mo><munder><mo>∑</mo><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>≤</mo><mi>x</mi></mrow></munder><mfrac><msub><mi>p</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>p</mi><mrow><mo>⋅</mo><mi>j</mi></mrow></msub></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mi mathvariant="normal">∣</mi><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>≤</mo><mi>y</mi><mtext>&nbsp;</mtext><mi mathvariant="normal">∣</mi><mtext>&nbsp;</mtext><mi>X</mi><mo>=</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><munder><mo>∑</mo><mrow><msub><mi>y</mi><mi>j</mi></msub><mo>≤</mo><mi>y</mi></mrow></munder><mfrac><msub><mi>p</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>p</mi><mrow><mi>i</mi><mo>⋅</mo></mrow></msub></mfrac></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}F(x|y_j) = P(X \le x\ | \ Y=y_j) = \sum _{x_i\le x} \frac{p_{ij}}{p_{\cdot j}} \\F(y|x_i) = P(Y \le y\ | \ X=x_i) = \sum _{y_j\le y} \frac{p_{ij}}{p_{i \cdot}}\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.6349em;vertical-align:-2.5675em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.0675em;"><span style="top:-5.0675em;"><span class="pstrut" style="height:3.1076em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mspace">&nbsp;</span><span class="mord">∣</span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.8888em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mrel mtight">≤</span><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3613em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">⋅</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9721em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.2986em;"><span class="pstrut" style="height:3.1076em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace">&nbsp;</span><span class="mord">∣</span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.8888em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:-0.0359em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2819em;"><span></span></span></span></span></span></span><span class="mrel mtight">≤</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4585em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mord mtight">⋅</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.5675em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>我们称：</p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><msub><mi>y</mi><mi>j</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(x|y_j)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 为在给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><msub><mi>y</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">Y=y_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 的条件下 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的条件分布函数</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mi mathvariant="normal">∣</mi><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(y|x_i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 为在给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>=</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">X=x_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的条件下 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 的条件分布函数</li></ul><h4 id="3-3-2-连续型随机向量的条件密度函数和条件分布函数">3.3.2 连续型随机向量的条件密度函数和条件分布函数</h4><p>条件密度函数，即联合分布的概率密度曲面上，约束了某一维度的随机变量为定值，于是条件密度函数的图像就是一个空间曲线：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi>p</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mfrac><mo separator="true">,</mo><mspace width="1em"></mspace><mo>−</mo><mi mathvariant="normal">∞</mi><mo>&lt;</mo><mi>x</mi><mo>&lt;</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>y</mi><mi mathvariant="normal">∣</mi><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi>p</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo separator="true">,</mo><mspace width="1em"></mspace><mo>−</mo><mi mathvariant="normal">∞</mi><mo>&lt;</mo><mi>y</mi><mo>&lt;</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}p(x|y) = \frac{p(x,y)}{p_Y(y)}, \quad -\infty &lt; x &lt; +\infty \\p(y|x) = \frac{p(x,y)}{p_X(x)}, \quad -\infty &lt; y &lt; +\infty\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.326em;vertical-align:-2.413em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.913em;"><span style="top:-4.913em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord">∞</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">+</span><span class="mord">∞</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord">∞</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">+</span><span class="mord">∞</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.413em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>我们称：</p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(x|y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> 为在给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">Y=y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> 的条件下 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的条件密度函数</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>y</mi><mi mathvariant="normal">∣</mi><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(y|x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 为在给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">X=x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 的条件下 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 的条件密度函数</li></ul><p>条件分布函数，即上述曲线的分段积分结果：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mi>x</mi><mtext>&nbsp;</mtext><mi mathvariant="normal">∣</mi><mtext>&nbsp;</mtext><mi>Y</mi><mo>=</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi>x</mi></msubsup><mfrac><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>u</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi>p</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mfrac><mi>d</mi><mi>u</mi><mo separator="true">,</mo><mspace width="1em"></mspace><mo>−</mo><mi mathvariant="normal">∞</mi><mo>&lt;</mo><mi>x</mi><mo>&lt;</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mi mathvariant="normal">∣</mi><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>≤</mo><mi>y</mi><mtext>&nbsp;</mtext><mi mathvariant="normal">∣</mi><mtext>&nbsp;</mtext><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi>y</mi></msubsup><mfrac><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi>p</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mi>d</mi><mi>v</mi><mo separator="true">,</mo><mspace width="1em"></mspace><mo>−</mo><mi mathvariant="normal">∞</mi><mo>&lt;</mo><mi>y</mi><mo>&lt;</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}F(x|y) = P(X \le x \ | \ Y=y) = \int_{-\infty}^x \frac{p(u,y)}{p_Y(y)} du,\quad -\infty &lt; x &lt; +\infty \\F(y|x) = P(Y \le y \ | \ X=x) = \int_{-\infty}^y \frac{p(x,v)}{p_X(x)} dv, \quad -\infty &lt; y &lt; +\infty\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.3946em;vertical-align:-2.4473em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9473em;"><span style="top:-4.9473em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mspace">&nbsp;</span><span class="mord">∣</span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">d</span><span class="mord mathnormal">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord">∞</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">+</span><span class="mord">∞</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace">&nbsp;</span><span class="mord">∣</span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord">∞</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">+</span><span class="mord">∞</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.4473em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>我们称：</p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(x|y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> 为在给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">Y=y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> 的条件下 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的条件分布函数</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mi mathvariant="normal">∣</mi><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(y|x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 为在给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">X=x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 的条件下 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 的条件分布函数</li></ul><h3 id="3-4-随机变量的独立性">3.4 随机变量的独立性</h3><div class="note note-light">            <p>本目主要介绍随机变量的独立性。我们知道随机事件之间是有独立性的，即满足 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(AB)=P(A)P(B)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span> 的事件，那么随机变量之间也有独立性吗？答案是有的，以生活中的例子为实例，比如我和某个同学进教室，就是独立的两个随机变量。下面开始介绍。</p>          </div><ul><li><p>定义：我们定义如果两个随机变量的分布函数满足下式，则两个随机变量相互独立：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(x,y)=F_X(x)F_Y(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span></span></p></li><li><p>性质：对于随机向量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span></p><ol><li><p>随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 相互独立的充分必要条件是：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mtext>离散型:</mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><msub><mi>x</mi><mi>i</mi></msub><mo separator="true">,</mo><mi>Y</mi><mo>=</mo><msub><mi>y</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><msub><mi>y</mi><mi>j</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mtext>连续型:</mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>p</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mi>p</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\text{离散型:}&amp; P(X=x_i,Y=y_j) = P(X=x_i)P(Y=y_j) \\\text{连续型:}&amp; p(x,y) = p_X(x)p_Y(y)\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">离散型</span><span class="mord">:</span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">连续型</span><span class="mord">:</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li><li><p>若随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 相互独立，且 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h(\cdot)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord">⋅</span><span class="mclose">)</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(\cdot)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord">⋅</span><span class="mclose">)</span></span></span></span> 连续，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>g</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h(X),g(Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span> 也相互独立</p></li></ol></li></ul><h3 id="3-5-随机向量函数的分布">3.5 随机向量函数的分布</h3><div class="note note-light">            <p>在 2.4 目中我们了解到了随机变量函数的分布，现在我们讨论随机向量函数的分布。在生活中，假设我们已经知道了一个人群中所有人的身高和体重的分布情况，现在想要血糖根据身高和体重的分布情况，就需要用到本目的理念。我们从离散型和连续型随机向量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span> 出发，讨论 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(X,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span> 的分布情况。</p>          </div><h4 id="3-5-1-离散型随机向量函数的分布">3.5.1 离散型随机向量函数的分布</h4><p>按照规则枚举即可。</p><h4 id="3-5-2-连续型随机向量函数的分布">3.5.2 连续型随机向量函数的分布</h4><p>与连续型随机变量函数的分布类似，这类题目一般也是：给定随机向量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span> 的密度函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> 和 映射函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span>，现在需要求解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z=g(X,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span> 的分布函数（若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> 二元连续，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span></span></span> 也是连续型随机变量）。方法同理，先求解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span></span></span> 的分布函数，再对 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span></span></span> 求导得到密度函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>Z</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p_Z(z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">Z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span></span>​。接下来我们介绍两种常见随机向量的分布。</p><p><strong>(1) 和的分布：</strong></p><ul><li><p>先求分布函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>F</mi><mi>Z</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F_Z(z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">Z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span></span>：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>F</mi><mi>Z</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>+</mo><mi>Y</mi><mo>≤</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mo>∬</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi><mo>≤</mo><mi>z</mi></mrow></munder><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mi>d</mi><mi>y</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi>z</mi></msubsup><mrow><mo fence="true">[</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>t</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo fence="true">]</mo></mrow><mi>d</mi><mi>t</mi></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi>z</mi></msubsup><mrow><mo fence="true">[</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>t</mi><mo>−</mo><mi>y</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mi>d</mi><mi>y</mi><mo fence="true">]</mo></mrow><mi>d</mi><mi>t</mi></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr></mtable></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}F_Z(z) &amp;= P(X+Y \le z) \\&amp;= \iint\limits_{x+y \le z} p(x,y) dxdy \\&amp;\begin{align}&amp;= \int _{-\infty}^z \left [ \int_{-\infty}^{+\infty} p(x,t-x)dx \right ] dt \\&amp;= \int _{-\infty}^z \left [ \int_{-\infty}^{+\infty} p(t-y,y)dy \right ] dt\end{align}\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:10.7523em;vertical-align:-5.1262em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.6262em;"><span style="top:-9.8277em;"><span class="pstrut" style="height:5.0415em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">Z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:-7.8067em;"><span class="pstrut" style="height:5.0415em;"></span><span class="mord"></span></span><span style="top:-2.7569em;"><span class="pstrut" style="height:5.0415em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.1262em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.6262em;"><span style="top:-9.8277em;"><span class="pstrut" style="height:5.0415em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:-7.8067em;"><span class="pstrut" style="height:5.0415em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="mspace" style="margin-right:0.4445em;"></span><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.361em;"><span style="top:-1.8878em;margin-left:-0.4445em;"><span class="pstrut" style="height:3.36em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mbin mtight">+</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="mrel mtight">≤</span><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span><span style="top:-3.361em;"><span class="pstrut" style="height:3.36em;"></span><span><span class="mop op-symbol large-op" style="margin-right:0.44445em;">∬</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.7083em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-2.7569em;"><span class="pstrut" style="height:5.0415em;"></span><span class="mord"><span class="mord"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.0415em;"><span style="top:-5.0415em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.5415em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.0415em;"><span style="top:-5.0415em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.5415em;"><span></span></span></span></span></span></span><span class="tag"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.0415em;"><span style="top:-5.0415em;"><span class="pstrut" style="height:3.5212em;"></span><span class="eqn-num"></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.5212em;"></span><span class="eqn-num"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.5415em;"><span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.1262em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li><li><p>由分布函数定义：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi>x</mi></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>d</mi><mi>u</mi></mrow><annotation encoding="application/x-tex">F_X(x) = \int_{-\infty}^xp(u)du</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3846em;vertical-align:-0.9703em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">u</span></span></span></span></span></p></li><li><p>所以可得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi><mo>=</mo><mi>X</mi><mo>+</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">Z=X+Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 的密度函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>Z</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p_Z(z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">Z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span></span> 为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>p</mi><mi>Z</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>z</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mspace width="1em"></mspace></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>p</mi><mi>Z</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mi>y</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mi>d</mi><mi>y</mi><mspace width="1em"></mspace></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}p_Z(z) = \int_{-\infty}^{+\infty} p(x,z-x)dx \quad &amp;(1) \\p_Z(z) = \int_{-\infty}^{+\infty} p(z-y,y)dy \quad &amp;(2) \\\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.583em;vertical-align:-2.5415em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.0415em;"><span style="top:-5.0415em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">Z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:1em;"></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">Z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:1em;"></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.5415em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.0415em;"><span style="top:-5.0415em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mopen">(</span><span class="mord">2</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.5415em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li><li><p>若 X 和 Y 相互独立，还可得卷积式：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left right" columnspacing="0em 1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>p</mi><mi>Z</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>z</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><msub><mi>p</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⋅</mo><msub><mi>p</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mspace width="1em"></mspace></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>p</mi><mi>Z</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mi>y</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mi>d</mi><mi>y</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><msub><mi>p</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mi>y</mi><mo stretchy="false">)</mo><mo>⋅</mo><msub><mi>p</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mi>d</mi><mi>y</mi><mspace width="1em"></mspace></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}p_Z(z) &amp;= \int_{-\infty}^{+\infty} p(x,z-x)dx \\&amp;= \int_{-\infty}^{+\infty} p_X(x)\cdot p_Y(z-x) dx \quad &amp;(1) \\p_Z(z) &amp;= \int_{-\infty}^{+\infty} p(z-y,y)dy \\&amp;= \int_{-\infty}^{+\infty} p_X(z-y)\cdot p_Y(y) dy \quad &amp;(2)\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:11.166em;vertical-align:-5.333em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.833em;"><span style="top:-7.833em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">Z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:-5.0415em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">Z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:0.5415em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.333em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.833em;"><span style="top:-7.833em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span><span style="top:-5.0415em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:1em;"></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:0.5415em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:1em;"></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.333em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.0415em;"><span style="top:-5.0415em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span></span></span><span style="top:0.5415em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">2</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.333em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ul><p><strong>(2) 次序统计量的分布（对于两个相互独立的随机变量 X 和 Y）：</strong></p><ul><li><p>对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mo>=</mo><mi>max</mi><mo>⁡</mo><mrow><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex">M=\max{(X,Y)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">max</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span></span> 的分布函数，有：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>F</mi><mi>M</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>M</mi><mo>≤</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>max</mi><mo>⁡</mo><mrow><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><mo>≤</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mi>z</mi><mo separator="true">,</mo><mi>Y</mi><mo>≤</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mi>z</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>≤</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>⋅</mo><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}F_M(z) &amp;= P(M \le z) \\&amp;= P(\max{(X,Y)} \le z) \\&amp;= P(X \le z, Y \le z) \\&amp;= P(X \le z) \cdot P(Y \le z) \\&amp;= F_X(z) \cdot F_Y(z)\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:7.5em;vertical-align:-3.5em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4em;"><span style="top:-6.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:-4.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-3.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-1.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-0.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.5em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4em;"><span style="top:-6.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:-4.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mop">max</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:-3.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:-1.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:-0.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.5em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li><li><p>对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mi>min</mi><mo>⁡</mo><mrow><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex">N=\min{(X,Y)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">min</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span></span> 的分布函数，有：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>F</mi><mi>N</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>N</mi><mo>≤</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>min</mi><mo>⁡</mo><mrow><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><mo>≤</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>1</mn><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mi>min</mi><mo>⁡</mo><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>+</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><mo>≥</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>1</mn><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≥</mo><mi>z</mi><mo separator="true">,</mo><mi>Y</mi><mo>≥</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>1</mn><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≥</mo><mi>z</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>≥</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>1</mn><mo>−</mo><mo stretchy="false">[</mo><mn>1</mn><mo>−</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>⋅</mo><mo stretchy="false">[</mo><mn>1</mn><mo>−</mo><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}F_N(z) &amp;= P(N \le z) \\&amp;= P(\min{(X,Y)} \le z) \\&amp;= 1 - P(\min{(X+Y)} \ge z) \\&amp;= 1 - P(X \ge z,Y \ge z) \\&amp;= 1 - P(X \ge z) \cdot P(Y \ge z) \\&amp;= 1 - [1 - F_X(z)] \cdot [1 - F_Y(z)]\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:9em;vertical-align:-4.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.75em;"><span style="top:-6.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:-5.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-0.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:0.59em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.25em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.75em;"><span style="top:-6.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:-5.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mop">min</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mop">min</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:-0.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:0.59em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">[</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">[</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)]</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.25em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li><li><p>若拓展到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个相互独立且同分布的随机变量，则有：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>F</mi><mi>M</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo stretchy="false">[</mo><mi>F</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">]</mo><mi>n</mi></msup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>p</mi><mi>M</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>n</mi><mi>p</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>F</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">]</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}F_M(z) &amp;= [F(z)]^n \\p_M(z) &amp;= np(z)[F(z)]^{n-1}\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0241em;vertical-align:-1.2621em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7621em;"><span style="top:-3.9221em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:-2.3979em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2621em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7621em;"><span style="top:-3.9221em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span><span style="top:-2.3979em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2621em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>F</mi><mi>N</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>1</mn><mo>−</mo><mo stretchy="false">[</mo><mn>1</mn><mo>−</mo><mi>F</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">]</mo><mi>n</mi></msup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>p</mi><mi>N</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>n</mi><mi>p</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mn>1</mn><mo>−</mo><mi>F</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">]</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}F_N(z) &amp;= 1 - [1-F(z)]^n \\p_N(z) &amp;= np(z)[1-F(z)]^{n-1}\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0241em;vertical-align:-1.2621em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7621em;"><span style="top:-3.9221em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:-2.3979em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2621em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7621em;"><span style="top:-3.9221em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">[</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span><span style="top:-2.3979em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mopen">[</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2621em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ul><h2 id="第4章-随机变量的数字特征">第4章 随机变量的数字特征</h2><div class="note note-light">            <p>本章我们将学习随机变量的一些数字特征。所谓的数字特征其实就是随机变量分布的一些内在属性，比如均值、方差、协方差等等，有些分布特性甚至可以通过某个数字特征而直接觉得。其中<strong>期望</strong>和<strong>方差</strong>往往用来衡量单个随机变量的特征，而<strong>协方差</strong>与<strong>相关系数</strong>则是用来衡量随机变量之间的数字特征。接下来开始介绍。</p>          </div><h3 id="4-1-数学期望">4.1 数学期望</h3><div class="note note-light">            <p><strong>加权平均</strong>概念的严格数学定义。</p>          </div><h4 id="4-1-1-随机变量的数学期望">4.1.1 随机变量的数学期望</h4><ul><li><p>离散型</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>E</mi><mi>X</mi><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><msub><mi>x</mi><mi>i</mi></msub><msub><mi>p</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">EX = \sum_{i=1}^{\infty} x_i p_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">EX</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p></li><li><p>连续型</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>E</mi><mi>X</mi><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>x</mi><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}&amp;EX = \int_{-\infty}^{+\infty} xp(x)dx\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.7915em;vertical-align:-1.1458em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6458em;"><span style="top:-3.6458em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.1458em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6458em;"><span style="top:-3.6458em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.07847em;">EX</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.1458em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ul><h4 id="4-1-2-随机变量函数的数学期望">4.1.2 随机变量函数的数学期望</h4><ul><li><p>离散型</p><ul><li><p>一元</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>E</mi><mi>g</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mi>g</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mi>p</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Eg(X) = \sum_{i=1}^{\infty}g(x_i)p_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p></li><li><p>二元</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>E</mi><mi>g</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mi>g</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>y</mi><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mi>p</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">Eg(X,Y) = \sum_{i=1}^{\infty}\sum_{j=1}^{\infty}g(x_i,y_i)p_{ij}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.0652em;vertical-align:-1.4138em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4138em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span></span></p></li></ul></li><li><p>连续型</p><ul><li><p>一元</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>E</mi><mi>g</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">Eg(X) = \int_{-\infty}^{+\infty}g(x)p(x)dx</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4915em;vertical-align:-0.9703em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span></span></span></p></li><li><p>二元</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>E</mi><mi>g</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>g</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>y</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mi>d</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">Eg(X,Y) = \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}g(x_i,y_i)p(x,y)dxdy</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4915em;vertical-align:-0.9703em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span></span></p></li></ul></li></ul><h4 id="4-1-3-数学期望的性质">4.1.3 数学期望的性质</h4><ol><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mi>C</mi><mo>=</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">EC=C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">EC</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span></li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>C</mi><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mi>C</mi><mi>E</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">E(CX)=CEX</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">CX</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">CEX</span></span></span></span></li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo>+</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mi>X</mi><mo>+</mo><mi>E</mi><mi>Y</mi></mrow><annotation encoding="application/x-tex">E(X+Y)=EX+EY</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">EX</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span></li><li>若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 相互独立，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mi>X</mi><mi>E</mi><mi>Y</mi></mrow><annotation encoding="application/x-tex">E(XY)=EXEY</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">EXE</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span></li></ol><h3 id="4-2-方差">4.2 方差</h3><div class="note note-light">            <p>随机变量的取值与均值之间的<strong>离散程度</strong>。</p>          </div><h4 id="4-2-1-方差的定义">4.2.1 方差的定义</h4><p>我们定义随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的方差 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(X)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span> 为：（全部可由期望的性质推导而来）</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>E</mi><mrow><mo fence="true">[</mo><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>E</mi><mi>X</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo fence="true">]</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>E</mi><mrow><mo fence="true">(</mo><msup><mi>X</mi><mn>2</mn></msup><mo fence="true">)</mo></mrow><mo>−</mo><mo stretchy="false">(</mo><mi>E</mi><mi>X</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}D(X) &amp;= E\left[(X-EX)^2\right ] \\&amp;= E\left ( X^2 \right ) - (EX)^2\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0482em;vertical-align:-1.2741em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7741em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span><span style="top:-2.3859em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2741em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7741em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">[</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">EX</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">]</span></span></span></span></span><span style="top:-2.3859em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">EX</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2741em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><h4 id="4-2-2-方差的性质">4.2.2 方差的性质</h4><p>下列方差的性质全部可由上述方差的定义式，结合期望的性质推导而来：</p><ol><li><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>a</mi><mi>X</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mi>D</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(aX+b) = a^2D(X)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span></p></li><li><p>若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>X</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>X</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>⋯</mo></mrow><annotation encoding="application/x-tex">X_1,X_2,\cdots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span></span></span></span> 相互独立，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>a</mi><msub><mi>X</mi><mn>1</mn></msub><mo>±</mo><mi>b</mi><msub><mi>X</mi><mn>2</mn></msub><mo>±</mo><mo>⋯</mo><mtext> </mtext><mo stretchy="false">)</mo><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mi>D</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mi>D</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mo>⋯</mo></mrow><annotation encoding="application/x-tex">D(aX_1 \pm bX_2 \pm \cdots) = a^2D(X_1) + b^2D(X_2) + \cdots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord mathnormal">b</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.313em;"></span><span class="minner">⋯</span></span></span></span></p></li><li><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mrow><mo fence="true">[</mo><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>E</mi><mi>X</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo fence="true">]</mo></mrow><mo>≤</mo><mi>E</mi><mrow><mo fence="true">[</mo><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>C</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">E\left[ (X-EX)^2 \right] \le E \left [ (X-C)^2 \right ]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">[</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">EX</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">]</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">[</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">]</span></span></span></span></span></span>​</p></li><li><p>切比雪夫不等式 <s>（本以为不要求掌握的，但是被小测拷打了，补一下）</s>：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">∀</mi><mi>ϵ</mi><mo>&gt;</mo><mn>0</mn><mo separator="true">,</mo><mi>P</mi><mo stretchy="false">(</mo><mi mathvariant="normal">∣</mi><mi>X</mi><mo>−</mo><mi>E</mi><mi>X</mi><mi mathvariant="normal">∣</mi><mo>&lt;</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mo>≥</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mi>D</mi><mi>X</mi></mrow><msup><mi>ϵ</mi><mn>2</mn></msup></mfrac></mrow><annotation encoding="application/x-tex">\forall \epsilon &gt;0, P(|X - EX| &lt; \epsilon) \ge 1 - \frac{DX}{\epsilon^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord">∀</span><span class="mord mathnormal">ϵ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">EX</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">ϵ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0463em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">ϵ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p></li></ol><h3 id="4-3-结论与推导（补）">4.3 结论与推导（补）</h3><table><thead><tr><th style="text-align:center">类型</th><th style="text-align:center">分布</th><th style="text-align:center">符号</th><th style="text-align:center">期望 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(X)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span></th><th style="text-align:center">方差 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(X)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span></th></tr></thead><tbody><tr><td style="text-align:center">离散型</td><td style="text-align:center">0-1 分布</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∼</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><mo>−</mo><mi>p</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>p</mi></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">X \sim \begin{pmatrix} 0 &amp; 1 \\ 1-p &amp; p \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(1-p)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mclose">)</span></span></span></span></td></tr><tr><td style="text-align:center">—</td><td style="text-align:center">*二项分布</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>B</mi><mo stretchy="false">(</mo><mi>n</mi><mo separator="true">,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sim B(n,p)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mclose">)</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>p</mi></mrow><annotation encoding="application/x-tex">np</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">p</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>p</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">np(1-p)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mclose">)</span></span></span></span></td></tr><tr><td style="text-align:center">—</td><td style="text-align:center">几何分布</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>G</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sim G(p)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">G</span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mclose">)</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mi>p</mi></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \frac{1}{p}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2019em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">p</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mn>1</mn><mo>−</mo><mi>p</mi></mrow><msup><mi>p</mi><mn>2</mn></msup></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \frac{1-p}{p^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2019em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td></tr><tr><td style="text-align:center">—</td><td style="text-align:center">*泊松分布</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>P</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sim P(\lambda)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal">λ</span><span class="mclose">)</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">λ</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">λ</span></span></span></span></td></tr><tr><td style="text-align:center">连续型</td><td style="text-align:center">均匀分布</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>U</mi><mo stretchy="false">[</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">X \sim U[a,b]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mopen">[</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mclose">]</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mn>2</mn></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \frac{a+b}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>−</mo><mi>a</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mn>12</mn></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \frac{(b-a)^2}{12}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">12</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">a</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td></tr><tr><td style="text-align:center">—</td><td style="text-align:center">指数分布</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>e</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sim e(\lambda)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">e</span><span class="mopen">(</span><span class="mord mathnormal">λ</span><span class="mclose">)</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mi>λ</mi></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \frac{1}{\lambda}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">λ</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><msup><mi>λ</mi><mn>2</mn></msup></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \frac{1}{\lambda^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td></tr><tr><td style="text-align:center">—</td><td style="text-align:center">*正态分布</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><mi>μ</mi><mo separator="true">,</mo><msup><mi>σ</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sim N(\mu,\sigma^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mopen">(</span><span class="mord mathnormal">μ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\sigma^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></td></tr></tbody></table><div class="note note-warning">            <p>注：打星号表示在两个随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo separator="true">,</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X,Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 相互独立时，具备可加性。具体的：</p><ol><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mi>μ</mi><mn>1</mn></msub><mo separator="true">,</mo><msubsup><mi>σ</mi><mn>1</mn><mn>2</mn></msubsup><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>Y</mi><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mi>μ</mi><mn>2</mn></msub><mo separator="true">,</mo><msubsup><mi>σ</mi><mn>2</mn><mn>2</mn></msubsup><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi><mo>±</mo><mi>Y</mi><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mi>μ</mi><mn>1</mn></msub><mo>±</mo><msub><mi>μ</mi><mn>2</mn></msub><mo separator="true">,</mo><msubsup><mi>σ</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mn>2</mn><mn>2</mn></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sim N(\mu_1,\sigma_1^2), Y \sim N(\mu_2,\sigma_2^2) \to X\pm Y\sim N(\mu_1\pm\mu_2,\sigma_1^2+\sigma_2^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4519em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4519em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0622em;vertical-align:-0.2481em;"></span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4519em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4519em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>B</mi><mo stretchy="false">(</mo><msub><mi>n</mi><mn>1</mn></msub><mo separator="true">,</mo><mi>p</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>Y</mi><mo>∼</mo><mi>B</mi><mo stretchy="false">(</mo><msub><mi>n</mi><mn>2</mn></msub><mo separator="true">,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi><mo>+</mo><mi>Y</mi><mo>∼</mo><mi>B</mi><mo stretchy="false">(</mo><msub><mi>n</mi><mn>1</mn></msub><mo>+</mo><msub><mi>n</mi><mn>2</mn></msub><mo separator="true">,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sim B(n_1,p), Y \sim B(n_2,p) \to X+Y\sim B(n_1+n_2,p)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mclose">)</span></span></span></span></li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>λ</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>Y</mi><mo>∼</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>λ</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi><mo>+</mo><mi>Y</mi><mo>∼</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>λ</mi><mn>1</mn></msub><mo>+</mo><msub><mi>λ</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sim P(\lambda_1),Y\sim P(\lambda_2) \to X+Y \sim P(\lambda_1+\lambda_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></li></ol>          </div>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-65e4fb32" role="button" aria-expanded="false" aria-controls="collapse-65e4fb32">        <div class="fold-arrow">▶</div>推导      </div>      <div class="fold-collapse collapse" id="collapse-65e4fb32">        <div class="fold-content">          <p>推导的根本方式还是从定义出发。当然为了省事也可以从性质出发。</p><p>0-1 分布</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404202313030.jpg" alt="0-1 分布"></p><p>二项分布</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404202313503.jpg" alt="二项分布"></p><p>几何分布</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404202313329.jpg" alt="几何分布"></p><p>泊松分布</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404202313316.jpg" alt="泊松分布"></p><p>均匀分布</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404202313239.jpg" alt="均匀分布"></p><p>指数分布</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404202313832.jpg" alt="指数分布"></p>        </div>      </div>    </div><h3 id="4-4-协方差与相关系数">4.4 协方差与相关系数</h3><h4 id="4-4-1-协方差">4.4.1 协方差</h4><p>定义：随机变量 X 与 Y 的协方差 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cov(X,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span> 为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>E</mi><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>E</mi><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>−</mo><mi>E</mi><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mi>Y</mi><mo stretchy="false">)</mo><mo>−</mo><mi>E</mi><mi>X</mi><mi>E</mi><mi>Y</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}Cov(X,Y)&amp;= E[(X-EX)(Y-EY)] \\&amp;= E(XY) - EXEY\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mopen">[(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">EX</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)]</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">EXE</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>特别的：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mi>D</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">Cov(X,X) = DX</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span></span></p><p>性质：</p><ol><li>交换律：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>Y</mi><mo separator="true">,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cov(X,Y)=Cov(Y,X)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span></li><li>提取率：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>a</mi><mi>X</mi><mo separator="true">,</mo><mi>b</mi><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mi>b</mi><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cov(aX,bY)=abCov(X,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">bY</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">ab</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span></li><li>分配率：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo>+</mo><msub><mi>X</mi><mn>2</mn></msub><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>+</mo><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>2</mn></msub><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cov(X_1+X_2,Y) = Cov(X_1,Y)+Cov(X_2,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span></li><li>独立性：若 X 与 Y 相互独立，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">Cov(X,Y)=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>；反之不一定成立</li><li>放缩性：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow><mo fence="true">[</mo><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo fence="true">]</mo></mrow><mn>2</mn></msup><mo>≤</mo><mi>D</mi><mi>X</mi><mo>⋅</mo><mi>D</mi><mi>Y</mi></mrow><annotation encoding="application/x-tex">\left[Cov(X,Y)\right]^2 \le DX \cdot DY</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.204em;vertical-align:-0.25em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mclose delimcenter" style="top:0em;">]</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span></li></ol><h4 id="4-4-2-相关系数">4.4.2 相关系数</h4><p>定义：相关系数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span></span></span></span> 是用来刻画两个随机变量之间<strong>线性</strong>相关关系强弱的一个数字特征，注意是线性关系。<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>ρ</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|\rho|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">ρ</span><span class="mord">∣</span></span></span></span> 越接近 0，则说明两个随机变量越不线性相关；<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>ρ</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|\rho|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">ρ</span><span class="mord">∣</span></span></span></span> 越接近 1，则说明两个随机变量越线性相关，定义式为</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>ρ</mi><mrow><mi>X</mi><mo separator="true">,</mo><mi>Y</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><mrow><msqrt><mrow><mi>D</mi><mi>X</mi></mrow></msqrt><msqrt><mrow><mi>D</mi><mi>Y</mi></mrow></msqrt></mrow></mfrac></mrow><annotation encoding="application/x-tex">\rho_{X,Y} = \frac{Cov(X,Y)}{\sqrt{DX}\sqrt{DY}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.357em;vertical-align:-0.93em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.1833em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9267em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-2.8867em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1133em;"><span></span></span></span></span></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9267em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span><span style="top:-2.8867em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1133em;"><span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.93em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><p>特别的：</p><ol><li>若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>&lt;</mo><mi>ρ</mi><mo>&lt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">0 &lt; \rho &lt; 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6835em;vertical-align:-0.0391em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，则称 X 与 Y 正相关</li><li>若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>1</mn><mo>&lt;</mo><mi>ρ</mi><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">-1&lt;\rho&lt;0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，则称 X 与 Y 负相关</li></ol><p>性质：</p><ol><li>放缩性（由协方差性质5可得）：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>ρ</mi><mi mathvariant="normal">∣</mi><mo>≤</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">|\rho| \le 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">ρ</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></li><li>独立性（由协方差性质4可得）：若 X 与 Y 相互独立，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">p=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>；反之不一定成立</li><li>线性相关性（不予证明）：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>ρ</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">|\rho|=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">ρ</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 的充分必要条件是存在常数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo stretchy="false">(</mo><mi>a</mi><mo mathvariant="normal">≠</mo><mn>0</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a(a\ne0),b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span></span></span></span> 使得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mi>a</mi><mi>X</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">P(Y=aX+b)=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">a</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></li></ol><h4 id="4-4-3-独立性与线性相关性（补）">4.4.3 独立性与线性相关性（补）</h4><p>一般的：对于两个随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span></p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 相互独立 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 线性无关（可以用线性相关的定义式结合协方差计算公式导出）</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 相互独立 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↚</mo></mrow><annotation encoding="application/x-tex">\nleftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel amsrm">↚</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 线性无关（因为有可能出现 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 非线性相关）</li></ul><p>特别的：对于满足二维正态分布的随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span>，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>∼</mo><mo stretchy="false">(</mo><msub><mi>μ</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>μ</mi><mn>2</mn></msub><mo separator="true">,</mo><msubsup><mi>σ</mi><mn>1</mn><mn>2</mn></msubsup><mo separator="true">,</mo><msubsup><mi>σ</mi><mn>2</mn><mn>2</mn></msubsup><mo separator="true">,</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,Y) \sim (\mu_1,\mu_2,\sigma_1^2,\sigma_2^2,\rho)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4519em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4519em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">ρ</span><span class="mclose">)</span></span></span></span></p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 相互独立 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 线性无关</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 相互独立 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>←</mo></mrow><annotation encoding="application/x-tex">\leftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">←</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span>​ 线性无关</li></ul>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-92251a21" role="button" aria-expanded="false" aria-controls="collapse-92251a21">        <div class="fold-arrow">▶</div>证明 - 二维正态分布的两个随机变量：相互独立 等价于 线性无关      </div>      <div class="fold-collapse collapse" id="collapse-92251a21">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404261112302.jpg" alt="二维正态分布的两个随机变量：相互独立 等价于 线性无关"></p><p>参考：<a href="https://www.zhihu.com/question/29641138">https://www.zhihu.com/question/29641138</a></p>        </div>      </div>    </div><h2 id="第5章-大数定律与中心极限定理">第5章 大数定律与中心极限定理</h2><div class="note note-light">            <p>本章只需要知道一个<strong>独立同分布中心极限定理</strong>即可，至于<strong>棣莫弗-拉普拉斯中心极限定理</strong>其实就是前者的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>X</mi><mi>i</mi></msub><msubsup><mo stretchy="false">}</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></msubsup></mrow><annotation encoding="application/x-tex">\{X_i\}_{i=1}^{\infty}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2587em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">}</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> 服从伯努利 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 重分布罢了。</p>          </div><h3 id="独立同分布中心极限定理">独立同分布中心极限定理</h3><p>定义：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>X</mi><mi>i</mi></msub><msubsup><mo stretchy="false">}</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></msubsup></mrow><annotation encoding="application/x-tex">\{X_i\}_{i=1}^{\infty}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2587em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">}</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> 独立同分布且非零方差，其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><msub><mi>X</mi><mi>i</mi></msub><mo>=</mo><mi>μ</mi><mo separator="true">,</mo><mi>D</mi><msub><mi>X</mi><mi>i</mi></msub><mo>=</mo><msup><mi>σ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">EX_i=\mu,DX_i=\sigma^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>，则有：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>X</mi><mi>i</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mo stretchy="false">(</mo><mi>E</mi><msub><mi>X</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo separator="true">,</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mo stretchy="false">(</mo><mi>D</mi><msub><mi>X</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><mi>n</mi><mi>μ</mi><mo separator="true">,</mo><mi>n</mi><msup><mi>σ</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\sum_{i=1}^n X_i &amp;\sim N(\sum_{i=1}^n(EX_i),\sum_{i=1}^n(DX_i)) \\&amp;\sim N(n\mu,n\sigma^2)\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.7532em;vertical-align:-2.1266em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6266em;"><span style="top:-4.6266em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.1848em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1266em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6266em;"><span style="top:-4.6266em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mopen">(</span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">))</span></span></span><span style="top:-2.1848em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mord mathnormal">μ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1266em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>解释：其实就是对于独立同分布的随机事件 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>X</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">X_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，在事件数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 足够大时，就近似为正态分布（术语叫做<strong>依分布</strong>）。这样就可以很方便利用正态分布的性质计算当前事件的概率。至于<strong>棣莫弗-拉普拉斯中心极限定理</strong>就是上述 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi><mo>=</mo><mi>p</mi><mo separator="true">,</mo><msup><mi>σ</mi><mn>2</mn></msup><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mu=p,\sigma^2=p(1-p)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mclose">)</span></span></span></span> 的特殊情况罢了</p><h2 id="第6章-数理统计的基本概念">第6章 数理统计的基本概念</h2><div class="note note-light">            <p>开始统计学之旅。</p>          </div><h3 id="6-1-总体与样本">6.1 总体与样本</h3><p><s>类比 ML：数据集=总体，样本=样本。</s></p><p>我们只研究一种样本：简单随机样本 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>X</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>X</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X_1,X_2,...,X_n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">...</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>。符合下列两种特点：</p><ol><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>X</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>X</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X_1,X_2,...,X_n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">...</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 相互独立</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>X</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>X</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X_1,X_2,...,X_n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">...</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 同分布</li></ol><p>同样的，我们研究总体 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的分布与概率密度，一般概率密度会直接给，需要我们在此基础之上研究所有样本的联合密度：</p><ul><li><p>分布：由于样本相互独立，故：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>⋯</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(x_1,x_2,...,x_n)=F(x_1)F(x_2) \cdots F(x_n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">...</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p></li><li><p>联合密度：同样由于样本相互独立，故：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>⋯</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(x_1,x_2,...,x_n)=p(x_1)p(x_2) \cdots p(x_n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">...</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p></li></ul><h3 id="6-2-经验分布与频率直方图">6.2 经验分布与频率直方图</h3><p>经验分布函数是利用样本得到的。也是给区间然后统计样本频度进而计算频率，只不过区间长度不是固定的。</p><p>频率直方图就是选定<strong>固定</strong>的区间长度，然后统计频度进而计算频率作图。</p><h3 id="6-3-统计量">6.3 统计量</h3><p><strong>统计量定义</strong>：关于样本不含未知数的表达式。</p><p><strong>常见统计量</strong>：假设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>X</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>X</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X_1,X_2,...,X_n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">...</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 为来自总体 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的简单随机样本</p><p>一、样本均值和样本方差</p><ul><li><p>样本均值：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>X</mi><mi>i</mi></msub></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \overline{X} = \frac{1}{n} \sum_{i=1}^n X_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p></li><li><p>样本方差：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><msubsup><mi>S</mi><mn>0</mn><mn>2</mn></msubsup><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mo stretchy="false">(</mo><msub><mi>X</mi><mi>i</mi></msub><mo>−</mo><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>X</mi><mi>i</mi><mn>2</mn></msubsup><mo>−</mo><msup><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover><mn>2</mn></msup></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle S_0^2 = \frac{1}{n} \sum_{i=1}^n (X_i - \overline{X})^2 = \frac{1}{n}\sum_{i=1}^n X_i^2 - \overline{X}^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1111em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1333em;vertical-align:-0.25em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0873em;"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0873em;"><span style="top:-3.3362em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></p></li><li><p>样本标准差：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><msub><mi>S</mi><mn>0</mn></msub><mo>=</mo><msqrt><msubsup><mi>S</mi><mn>0</mn><mn>2</mn></msubsup></msqrt></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle S_0 = \sqrt{S_0^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.84em;vertical-align:-0.5413em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2987em;"><span class="svg-align" style="top:-3.8em;"><span class="pstrut" style="height:3.8em;"></span><span class="mord" style="padding-left:1em;"><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7959em;"><span style="top:-2.4337em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2663em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.2587em;"><span class="pstrut" style="height:3.8em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.88em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.88em" viewBox="0 0 400000 1944" preserveAspectRatio="xMinYMin slice"><path d="M983 90l0 -0c4,-6.7,10,-10,18,-10 H400000v40H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5c53.7,-170.3,84.5,-266.8,92.5,-289.5zM1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.5413em;"><span></span></span></span></span></span></span></span></span></p></li><li><p>修正样本方差：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><msup><mi>S</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mo stretchy="false">(</mo><msub><mi>X</mi><mi>i</mi></msub><mo>−</mo><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle S^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \overline{X})^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1333em;vertical-align:-0.25em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></p></li><li><p>修正样本标准差：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mi>S</mi><mo>=</mo><msqrt><msup><mi>S</mi><mn>2</mn></msup></msqrt></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle S = \sqrt{S^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.0361em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0039em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-2.9639em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.0361em;"><span></span></span></span></span></span></span></span></span></p>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-0d019804" role="button" aria-expanded="false" aria-controls="collapse-0d019804">        <div class="fold-arrow">▶</div>推导      </div>      <div class="fold-collapse collapse" id="collapse-0d019804">        <div class="fold-content">          <p>设总体 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的数学期望和方差分别为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\sigma^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>X</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>X</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X_1,X_2,...,X_n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">...</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 是简单随机样本，则：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202405101145088.png" alt="样本均值的数学期望与总体的数学期望相等"></p><p>即：样本均值的数学期望 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo></mrow><annotation encoding="application/x-tex">=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">=</span></span></span></span> 总体的数学期望</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202405101148690.png" alt="样本方差的数学期望与总体的数学期望 不相等"></p><p>即：样本方差的数学期望 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo mathvariant="normal">≠</mo></mrow><annotation encoding="application/x-tex">\ne</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span></span></span></span> 总体的数学期望</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202405101148126.png" alt="修正样本方差推导"></p><p>上图即：修正样本方差推导</p>        </div>      </div>    </div></li><li><p>样本 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 阶原点矩：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><msub><mi>A</mi><mi>k</mi></msub><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>X</mi><mi>i</mi><mi>k</mi></msubsup><mo separator="true">,</mo><mspace width="1em"></mspace><mi>k</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle A_k = \frac{1}{n} \sum_{i=1}^n X_i^k,\quad k=1,2,\cdots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-2.453em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span></span></span></span></p></li><li><p>样本 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 阶中心矩：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><msub><mi>B</mi><mi>k</mi></msub><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mo stretchy="false">(</mo><msub><mi>X</mi><mi>i</mi></msub><mo>−</mo><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover><msup><mo stretchy="false">)</mo><mi>k</mi></msup><mo separator="true">,</mo><mspace width="1em"></mspace><mi>k</mi><mo>=</mo><mn>2</mn><mo separator="true">,</mo><mn>3</mn><mo separator="true">,</mo><mo>⋯</mo></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle B_k = \frac{1}{n} \sum_{i=1}^n (X_i-\overline{X})^k,\quad k=2,3,\cdots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1491em;vertical-align:-0.25em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span></span></span></span></p></li></ul><p>二、次序统计量</p><ul><li>序列最小值</li><li>序列最大值</li><li>极差 = 序列最大值 - 序列最小值</li></ul><h3 id="6-4-正态总体抽样分布定理">6.4 正态总体抽样分布定理</h3><div class="note note-light">            <p>时刻牢记一句话：构造性定义！</p>          </div><h4 id="6-4-1-卡方分布、t-分布、F-分布">6.4.1 卡方分布、t 分布、F 分布</h4><p><strong>分位数</strong></p><ul><li>我们定义实数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mi>α</mi></msub></mrow><annotation encoding="application/x-tex">\lambda_\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 为随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的上侧 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 分位数（点）当且仅当 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>&gt;</mo><msub><mi>λ</mi><mi>α</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>α</mi></mrow><annotation encoding="application/x-tex">P(X &gt; \lambda_\alpha) = \alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span>​</li><li>我们定义实数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mrow><mn>1</mn><mo>−</mo><mi>β</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\lambda_{1-\beta}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.05278em;">β</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 为随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的下侧 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span> 分位数（点）当且仅当 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>&lt;</mo><msub><mi>λ</mi><mrow><mn>1</mn><mo>−</mo><mi>β</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mi>β</mi></mrow><annotation encoding="application/x-tex">P(X &lt; \lambda_{1-\beta})=\beta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.05278em;">β</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span></li></ul><p><strong><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>χ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\chi^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 分布</strong></p>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-dd3d75fc" role="button" aria-expanded="false" aria-controls="collapse-dd3d75fc">        <div class="fold-arrow">▶</div>密度函数图像      </div>      <div class="fold-collapse collapse" id="collapse-dd3d75fc">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202405171013305.webp" alt="密度函数图像"></p>        </div>      </div>    </div><p>定义：</p><ul><li>对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个独立同分布的标准正态随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>X</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>X</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><msub><mi>X</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X_1,X_2,\cdots ,X_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><msubsup><mi>X</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>X</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><mo>⋯</mo><mo>+</mo><msubsup><mi>X</mi><mi>n</mi><mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">Y = X_1^2 + X_2^2 + \cdots + X_n^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0622em;vertical-align:-0.2481em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4519em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0622em;vertical-align:-0.2481em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4519em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0611em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span></li><li>则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 服从自由度为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>χ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\chi^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 分布，记作：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>∼</mo><msup><mi>χ</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Y \sim \chi^2(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li></ul><p>性质：</p><ul><li><p>可加性：若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Y</mi><mn>1</mn></msub><mo>∼</mo><msup><mi>χ</mi><mn>2</mn></msup><mo stretchy="false">(</mo><msub><mi>n</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo separator="true">,</mo><msub><mi>Y</mi><mn>2</mn></msub><mo>∼</mo><msup><mi>χ</mi><mn>2</mn></msup><mo stretchy="false">(</mo><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Y_1 \sim \chi^2(n_1), Y_2 \sim \chi^2(n_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 且 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Y</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>Y</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">Y_1,Y_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 相互独立，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Y</mi><mn>1</mn></msub><mo>+</mo><msub><mi>Y</mi><mn>2</mn></msub><mo>∼</mo><msup><mi>χ</mi><mn>2</mn></msup><mo stretchy="false">(</mo><msub><mi>n</mi><mn>1</mn></msub><mo>+</mo><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Y_1+Y_2 \sim \chi^2(n_1+n_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></p></li><li><p>统计性：对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>∼</mo><msup><mi>χ</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Y \sim \chi^2(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>，有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mi>Y</mi><mo>=</mo><mi>n</mi><mo separator="true">,</mo><mi>D</mi><mi>Y</mi><mo>=</mo><mn>2</mn><mi>n</mi></mrow><annotation encoding="application/x-tex">EY = n, DY = 2n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">n</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span><span class="mord mathnormal">n</span></span></span></span></p>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-e5c1222d" role="button" aria-expanded="false" aria-controls="collapse-e5c1222d">        <div class="fold-arrow">▶</div>推导      </div>      <div class="fold-collapse collapse" id="collapse-e5c1222d">        <div class="fold-content">          <p>EY 的推导利用：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><msup><mi>X</mi><mn>2</mn></msup><mo>=</mo><mi>D</mi><mi>X</mi><mo>−</mo><mo stretchy="false">(</mo><mi>E</mi><mi>X</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">EX^2 = DX - (EX)^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">EX</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202405211524059.png" alt="EY 的推导"></p><p>DY 的推导利用：方差计算公式、随机变量函数的数学期望进行计算</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202405211524574.png" alt="DY 的推导"></p>        </div>      </div>    </div></li></ul><p><strong><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span> 分布</strong></p>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-bfc3604c" role="button" aria-expanded="false" aria-controls="collapse-bfc3604c">        <div class="fold-arrow">▶</div>密度函数图像      </div>      <div class="fold-collapse collapse" id="collapse-bfc3604c">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202405171014091.webp" alt="密度函数图像"></p>        </div>      </div>    </div><p>定义：</p><ul><li>若随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>Y</mi><mo>∼</mo><msup><mi>χ</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sim N(0, 1),Y \sim \chi^2 (n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 且 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo separator="true">,</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X,Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 相互独立</li><li>则称随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mi>X</mi><msqrt><mrow><mi>Y</mi><mi mathvariant="normal">/</mi><mi>n</mi></mrow></msqrt></mfrac></mstyle></mrow><annotation encoding="application/x-tex">T = \displaystyle \frac{X}{\sqrt{Y/n}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4903em;vertical-align:-1.13em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.175em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.935em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mord">/</span><span class="mord mathnormal">n</span></span></span><span style="top:-2.895em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.28em" viewBox="0 0 400000 1296" preserveAspectRatio="xMinYMin slice"><path d="M263,681c0.7,0,18,39.7,52,119c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120c340,-704.7,510.7,-1060.3,512,-1067l0 -0c4.7,-7.3,11,-11,19,-11H40000v40H1012.3s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60zM1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305em;"><span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.13em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 为服从自由度为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span> 分布，记作 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi><mo>∼</mo><mi>t</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T \sim t(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">t</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>​</li></ul><p>性质：</p><ul><li>密度函数是偶函数，具备对称性</li></ul><p><strong><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span> 分布</strong></p>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-670e1898" role="button" aria-expanded="false" aria-controls="collapse-670e1898">        <div class="fold-arrow">▶</div>密度函数图像      </div>      <div class="fold-collapse collapse" id="collapse-670e1898">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202405171015933.webp" alt="密度函数图像"></p>        </div>      </div>    </div><p>定义：</p><ul><li>若随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∼</mo><msup><mi>χ</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>Y</mi><mo>∼</mo><msup><mi>χ</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sim \chi^2(m), Y \sim \chi^2(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">m</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 且相互独立</li><li>则称随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi>X</mi><mi mathvariant="normal">/</mi><mi>m</mi></mrow><mrow><mi>Y</mi><mi mathvariant="normal">/</mi><mi>n</mi></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">G=\displaystyle \frac{X/m}{Y/n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">G</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mord">/</span><span class="mord mathnormal">n</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">/</span><span class="mord mathnormal">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 服从自由度为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>m</mi><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(m,n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">m</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span> 分布，记作 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mo>∼</mo><mi>F</mi><mo stretchy="false">(</mo><mi>m</mi><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G \sim F(m, n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">G</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">m</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>​</li></ul><p>性质：</p><ul><li>倒数自由度转换：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mi>G</mi></mfrac><mo>∼</mo><mi>F</mi><mo stretchy="false">(</mo><mi>n</mi><mo separator="true">,</mo><mi>m</mi><mo stretchy="false">)</mo></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \frac{1}{G} \sim F(n, m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">G</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span></li><li><a href="https://zhuanlan.zhihu.com/p/382940609">三变性质</a>：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><msub><mi>F</mi><mrow><mn>1</mn><mo>−</mo><mi>α</mi></mrow></msub><mo stretchy="false">(</mo><mi>m</mi><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mo fence="true">[</mo><msub><mi>F</mi><mi>α</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo separator="true">,</mo><mi>m</mi><mo stretchy="false">)</mo><mo fence="true">]</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle F_{1-\alpha}(m, n) = \left [F_\alpha (n, m)\right]^{-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">m</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.204em;vertical-align:-0.25em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span><span class="mclose delimcenter" style="top:0em;">]</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span></li></ul><h4 id="6-4-2-正态总体抽样分布基本定理">6.4.2 正态总体抽样分布基本定理</h4><p>设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>X</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>X</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><msub><mi>X</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X_1,X_2,\cdots ,X_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是来自正态总体 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>μ</mi><mo separator="true">,</mo><msup><mi>σ</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(\mu, \sigma^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mopen">(</span><span class="mord mathnormal">μ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 的简单随机样本，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover><mo separator="true">,</mo><msup><mi>S</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\overline{X},S^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0778em;vertical-align:-0.1944em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 分别是样本均值和修正样本方差。则有：</p><p>定理：</p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><mi>μ</mi><mo separator="true">,</mo><mfrac><msup><mi>σ</mi><mn>2</mn></msup><mi>n</mi></mfrac><mo stretchy="false">)</mo></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \overline{X} \sim N(\mu, \frac{\sigma^2}{n})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mopen">(</span><span class="mord mathnormal">μ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span></span></span></span></li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>S</mi><mn>2</mn></msup></mrow><msup><mi>σ</mi><mn>2</mn></msup></mfrac><mo>∼</mo><msup><mi>χ</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \frac{(n-1)S^2}{\sigma^2} \sim \chi^2(n-1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\overline{X}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>S</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 相互独立</li></ul><p>推论：</p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><msqrt><mi>n</mi></msqrt><mo stretchy="false">(</mo><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover><mo>−</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow><mi>S</mi></mfrac><mo>∼</mo><mi>t</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \frac{\sqrt{n}(\overline{X} - \mu)}{S} \sim t(n-1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2463em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8003em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">n</span></span></span><span style="top:-2.7603em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2397em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">μ</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">t</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></li></ul><h2 id="第7章-参数估计">第7章 参数估计</h2><div class="note note-light">            <p>有些时候我们知道数据的分布类型，但是不清楚表达式中的某些参数，这就需要我们利用「已有的样本」对分布表达式中的参数进行估计。本章我们将从点估计、估计评价、区间估计三部分出发进行介绍。</p>          </div><h3 id="7-1-点估计">7.1 点估计</h3><div class="note note-light">            <p>所谓点估计策略，就是直接给出参数的一个估计值。本目我们介绍点估计策略中的两个方法：矩估计法、极大似然估计法。</p>          </div><h4 id="7-1-1-矩估计法">7.1.1 矩估计法</h4><p>其实就一句话：我们用样本的原点矩 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">A_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 来代替总体 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><msup><mi>X</mi><mi>k</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(X^k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0991em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个未知参数就需要用到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个原点矩：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><msup><mi>X</mi><mi>k</mi></msup><mo stretchy="false">)</mo><mo>=</mo><msub><mi>A</mi><mi>k</mi></msub><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>X</mi><mi>i</mi><mi>k</mi></msubsup></mrow><annotation encoding="application/x-tex">E(X^k) = A_k =  \frac{1}{n}\sum_{i=1}^nX_i^k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1491em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-2.453em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span></span></p><h4 id="7-1-2-极大似然估计法">7.1.2 极大似然估计法</h4><p>基本原理是：在当前样本数据的局面下，我们希望<strong>找到合适的参数使得当前的样本分布情况发生的概率最大</strong>。由于各样本相互独立，因此我们可以用连乘的概率公式来计算当前局面的概率值：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>θ</mi><mo separator="true">;</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(\theta;x_1,x_2,\cdots,x_n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p><p>上述 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>θ</mi><mo separator="true">;</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(\theta;x_1,x_2,\cdots,x_n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 即似然函数，目标就是选择适当的参数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span></span> 来最大化似然函数。无论是离散性还是连续型，都可以采用下面的方式来计算极大似然估计：</p><ol><li>写出似然函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(\theta)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span></li><li>将上述似然函数取对数</li><li>求对数似然函数关于所有未知参数的偏导并计算极值点</li><li>解出参数关于样本统计量的表达式</li></ol><p>离散型随机变量的似然函数表达式</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo separator="true">;</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi>P</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mi>i</mi></msub><mo>=</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(\theta) = \prod_{i=1}^n p(x_i;\theta) = \prod_{i=1}^n P(X_i = x_i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∏</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∏</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p><p>连续型随机变量的似然函数表达式</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo separator="true">;</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(\theta) = \prod_{i=1}^n p(x_i;\theta)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∏</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span></span></p><p>可以看出极大似然估计本质上就是一个多元函数求极值的问题。特别地，当我们没法得到参数关于样本统计量的表达式 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(\theta)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span> 时，可以直接从定义域、原函数恒增或恒减等角度出发求解这个多元函数的极值。</p><h3 id="7-2-估计量的评价标准">7.2 估计量的评价标准</h3><div class="note note-light">            <p>如何衡量不同的点估计方法好坏？我们引入三种点估计量的评价指标：无偏性、有效性、一致性。其中一致性一笔带过，不做详细讨论。补充一点，参数的估计量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span></span> 是关于样本的统计量，因此可以对其进行求期望、方差等操作。</p>          </div><h4 id="7-2-1-无偏性">7.2.1 无偏性</h4><p>顾名思义，就是希望估计出来的参数量尽可能不偏离真实值。我们定义满足下式的估计量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><annotation encoding="application/x-tex">\hat \theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9579em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9579em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span></span></span></span> 为真实参数的无偏估计：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>E</mi><mover accent="true"><mi>θ</mi><mo>^</mo></mover><mo>=</mo><mi>θ</mi></mrow><annotation encoding="application/x-tex">E\hat \theta =\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9579em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9579em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span></span></span></p><h4 id="7-2-2-有效性">7.2.2 有效性</h4><p>有效性是基于比较的定义方法。对于两个无偏估计 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>θ</mi><mo>^</mo></mover><mn>1</mn></msub><mo separator="true">,</mo><msub><mover accent="true"><mi>θ</mi><mo>^</mo></mover><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\hat\theta_1,\hat\theta_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1523em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9579em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9579em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，谁的方差越小谁就越有效。即若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><msub><mover accent="true"><mi>θ</mi><mo>^</mo></mover><mn>1</mn></msub><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>D</mi><mo stretchy="false">(</mo><msub><mover accent="true"><mi>θ</mi><mo>^</mo></mover><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(\hat\theta_1),D(\hat\theta_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2079em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9579em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9579em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 满足下式，则称 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>θ</mi><mo>^</mo></mover><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\hat\theta_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1079em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9579em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 更有效</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><msub><mover accent="true"><mi>θ</mi><mo>^</mo></mover><mn>1</mn></msub><mo stretchy="false">)</mo><mo>&lt;</mo><mi>D</mi><mo stretchy="false">(</mo><msub><mover accent="true"><mi>θ</mi><mo>^</mo></mover><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(\hat\theta_1) &lt; D(\hat\theta_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2079em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9579em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2079em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9579em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p><h4 id="7-2-3-一致性">7.2.3 一致性</h4><p>即当样本容量 n 趋近于无穷时，参数的估计值也能趋近于真实值，则称该估计量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><annotation encoding="application/x-tex">\hat\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9579em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9579em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span></span></span></span> 为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span></span> 的一致估计量</p><h3 id="7-3-区间估计">7.3 区间估计</h3><div class="note note-light">            <p>由于点估计只能进行比较，无法对单一估计进行性能度量。因此引入「主元法」的概念与「区间估计」策略</p>          </div><h4 id="7-3-1-基本概念">7.3.1 基本概念</h4><p>可靠程度：参数估计区间越长，可靠程度越高</p><p>精确程度：参数估计区间越短，可靠程度越高</p><h4 id="7-3-2-区间估计常用方法之主元法">7.3.2 区间估计常用方法之主元法</h4><p>主元法的核心逻辑就一个：在已知数据总体分布的情况下，构造一个关于样本 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 和待估参数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span></span> 的函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z(X,\theta)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span>，然后利用置信度和总体分布函数，通过查表得到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z(X,\theta)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span> 的取值范围，最后通过移项变形得到待估参数的区间，也就是估计区间。</p><h4 id="7-3-3-正态总体的区间估计">7.3.3 正态总体的区间估计</h4><p>我们只需要掌握「一个总体服从正态分布」的情况。这种情况下的区间估计分为三种，其中估计均值 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span> 有 2 种，估计方差 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\sigma^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 有 1 种。估计的逻辑我总结为了以下三步：</p><ol><li>构造主元 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z(X,\theta)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span></li><li>利用置信度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>−</mo><mi>α</mi></mrow><annotation encoding="application/x-tex">1-\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 计算主元 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span></span></span> 的取值范围</li><li>对主元 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span></span></span> 的取值范围移项得到参数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span></span> 的取值范围</li></ol><p>为了提升区间估计的可信度，我们希望上述第 2 步计算出来的关于主元的取值范围尽可能准确。我们不加证明的给出以下结论：取主元的取值范围为 <strong>主元服从的分布的上下 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mi>α</mi><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{\alpha}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 分位数之间</strong>。</p><p><strong>(一) 求 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span> 的置信区间，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\sigma^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 已知</strong></p><p>构造主元 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z(X,\theta)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span>：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>Z</mi><mo>=</mo><mfrac><mrow><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover><mo>−</mo><mi>μ</mi></mrow><mrow><mi>σ</mi><mi mathvariant="normal">/</mi><msqrt><mi>n</mi></msqrt></mrow></mfrac><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z = \frac{\overline{X} - \mu}{\sigma / \sqrt{n}} \sim N(0,1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.5006em;vertical-align:-0.9403em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5603em;"><span style="top:-2.3097em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mord">/</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8003em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">n</span></span></span><span style="top:-2.7603em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path 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class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9403em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></span></p><p>利用置信度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>−</mo><mi>α</mi></mrow><annotation encoding="application/x-tex">1-\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 计算主元 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span></span></span> 的取值范围：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>P</mi><mo stretchy="false">(</mo><mi mathvariant="normal">∣</mi><mi>Z</mi><mi mathvariant="normal">∣</mi><mo>≤</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>1</mn><mo>−</mo><mi>α</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>↓</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>Z</mi><mo>∈</mo><mo stretchy="false">[</mo><mo>−</mo><mi>λ</mi><mo separator="true">,</mo><mi>λ</mi><mo stretchy="false">]</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo stretchy="false">[</mo><mo>−</mo><msub><mi>u</mi><mfrac><mi>α</mi><mn>2</mn></mfrac></msub><mo separator="true">,</mo><msub><mi>u</mi><mfrac><mi>α</mi><mn>2</mn></mfrac></msub><mo stretchy="false">]</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}P(|Z| \le \lambda) &amp;= 1-\alpha \\&amp;\downarrow\\Z \in [-\lambda,\lambda] &amp;= [-u_{\frac{\alpha}{2}},u_\frac{\alpha}{2}]\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.5308em;vertical-align:-2.0154em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5154em;"><span style="top:-4.6754em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">λ</span><span class="mclose">)</span></span></span><span style="top:-3.1754em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-1.6754em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">[</span><span class="mord">−</span><span class="mord mathnormal">λ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">λ</span><span class="mclose">]</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.0154em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5154em;"><span style="top:-4.6754em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span><span style="top:-3.1754em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">↓</span></span></span><span style="top:-1.6754em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">[</span><span class="mord">−</span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3341em;"><span style="top:-2.85em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3908em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3341em;"><span style="top:-2.85em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3908em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.0154em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>对主元 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span></span></span> 的取值范围移项得到参数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span></span> 的取值范围：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover><mo>−</mo><mfrac><mi>σ</mi><msqrt><mi>n</mi></msqrt></mfrac><msub><mi>u</mi><mfrac><mi>α</mi><mn>2</mn></mfrac></msub><mo>≤</mo><mi>μ</mi><mo>≤</mo><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover><mo>+</mo><mfrac><mi>σ</mi><msqrt><mi>n</mi></msqrt></mfrac><msub><mi>u</mi><mfrac><mi>α</mi><mn>2</mn></mfrac></msub></mrow><annotation encoding="application/x-tex">\overline{X} - \frac{\sigma}{\sqrt{n}} u_\frac{\alpha}{2} \le \mu \le \overline{X} + \frac{\sigma}{\sqrt{n}} u_\frac{\alpha}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9667em;vertical-align:-0.0833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0376em;vertical-align:-0.93em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.3097em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8003em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">n</span></span></span><span style="top:-2.7603em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2397em;"><span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.93em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3341em;"><span style="top:-2.85em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3908em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9667em;vertical-align:-0.0833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0376em;vertical-align:-0.93em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.3097em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8003em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">n</span></span></span><span style="top:-2.7603em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2397em;"><span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.93em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3341em;"><span style="top:-2.85em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3908em;"><span></span></span></span></span></span></span></span></span></span></span></p><p><strong>(二) 求 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span> 的置信区间，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\sigma^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 未知</strong></p><p>构造主元 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z(X,\theta)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span>：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>Z</mi><mo>=</mo><mfrac><mrow><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover><mo>−</mo><mi>μ</mi></mrow><mrow><mi>S</mi><mi mathvariant="normal">/</mi><msqrt><mi>n</mi></msqrt></mrow></mfrac><mo>∼</mo><mi>t</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z = \frac{\overline{X} - \mu}{S / \sqrt{n}} \sim t(n-1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.5006em;vertical-align:-0.9403em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5603em;"><span style="top:-2.3097em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mord">/</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8003em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">n</span></span></span><span style="top:-2.7603em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2397em;"><span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9403em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">t</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></span></p><p>利用置信度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>−</mo><mi>α</mi></mrow><annotation encoding="application/x-tex">1-\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 计算主元 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span></span></span> 的取值范围：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>P</mi><mo stretchy="false">(</mo><mi mathvariant="normal">∣</mi><mi>Z</mi><mi mathvariant="normal">∣</mi><mo>≤</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>1</mn><mo>−</mo><mi>α</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>↓</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>Z</mi><mo>∈</mo><mo stretchy="false">[</mo><mo>−</mo><mi>λ</mi><mo separator="true">,</mo><mi>λ</mi><mo stretchy="false">]</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo stretchy="false">[</mo><mo>−</mo><msub><mi>t</mi><mfrac><mi>α</mi><mn>2</mn></mfrac></msub><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><msub><mi>t</mi><mfrac><mi>α</mi><mn>2</mn></mfrac></msub><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}P(|Z| \le \lambda) &amp;= 1-\alpha \\&amp;\downarrow\\Z \in [-\lambda,\lambda] &amp;= [-t_{\frac{\alpha}{2}}(n-1),t_\frac{\alpha}{2}(n-1)]\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.5308em;vertical-align:-2.0154em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5154em;"><span style="top:-4.6754em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">λ</span><span class="mclose">)</span></span></span><span style="top:-3.1754em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-1.6754em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">[</span><span class="mord">−</span><span class="mord mathnormal">λ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">λ</span><span class="mclose">]</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.0154em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5154em;"><span style="top:-4.6754em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span><span style="top:-3.1754em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">↓</span></span></span><span style="top:-1.6754em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">[</span><span class="mord">−</span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3341em;"><span style="top:-2.85em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3908em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3341em;"><span style="top:-2.85em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3908em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose">)]</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.0154em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>对主元 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span></span></span> 的取值范围移项得到参数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span></span> 的取值范围：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover><mo>−</mo><mfrac><mi>S</mi><msqrt><mi>n</mi></msqrt></mfrac><msub><mi>t</mi><mfrac><mi>α</mi><mn>2</mn></mfrac></msub><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≤</mo><mi>μ</mi><mo>≤</mo><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover><mo>+</mo><mfrac><mi>S</mi><msqrt><mi>n</mi></msqrt></mfrac><msub><mi>t</mi><mfrac><mi>α</mi><mn>2</mn></mfrac></msub><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\overline{X} - \frac{S}{\sqrt{n}} t_\frac{\alpha}{2}(n-1) \le \mu \le \overline{X} + \frac{S}{\sqrt{n}} t_\frac{\alpha}{2}(n-1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9667em;vertical-align:-0.0833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.2903em;vertical-align:-0.93em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.3097em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8003em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">n</span></span></span><span style="top:-2.7603em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2397em;"><span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.93em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3341em;"><span style="top:-2.85em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3908em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9667em;vertical-align:-0.0833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.2903em;vertical-align:-0.93em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.3097em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8003em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">n</span></span></span><span style="top:-2.7603em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2397em;"><span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.93em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3341em;"><span style="top:-2.85em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3908em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></span></p><p><strong>(三) 求 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\sigma^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 的置信区间，构造的主元与总体均值无关，因此不需要考虑 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span> 的情况：</strong></p><p>构造主元 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z(X,\theta)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span>：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>Z</mi><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>S</mi><mn>2</mn></msup></mrow><msup><mi>σ</mi><mn>2</mn></msup></mfrac><mo>∼</mo><msup><mi>χ</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z = \frac{(n-1)S^2}{\sigma^2}\sim \chi^2(n-1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></span></p><p>利用置信度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>−</mo><mi>α</mi></mrow><annotation encoding="application/x-tex">1-\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 计算主元 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span></span></span> 的取值范围：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>λ</mi><mn>1</mn></msub><mo>≤</mo><mi>Z</mi><mo>≤</mo><msub><mi>λ</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>1</mn><mo>−</mo><mi>α</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>↓</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>Z</mi><mo>∈</mo><mo stretchy="false">[</mo><msub><mi>λ</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>λ</mi><mn>2</mn></msub><mo stretchy="false">]</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo stretchy="false">[</mo><msubsup><mi>χ</mi><mrow><mn>1</mn><mo>−</mo><mfrac><mi>α</mi><mn>2</mn></mfrac></mrow><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><msubsup><mi>χ</mi><mfrac><mi>α</mi><mn>2</mn></mfrac><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}P(\lambda_1 \le Z \le \lambda_2) &amp;= 1-\alpha \\&amp;\downarrow\\Z \in [\lambda_1,\lambda_2] &amp;= [\chi^2_{1-\frac{\alpha}{2}}(n-1),\chi^2_\frac{\alpha}{2}(n-1)]\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.6519em;vertical-align:-2.076em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.576em;"><span style="top:-4.736em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-3.236em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-1.7118em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.076em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.576em;"><span style="top:-4.736em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span><span style="top:-3.236em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">↓</span></span></span><span style="top:-1.7118em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.753em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mbin mtight">−</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span><span style="top:-3.413em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4878em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.753em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span><span style="top:-3.413em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4878em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose">)]</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.076em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>对主元 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span></span></span> 的取值范围移项得到参数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span></span> 的取值范围：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>S</mi><mn>2</mn></msup></mrow><mrow><msubsup><mi>χ</mi><mfrac><mi>α</mi><mn>2</mn></mfrac><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><mo>≤</mo><msup><mi>σ</mi><mn>2</mn></msup><mo>≤</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>S</mi><mn>2</mn></msup></mrow><mrow><msubsup><mi>χ</mi><mrow><mn>1</mn><mo>−</mo><mfrac><mi>α</mi><mn>2</mn></mfrac></mrow><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{(n-1)S^2}{\chi^2_\frac{\alpha}{2}(n-1)} \le \sigma^2 \le \frac{(n-1)S^2}{\chi^2_{1-\frac{\alpha}{2}}(n-1)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.7172em;vertical-align:-1.2261em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7959em;"><span style="top:-2.7007em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span><span style="top:-3.3448em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.5401em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2261em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0001em;vertical-align:-0.136em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.7172em;vertical-align:-1.2261em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7959em;"><span style="top:-2.7007em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mbin mtight">−</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span><span style="top:-3.3448em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.5401em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2261em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><h2 id="第8章-假设检验">第8章 假设检验</h2><div class="note note-light">            <p>第 7 章的参数估计是在总体分布已知且未知分布表达式中某些参数的情况下，基于「抽取的少量样本」进行的参数估计。</p><p>现在的局面同样，我们已知总体分布和不完整的分布表达式参数。现在需要我们利用抽取的少量样本判断样本所在向量空间是否符合某种性质。本章的「假设检验」策略就是为了解决上述情况而诞生的。我们主要讨论单个正态总体的情况并针对均值和方差两个参数进行假设和检验：</p><ul><li>假设均值满足某种趋势，利用已知数据判断假设是否成立</li><li>假设方差满足某种趋势，利用已知数据判断假设是否成立</li></ul>          </div><h3 id="8-1-假设检验的基本概念">8.1 假设检验的基本概念</h3><p>基本思想：首先做出假设并构造一个关于样本观察值和已知参数的检验统计量，接着计算假设发生的情况下小概率事件发生时该检验统计量的取值范围（拒绝域），最终代入已知样本数据判断计算结果是否在拒绝域内。如果在，则说明在当前假设的情况下小概率事件发生了，对应的假设为假；反之说明假设为真。</p><p>为了量化「小概率事件发生」这个指标，我们引入显著性水平 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 这一概念。该参数为一个很小的正数，定义为「小概率事件发生」的概率上界。</p><p>基于数据的实验导致我们无法避免错误，因此我们定义以下两类错误：</p><ul><li>第一类错误：弃真错误。即假设正确，但由于数据采样不合理导致拒绝了真实的假设</li><li>第二类错误：存伪错误。即假设错误，同样因为数据的不合理导致接受了错误的假设</li></ul><h3 id="8-2-单个正态总体均值的假设检验">8.2 单个正态总体均值的假设检验</h3><p>设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>X</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>X</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><msub><mi>X</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X_1,X_2,\cdots ,X_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是来自正态总体 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>μ</mi><mo separator="true">,</mo><msup><mi>σ</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(\mu, \sigma^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mopen">(</span><span class="mord mathnormal">μ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 的简单随机样本。后续进行假设判定计算统计量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span></span></span> 的真实值时，若总体均值 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span> 已知就直接代入，若未知题目也一定会给一个阈值，代这个阈值即可。</p><p>当总体方差 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\sigma^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 已知时，我们构造样本统计量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span></span></span> 为正态分布：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>Z</mi><mo>=</mo><mfrac><mrow><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover><mo>−</mo><mi>μ</mi></mrow><mrow><mi>σ</mi><mi mathvariant="normal">/</mi><msqrt><mi>n</mi></msqrt></mrow></mfrac><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z = \frac{\overline{X} - \mu}{\sigma / \sqrt{n}} \sim N(0,1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.5006em;vertical-align:-0.9403em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5603em;"><span style="top:-2.3097em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mord">/</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8003em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">n</span></span></span><span style="top:-2.7603em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2397em;"><span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9403em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></span></p><ul><li>检验是否则求解双侧 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 分位数</li><li>检验单边则求解单侧 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 分位数</li></ul><p>当总体方差 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\sigma^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 未知时，我们构造样本统计量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span></span></span> 为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span> 分布：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>Z</mi><mo>=</mo><mfrac><mrow><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover><mo>−</mo><mi>μ</mi></mrow><mrow><mi>S</mi><mi mathvariant="normal">/</mi><msqrt><mi>n</mi></msqrt></mrow></mfrac><mo>∼</mo><mi>t</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z = \frac{\overline{X} - \mu}{S / \sqrt{n}} \sim t(n-1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.5006em;vertical-align:-0.9403em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5603em;"><span style="top:-2.3097em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mord">/</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8003em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">n</span></span></span><span style="top:-2.7603em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2397em;"><span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9403em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">t</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></span></p><div class="note note-warning">            <p>注：之所以这样构造是因为当总体 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> 未知时，上一个方法构造的主元已经不再是统计量，我们需要找到能够代替未知参数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> 的变量，这里就采用其无偏估计「修正样本方差 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>S</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>」来代替 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\sigma^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>。也是说直接拿样本的修正方差来代替总体的方差了。</p>          </div><ul><li>检验是否则求解双侧 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 分位数</li><li>检验单边则求解单侧 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 分位数</li></ul><h3 id="8-3-单个正态总体方差的假设检验">8.3 单个正态总体方差的假设检验</h3><p>设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>X</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>X</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><msub><mi>X</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X_1,X_2,\cdots ,X_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是来自正态总体 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>μ</mi><mo separator="true">,</mo><msup><mi>σ</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(\mu, \sigma^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mopen">(</span><span class="mord mathnormal">μ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 的简单随机样本。后续进行假设判定计算统计量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span></span></span> 的真实值时，若总体方差 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\sigma^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 已知就直接代入，若未知题目也一定会给一个阈值，代这个阈值即可。</p><p>我们直接构造样本统计量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span></span></span> 为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>χ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\chi^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 分布：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>Z</mi><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>S</mi><mn>2</mn></msup></mrow><msup><mi>σ</mi><mn>2</mn></msup></mfrac><mo>∼</mo><msup><mi>χ</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z = \frac{(n-1)S^2}{\sigma^2} \sim \chi^2(n-1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></span></p><ul><li>检验是否则求解双侧 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 分位数</li><li>检验单边则求解单侧 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 分位数</li></ul>]]></content>
+    <content type="html"><![CDATA[<h2 id="算法设计与分析">算法设计与分析</h2><h2 id="前言">前言</h2><p>学科地位：</p><table><thead><tr><th style="text-align:center">主讲教师</th><th style="text-align:center">学分配额</th><th style="text-align:center">学科类别</th></tr></thead><tbody><tr><td style="text-align:center">段博佳</td><td style="text-align:center">3</td><td style="text-align:center">自发课</td></tr></tbody></table><p>成绩组成：</p><table><thead><tr><th style="text-align:center">平时（5次作业+签到）</th><th style="text-align:center">期末（书本+讲义）</th></tr></thead><tbody><tr><td style="text-align:center">60%</td><td style="text-align:center">40%</td></tr></tbody></table><p>教材情况：</p><table><thead><tr><th style="text-align:center">课程名称</th><th style="text-align:center">选用教材</th><th style="text-align:center">版次</th><th style="text-align:center">作者</th><th style="text-align:center">出版社</th><th style="text-align:center">ISBN号</th></tr></thead><tbody><tr><td style="text-align:center">算法设计与分析</td><td style="text-align:center">《算法设计与分析(python版)》</td><td style="text-align:center">1</td><td style="text-align:center">王秋芬</td><td style="text-align:center">清华大学出版社</td><td style="text-align:center">978-7-302-57072-1</td></tr></tbody></table><h2 id="一、贪心">一、贪心</h2><p>Dijkstra、Prim、Kruskal、Huffman</p><h2 id="二、分治">二、分治</h2><p>二分、归并、快排（第 k 个数）</p><h2 id="三、动规">三、动规</h2><ul><li>区间dp：矩阵连乘</li><li>网格图dp：LCS</li><li>背包：完全背包、多重背包</li></ul><h2 id="四、深搜">四、深搜</h2><ul><li>排列树：TSP、图染色、调度、n 皇后</li><li>子集树：01背包</li></ul><h2 id="考后碎碎念">考后碎碎念</h2><p>考的中规中矩，但是忘带修正带导致卷面一坨hh。按照试卷顺序罗列一下：表达式树、大根堆、Dijkstra、图染色、第 k 个数、Kruskal、完全背包。</p><p>试卷一看就是 wq 出的，但还是用 python 写的代码。有一说一这次考试还加深了我对快排的理解，尤其是在让我给出分治后的序列时hh。硬要按照上面四个标签分类的话大概就是：</p><ul><li>贪心：Dijkstra、Kruskal</li><li>分治：第 k 个数</li><li>动规：完全背包</li><li>深搜：表达式树、大根堆、图染色</li></ul>]]></content>
     
     
     <categories>
@@ -1010,11 +1010,11 @@
   
   
   <entry>
-    <title>OptMethod</title>
-    <link href="/GPA/4th-term/OptMethod/"/>
-    <url>/GPA/4th-term/OptMethod/</url>
+    <title>ProbAndStat</title>
+    <link href="/GPA/4th-term/ProbAndStat/"/>
+    <url>/GPA/4th-term/ProbAndStat/</url>
     
-    <content type="html"><![CDATA[<h2 id="最优化方法">最优化方法</h2><h2 id="前言">前言</h2><p>学科地位：</p><table><thead><tr><th style="text-align:center">主讲教师</th><th style="text-align:center">学分配额</th><th style="text-align:center">学科类别</th></tr></thead><tbody><tr><td style="text-align:center">王启春</td><td style="text-align:center">4</td><td style="text-align:center">专业课</td></tr></tbody></table><p>成绩组成：</p><table><thead><tr><th style="text-align:center">平时（作业+考勤）</th><th style="text-align:center">期中（大作业）</th><th style="text-align:center">期末（闭卷）</th></tr></thead><tbody><tr><td style="text-align:center">20%</td><td style="text-align:center">30%</td><td style="text-align:center">50%</td></tr></tbody></table><p>教材情况：</p><table><thead><tr><th style="text-align:center">课程名称</th><th style="text-align:center">选用教材</th><th style="text-align:center">版次</th><th style="text-align:center">作者</th><th style="text-align:center">出版社</th><th style="text-align:center">ISBN号</th></tr></thead><tbody><tr><td style="text-align:center">最优化算法</td><td style="text-align:center">《最优化方法》</td><td style="text-align:center">2</td><td style="text-align:center">孙文瑜</td><td style="text-align:center">高等教育出版社</td><td style="text-align:center">978-7-04-029763-8</td></tr></tbody></table><h2 id="第一章-基本概念">第一章 基本概念</h2><h3 id="1-1-最优化问题简介">1.1 最优化问题简介</h3><p>本目主要讲解最优化问题的一些分类，下附脑图（由 Xmind 软件制作）：</p>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-cb469c68" role="button" aria-expanded="false" aria-controls="collapse-cb469c68">        <div class="fold-arrow">▶</div>分类脑图      </div>      <div class="fold-collapse collapse" id="collapse-cb469c68">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403091214182.png" alt="分类脑图"></p>        </div>      </div>    </div><h3 id="1-2-凸集和凸函数">1.2 凸集和凸函数</h3><div class="note note-light">            <p>由于本书介绍的最优化求解方法一般只适用于求解<strong>局部最优解</strong>，那么如何确定<strong>全局最优解</strong>呢？以及如何确定<strong>唯一的全局最优解</strong>呢？本目揭晓答案：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">(</mo><mtext>唯一</mtext><mo stretchy="false">)</mo><mtext>全局最优解</mtext><mo>=</mo><mtext>局部最优解</mtext><mo>+</mo><mo stretchy="false">(</mo><mtext>严格</mtext><mo stretchy="false">)</mo><mtext>凸目标函数</mtext><mo>+</mo><mtext>凸可行域</mtext></mrow><annotation encoding="application/x-tex">(\text{唯一})\text{全局最优解}=\text{局部最优解}+(\text{严格})\text{凸目标函数}+\text{凸可行域}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord text"><span class="mord cjk_fallback">唯一</span></span><span class="mclose">)</span><span class="mord text"><span class="mord cjk_fallback">全局最优解</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord text"><span class="mord cjk_fallback">局部最优解</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord text"><span class="mord cjk_fallback">严格</span></span><span class="mclose">)</span><span class="mord text"><span class="mord cjk_fallback">凸目标函数</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord cjk_fallback">凸可行域</span></span></span></span></span></span></p><p>其中凸可行域包含于凸集，凸目标函数包含于凸函数，凸目标函数+凸可行域的问题称为凸规划问题。因此本目将分别介绍凸集、凸函数和凸规划三个概念。</p>          </div><h4 id="1-2-1-凸集">1.2.1 凸集</h4><p><strong>凸集的定义</strong>：若集合中任意两点的线性组合都包含于集合，则称该集合为凸集。符号化即：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="normal">∀</mi><mspace width="1em"></mspace><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>∈</mo><mi>D</mi><mtext>&nbsp;</mtext><mo>∧</mo><mtext>&nbsp;</mtext><mi>λ</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>s</mi><mi mathvariant="normal">.</mi><mi>t</mi><mi mathvariant="normal">.</mi><mspace width="1em"></mspace><mi>λ</mi><mi>x</mi><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>λ</mi><mo stretchy="false">)</mo><mi>y</mi><mo>∈</mo><mi>D</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\forall \quad x,y \in D \ \land \ \lambda \in [0,1] \\s.t. \quad \lambda x+(1-\lambda)y \in D \\\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∀</span><span class="mspace" style="margin-right:1em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mspace">&nbsp;</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∧</span><span class="mspace">&nbsp;</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">λ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">]</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="mord">.</span><span class="mord mathnormal">t</span><span class="mord">.</span><span class="mspace" style="margin-right:1em;"></span><span class="mord mathnormal">λ</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">λ</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p><strong>凸集的性质</strong>：设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>D</mi><mn>2</mn></msub><mo>⊂</mo><msup><mi>R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">D_1,D_2 \subset R^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> 均为凸集（下列 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo separator="true">,</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x,y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> 均表示向量），则：</p><ol><li>两凸集的交 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mn>1</mn></msub><mo>∩</mo><msub><mi>D</mi><mn>2</mn></msub><mo>=</mo><mo stretchy="false">{</mo><mi>x</mi><mtext>&nbsp;</mtext><mi mathvariant="normal">∣</mi><mtext>&nbsp;</mtext><mi>x</mi><mo>∈</mo><msub><mi>D</mi><mn>1</mn></msub><mo>∧</mo><mi>x</mi><mo>∈</mo><msub><mi>D</mi><mn>2</mn></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">D_1 \cap D_2 = \{x\ |\ x \in D_1 \land x \in D_2\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal">x</span><span class="mspace">&nbsp;</span><span class="mord">∣</span><span class="mspace">&nbsp;</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span> 是凸集</li><li>两凸集的和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mn>1</mn></msub><mo>+</mo><msub><mi>D</mi><mn>2</mn></msub><mo>=</mo><mo stretchy="false">{</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mtext>&nbsp;</mtext><mi mathvariant="normal">∣</mi><mtext>&nbsp;</mtext><mi>x</mi><mo>∈</mo><msub><mi>D</mi><mn>1</mn></msub><mo separator="true">,</mo><mi>y</mi><mo>∈</mo><msub><mi>D</mi><mn>2</mn></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">D_1 + D_2 = \{x,y\ |\ x \in D_1 , y \in D_2\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace">&nbsp;</span><span class="mord">∣</span><span class="mspace">&nbsp;</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span> 是凸集</li><li>两凸集的差 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mn>1</mn></msub><mo>−</mo><msub><mi>D</mi><mn>2</mn></msub><mo>=</mo><mo stretchy="false">{</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mtext>&nbsp;</mtext><mi mathvariant="normal">∣</mi><mtext>&nbsp;</mtext><mi>x</mi><mo>∈</mo><msub><mi>D</mi><mn>1</mn></msub><mo separator="true">,</mo><mi>y</mi><mo>∈</mo><msub><mi>D</mi><mn>2</mn></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">D_1 - D_2 = \{x,y\ |\ x \in D_1 , y \in D_2\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace">&nbsp;</span><span class="mord">∣</span><span class="mspace">&nbsp;</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span> 是凸集</li><li>对于任意非零实数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span>，集合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><msub><mi>D</mi><mn>1</mn></msub><mo>=</mo><mo stretchy="false">{</mo><mi>α</mi><mi>x</mi><mtext>&nbsp;</mtext><mi mathvariant="normal">∣</mi><mtext>&nbsp;</mtext><mi>x</mi><mo>∈</mo><msub><mi>D</mi><mn>1</mn></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\alpha D_1 = \{ \alpha x \ |\ x \in D_1 \}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal">αx</span><span class="mspace">&nbsp;</span><span class="mord">∣</span><span class="mspace">&nbsp;</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span>​ 是凸集</li></ol>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-8f34ca46" role="button" aria-expanded="false" aria-controls="collapse-8f34ca46">        <div class="fold-arrow">▶</div>证明      </div>      <div class="fold-collapse collapse" id="collapse-8f34ca46">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403101941905.png" alt="凸集的性质证明"></p>        </div>      </div>    </div><p><strong>凸集的应用</strong></p><ol><li><p>刻画可行域</p><ul><li><p>凸组合定义</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>设&nbsp;</mtext><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mo separator="true">,</mo><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msup><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></msup><mo>∈</mo><msup><mi>R</mi><mi>n</mi></msup><mo separator="true">,</mo><mtext>且&nbsp;</mtext><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>p</mi></munderover><msub><mi>α</mi><mi>i</mi></msub><mo>=</mo><mn>1</mn><mo stretchy="false">(</mo><msub><mi>a</mi><mi>i</mi></msub><mo>≥</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>s</mi><mi mathvariant="normal">.</mi><mi>t</mi><mi mathvariant="normal">.</mi><mspace width="1em"></mspace><mi>x</mi><mo>=</mo><msub><mi>α</mi><mn>1</mn></msub><msup><mi>x</mi><mn>1</mn></msup><mo>+</mo><msub><mi>α</mi><mn>2</mn></msub><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>α</mi><mi>p</mi></msub><msup><mi>x</mi><mi>p</mi></msup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>则称&nbsp;</mtext><mi>x</mi><mtext>&nbsp;为向量&nbsp;</mtext><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mo separator="true">,</mo><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msup><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></msup><mtext>&nbsp;的凸组合</mtext></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\text{设}\ x^{(1)},x^{(2)}, \cdots, x^{(p)} \in R^n ,\text{且}\ \sum_{i=1}^p \alpha_i = 1(a_i \ge 0)  \\s.t. \quad x = \alpha_1x^1 + \alpha_2x^2 + \cdots + \alpha_px^p  \\\text{则称}\ x\ \text{为向量}\ x^{(1)},x^{(2)}, \cdots, x^{(p)}\ \text{的凸组合}\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:6.3983em;vertical-align:-2.9491em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.4491em;"><span style="top:-5.4491em;"><span class="pstrut" style="height:3.6985em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">设</span></span><span class="mspace">&nbsp;</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">2</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight">p</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord text"><span class="mord cjk_fallback">且</span></span><span class="mspace">&nbsp;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6985em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3471em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span><span style="top:-3.0074em;"><span class="pstrut" style="height:3.6985em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="mord">.</span><span class="mord mathnormal">t</span><span class="mord">.</span><span class="mspace" style="margin-right:1em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span></span></span></span></span></span></span><span style="top:-1.4094em;"><span class="pstrut" style="height:3.6985em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">则称</span></span><span class="mspace">&nbsp;</span><span class="mord mathnormal">x</span><span class="mspace">&nbsp;</span><span class="mord text"><span class="mord cjk_fallback">为向量</span></span><span class="mspace">&nbsp;</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">2</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight">p</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace">&nbsp;</span><span class="mord text"><span class="mord cjk_fallback">的凸组合</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.9491em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li><li><p>凸组合定理：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo>∈</mo><msup><mi>R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">D \in R^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> 是凸集的充分必要条件是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 中任取 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 个点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mi>i</mi></msup><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x^i(1,2,\cdots m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0747em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span> 的凸组合任属于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span>，即：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><msub><mi>α</mi><mi>i</mi></msub><msub><mi>x</mi><mi>i</mi></msub><mo>∈</mo><mi>D</mi><mrow><mo fence="true">(</mo><msub><mi>α</mi><mi>i</mi></msub><mo>≥</mo><mn>0</mn><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mi>m</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><msub><mi>α</mi><mi>i</mi></msub><mo>=</mo><mn>1</mn><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\sum_{i=1}^m \alpha_ix_i \in D\left( \alpha_i \ge 0(i=1,2,\cdots,m),\sum_{i=1}^m \alpha_i = 1 \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.0277em;vertical-align:-1.2777em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">(</span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size4">)</span></span></span></span></span></span></span></p>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-e771bc53" role="button" aria-expanded="false" aria-controls="collapse-e771bc53">        <div class="fold-arrow">▶</div>凸组合定理证明      </div>      <div class="fold-collapse collapse" id="collapse-e771bc53">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403102025159.png" alt="凸组合定理证明"></p>        </div>      </div>    </div></li></ul></li><li><p>分析最优解的最优性条件</p><ul><li><p>超平面定义（凸集分离定理）：设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>D</mi><mn>2</mn></msub><mo>⊂</mo><msup><mi>R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">D_1,D_2 \subset R^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> 为两非空凸集，若存在非零向量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo>∈</mo><msup><mi>R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\alpha \in R^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> 和实数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span>，使得</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>D</mi><mn>1</mn></msub><mo>⊂</mo><msup><mi>H</mi><mo>+</mo></msup><mo>=</mo><mo stretchy="false">{</mo><mi>x</mi><mo>∈</mo><msup><mi>R</mi><mi>n</mi></msup><mtext>&nbsp;</mtext><mi mathvariant="normal">∣</mi><mtext>&nbsp;</mtext><msup><mi>α</mi><mi>T</mi></msup><mi>x</mi><mo>≥</mo><mi>β</mi><mo stretchy="false">}</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>D</mi><mn>2</mn></msub><mo>⊂</mo><msup><mi>H</mi><mo>−</mo></msup><mo>=</mo><mo stretchy="false">{</mo><mi>x</mi><mo>∈</mo><msup><mi>R</mi><mi>n</mi></msup><mtext>&nbsp;</mtext><mi mathvariant="normal">∣</mi><mtext>&nbsp;</mtext><msup><mi>α</mi><mi>T</mi></msup><mi>x</mi><mo>≤</mo><mi>β</mi><mo stretchy="false">}</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}D_1 \subset H^+ = \{ x \in R^n \ | \ \alpha^T x \ge \beta\} \\D_2 \subset H^- = \{ x \in R^n \ | \ \alpha^T x \le \beta\}\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.1027em;vertical-align:-1.3013em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8013em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8213em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">{</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace">&nbsp;</span><span class="mord">∣</span><span class="mspace">&nbsp;</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mclose">}</span></span></span><span style="top:-2.3587em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8213em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">−</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">{</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace">&nbsp;</span><span class="mord">∣</span><span class="mspace">&nbsp;</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mclose">}</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3013em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>则称超平面 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi><mo>=</mo><mo stretchy="false">{</mo><mi>x</mi><mo>∈</mo><msup><mi>R</mi><mi>n</mi></msup><mtext>&nbsp;</mtext><mi mathvariant="normal">∣</mi><mtext>&nbsp;</mtext><msup><mi>α</mi><mi>T</mi></msup><mi>x</mi><mo>=</mo><mi>β</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">H = \{ x \in R^n \ | \ \alpha^Tx=\beta \}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0913em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace">&nbsp;</span><span class="mord">∣</span><span class="mspace">&nbsp;</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mclose">}</span></span></span></span> <strong>分离</strong>集合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">D_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">D_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，<strong>严格分离</strong>时上述不等式无法取等</p></li><li><p>投影定理：设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo>∈</mo><msup><mi>R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">D \in R^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> 是非空闭凸集，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>∈</mo><msup><mi>R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">y \in R^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> 但 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo mathvariant="normal">∉</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">y \notin D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mord"><span class="mrel">∈</span></span><span class="mord vbox"><span class="thinbox"><span class="llap"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="inner"><span class="mord"><span class="mord">/</span><span class="mspace" style="margin-right:0.0556em;"></span></span></span><span class="fix"></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span>，则</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mtext>存在唯一的点&nbsp;</mtext><mover accent="true"><mi>x</mi><mo stretchy="true">‾</mo></mover><mo>∈</mo><mi>D</mi><mo separator="true">,</mo><mtext>使得集合</mtext><mi>D</mi><mtext>到点</mtext><mi>y</mi><mtext>的距离最小</mtext></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mover accent="true"><mi>x</mi><mo stretchy="true">‾</mo></mover><mo>∈</mo><mi>D</mi><mtext>是点</mtext><mi>y</mi><mtext>到集合D的最短距离点的充分必要条件为</mtext><mo>:</mo><mi mathvariant="normal">∀</mi><mi>x</mi><mo>∈</mo><mi>D</mi><mo separator="true">,</mo><mo>&lt;</mo><mi>x</mi><mo>−</mo><mover accent="true"><mi>x</mi><mo stretchy="true">‾</mo></mover><mo separator="true">,</mo><mi>y</mi><mo>−</mo><mover accent="true"><mi>x</mi><mo stretchy="true">‾</mo></mover><mo>&gt;</mo><mo>≤</mo><mn>0</mn></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{align*}    (1)&amp; \text{存在唯一的点} \ \overline x \in D,\text{使得集合}D\text{到点} y \text{的距离最小} \\    (2)&amp; \overline x \in D \text{是点} y \text{到集合D的最短距离点的充分必要条件为}:\forall x \in D,&lt;x-\overline x,y - \overline x&gt; \le 0\end{align*}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">2</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord text"><span class="mord cjk_fallback">存在唯一的点</span></span><span class="mspace">&nbsp;</span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6306em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span></span></span><span style="top:-3.5506em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord text"><span class="mord cjk_fallback">使得集合</span></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord text"><span class="mord cjk_fallback">到点</span></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mord text"><span class="mord cjk_fallback">的距离最小</span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6306em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span></span></span><span style="top:-3.5506em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord text"><span class="mord cjk_fallback">是点</span></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mord text"><span class="mord cjk_fallback">到集合</span><span class="mord">D</span><span class="mord cjk_fallback">的最短距离点的充分必要条件为</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">∀</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6306em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span></span></span><span style="top:-3.5506em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6306em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span></span></span><span style="top:-3.5506em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ul></li></ol><h4 id="1-2-2-凸函数">1.2.2 凸函数</h4><p>凸函数的定义：设函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在凸集 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上有定义</p><ul><li>若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∀</mi><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>∈</mo><mi>D</mi><mtext>&nbsp;和&nbsp;</mtext><mi>λ</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\forall x,y \in D\ \text{和}\ \lambda \in [0,1]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">∀</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mspace">&nbsp;</span><span class="mord text"><span class="mord cjk_fallback">和</span></span><span class="mspace">&nbsp;</span><span class="mord mathnormal">λ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">]</span></span></span></span> 有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>λ</mi><mi>x</mi><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>λ</mi><mo stretchy="false">)</mo><mi>y</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>λ</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>λ</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(\lambda x + (1-\lambda)y) \le \lambda f(x) + (1-\lambda)f(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">λ</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">λ</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">λ</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">λ</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span>，则称 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 是凸集 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上的凸函数</li><li>若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∀</mi><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>∈</mo><mi>D</mi><mtext>&nbsp;和&nbsp;</mtext><mi>λ</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\forall x,y \in D\ \text{和}\ \lambda \in (0,1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">∀</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mspace">&nbsp;</span><span class="mord text"><span class="mord cjk_fallback">和</span></span><span class="mspace">&nbsp;</span><span class="mord mathnormal">λ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>λ</mi><mi>x</mi><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>λ</mi><mo stretchy="false">)</mo><mi>y</mi><mo stretchy="false">)</mo><mo>&lt;</mo><mi>λ</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>λ</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(\lambda x + (1-\lambda)y) &lt; \lambda f(x) + (1-\lambda)f(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">λ</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">λ</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">λ</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">λ</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span>，则称 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 是凸集 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上的严格凸函数</li></ul><p>凸函数的性质：</p><ol><li>如果 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span> 是定义在凸集 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上的凸函数，实数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\alpha \ge 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">\alpha f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span> 也是凸集 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上的凸函数</li><li>如果 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>f</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">f_1,f_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是定义在凸集 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上的凸函数，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mn>1</mn></msub><mo>+</mo><msub><mi>f</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">f_1+f_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 也是凸集 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上的凸函数</li><li>如果 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f_i(x)(i=1,2,\cdots,m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span> 是非空凸集 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上的凸函数，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>max</mi><mo>⁡</mo></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>m</mi></mrow></msub><mi mathvariant="normal">∣</mi><msub><mi>f</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">f(x) = \max_{1 \le i \le m} |f_i(x)|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop"><span class="mop">max</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mrel mtight">≤</span><span class="mord mathnormal mtight">i</span><span class="mrel mtight">≤</span><span class="mord mathnormal mtight">m</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2452em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord">∣</span></span></span></span> 也是凸集 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上的凸函数</li><li>如果 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f_i(x)(i=1,2,\cdots,m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span> 是非空凸集 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上的凸函数，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><msub><mi>α</mi><mi>i</mi></msub><msub><mi>f</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="1em"></mspace><mo stretchy="false">(</mo><msub><mi>α</mi><mi>i</mi></msub><mo>≥</mo><mn>0</mn><mo stretchy="false">)</mo></mstyle></mrow><annotation encoding="application/x-tex">f(x) = \displaystyle \sum_{i=1}^m \alpha_i f_i(x)\quad(\alpha_i \ge 0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:1em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span> 也是凸集 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span>​ 上的凸函数</li></ol>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-dcc9b38c" role="button" aria-expanded="false" aria-controls="collapse-dcc9b38c">        <div class="fold-arrow">▶</div>凸函数的性质证明      </div>      <div class="fold-collapse collapse" id="collapse-dcc9b38c">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403112030190.png" alt="第1、2条"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403112030160.png" alt="第4条"></p>        </div>      </div>    </div><p><strong>凸函数的判定定理</strong>：</p><ol><li><p>函数值角度：函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">R^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> 上的凸函数的充分必要条件是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∀</mi><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>∈</mo><msup><mi>R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\forall x,y \in R^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">∀</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span>，单变量函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>α</mi><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(\alpha)=f(x + \alpha y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">ϕ</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> 是关于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 的凸函数</p>     <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-d7d004d8" role="button" aria-expanded="false" aria-controls="collapse-d7d004d8">        <div class="fold-arrow">▶</div>凸函数的判别定理证明：函数值角度      </div>      <div class="fold-collapse collapse" id="collapse-d7d004d8">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403131019868.png" alt="凸函数的判别定理证明 - 函数值角度"></p>        </div>      </div>    </div></li><li><p>一阶梯度角度：设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 是定义在非空开凸集 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上的可微函数，则：</p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上凸函数的充分必要条件是：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>≥</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mi>T</mi></msup><mo stretchy="false">(</mo><mi>y</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mspace width="1em"></mspace><mi mathvariant="normal">∀</mi><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>∈</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">f(y) \ge f(x)+\nabla f(x)^T(y-x), \quad \forall x,y \in D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0913em;vertical-align:-0.25em;"></span><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∀</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span></li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上严格凸函数的充分必要条件是：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>&gt;</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo><mi>T</mi></msup><mo stretchy="false">(</mo><mi>y</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mspace width="1em"></mspace><mi mathvariant="normal">∀</mi><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>∈</mo><mi>D</mi><mo separator="true">,</mo><mspace width="1em"></mspace><mi>x</mi><mo mathvariant="normal">≠</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">f(y) &gt; f(x)+\nabla f(x)^T(y-x), \quad \forall x,y \in D,\quad x \ne y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0913em;vertical-align:-0.25em;"></span><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∀</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span></li></ul>     <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-9d403b87" role="button" aria-expanded="false" aria-controls="collapse-9d403b87">        <div class="fold-arrow">▶</div>凸函数的判别定理证明：一阶导数角度      </div>      <div class="fold-collapse collapse" id="collapse-9d403b87">        <div class="fold-content">          <p>无需掌握证明，但是为了便于理解性记忆，可以从<strong>二次凸函数</strong>进行辅助理解记忆。<br><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403131018032.png" alt="凸函数的判别定理证明 - 一阶导数角度"></p>        </div>      </div>    </div></li><li><p>二阶梯度角度：设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 是定义在非空开凸集 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上的二阶可微函数，则：</p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上凸函数的充分必要条件是：海塞矩阵正半定</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上严格凸函数的充分必要条件是：海塞矩阵正定？其实不一定，例如 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mn>4</mn></msup></mrow><annotation encoding="application/x-tex">x^4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span> 是严格凸函数，但是其海塞矩阵是正半定的并非正定。因此更严谨的定义应该是：海塞矩阵正定的是严格凸函数，严格凸函数的海塞矩阵是正半定的。</li></ul></li></ol><h4 id="1-2-3-凸规划（个人补充）">1.2.3 凸规划（个人补充）</h4><p>本目为个人补充内容，用于整合上述 1.2.1 与 1.2.2 内容。我们知道，学习凸集和凸函数的终极目标是为了求解凸规划问题，凸规划问题可以简述为 <strong>凸可行域 + 凸目标函数 + 局部最优解 = 全局最优解</strong>，那么如何证明这个定理是正确的呢？局部最优解求出来以后，目标函数也确定为凸函数以后，如何确定可行域是凸集呢？下面揭晓：</p><p><strong>证明凸规划问题的正确性</strong></p><ol><li><p>定理：在可行域是凸集，目标函数非严格凸的情况下，局部最优解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">x^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span>​ 也是全局最优解</p>     <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-01c9e6ee" role="button" aria-expanded="false" aria-controls="collapse-01c9e6ee">        <div class="fold-arrow">▶</div>证明（反证法）      </div>      <div class="fold-collapse collapse" id="collapse-01c9e6ee">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403122016285.png" alt="证明（反证法）"></p>        </div>      </div>    </div></li><li><p>定理：在可行域是凸集，目标函数是严格凸的情况下，局部最优解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">x^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span> 也是唯一的全局最优解</p>     <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-858dd40d" role="button" aria-expanded="false" aria-controls="collapse-858dd40d">        <div class="fold-arrow">▶</div>证明（反证法）      </div>      <div class="fold-collapse collapse" id="collapse-858dd40d">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403122017101.png" alt="证明（反证法）"></p>        </div>      </div>    </div></li></ol><p><strong>确定可行域是否为凸集</strong></p><ol><li><p>定理：若约束条件 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">c_i(x) \ge 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 中每一个约束函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c_i(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 都是凹函数，则可行域 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span> 是凸集</p>     <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-4ecb9cd4" role="button" aria-expanded="false" aria-controls="collapse-4ecb9cd4">        <div class="fold-arrow">▶</div>证明      </div>      <div class="fold-collapse collapse" id="collapse-4ecb9cd4">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403122019899.png" alt="证明"></p>        </div>      </div>    </div></li><li><p>定理：若约束条件 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≤</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">c_i(x) \le 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 中每一个约束函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c_i(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 都是凸函数，则可行域 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span> 是凸集</p>     <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-8354fb68" role="button" aria-expanded="false" aria-controls="collapse-8354fb68">        <div class="fold-arrow">▶</div>证明      </div>      <div class="fold-collapse collapse" id="collapse-8354fb68">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403122019572.png" alt="证明"></p>        </div>      </div>    </div></li><li><p>定理：若约束条件中每一个约束函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c_i(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 都恒等于零，则可行域 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span> 是凸集</p>     <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-b528b2da" role="button" aria-expanded="false" aria-controls="collapse-b528b2da">        <div class="fold-arrow">▶</div>证明      </div>      <div class="fold-collapse collapse" id="collapse-b528b2da">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403122019751.png" alt="证明"></p>        </div>      </div>    </div></li></ol><h3 id="1-3-最优性条件">1.3 最优性条件</h3><div class="note note-light">            <p>最优性条件是指最优化问题的最优解所必须满足的条件，本目介绍求解<strong>无约束最优化问题</strong>的一阶必要条件和二阶必要条件，再补充介绍二阶充分条件以及针对凸规划问题的二阶充分必要条件。最后简单介绍求解<strong>等式约束最优化问题</strong>的拉格朗日乘子法。</p>          </div><h4 id="1-3-1-下降方向">1.3.1 下降方向</h4><p>在开始之前先简单介绍一下「下降方向」这个概念。</p><ul><li><p><strong>下降方向</strong>定义：设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 为定义在空间 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">R^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> 的连续函数，点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>x</mi><mo>ˉ</mo></mover><mo>∈</mo><msup><mi>R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\bar x \in R^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6069em;vertical-align:-0.0391em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">x</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span>，若对于方向 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi><mo>∈</mo><msup><mi>R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">s \in R^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> 存在数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>δ</mi><mo>&gt;</mo></mrow><annotation encoding="application/x-tex">\delta &gt;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.03785em;">δ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span></span></span></span> 0 使</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mover accent="true"><mi>x</mi><mo>ˉ</mo></mover><mo>+</mo><mi>α</mi><mi>s</mi><mo stretchy="false">)</mo><mo>&lt;</mo><mi>f</mi><mo stretchy="false">(</mo><mover accent="true"><mi>x</mi><mo>ˉ</mo></mover><mo stretchy="false">)</mo><mo separator="true">,</mo><mspace width="1em"></mspace><mi mathvariant="normal">∀</mi><mi>α</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mi>δ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(\bar x+\alpha s) &lt; f(\bar x),\quad \forall \alpha \in (0,\delta)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">x</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord mathnormal">s</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">x</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∀</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03785em;">δ</span><span class="mclose">)</span></span></span></span></span></p><p>成立，则称 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">s</span></span></span></span> 为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>x</mi><mo>ˉ</mo></mover></mrow><annotation encoding="application/x-tex">\bar x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5678em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">x</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">ˉ</span></span></span></span></span></span></span></span></span></span> 处的一个下降方向。在 点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>x</mi><mo>ˉ</mo></mover></mrow><annotation encoding="application/x-tex">\bar x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5678em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">x</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">ˉ</span></span></span></span></span></span></span></span></span></span> 处的所有下降方向的全体记为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mover accent="true"><mi>x</mi><mo>ˉ</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(\bar x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">x</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></p></li><li><p><strong>下降方向</strong>定理：设函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>x</mi><mo>ˉ</mo></mover></mrow><annotation encoding="application/x-tex">\bar x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5678em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">x</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">ˉ</span></span></span></span></span></span></span></span></span></span> 处连续可微，如存在非零向量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi><mo>∈</mo><msup><mi>R</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">s  \in R^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> 使</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><mover accent="true"><mi>x</mi><mo>ˉ</mo></mover><msup><mo stretchy="false">)</mo><mi>T</mi></msup><mi>s</mi><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\nabla f(\bar x)^Ts &lt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1413em;vertical-align:-0.25em;"></span><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">x</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p><p>成立，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">s</span></span></span></span> 是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>x</mi><mo>ˉ</mo></mover></mrow><annotation encoding="application/x-tex">\bar x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5678em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5678em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">x</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2222em;"><span class="mord">ˉ</span></span></span></span></span></span></span></span></span></span> 处的一个下降方向</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-b02d0016" role="button" aria-expanded="false" aria-controls="collapse-b02d0016">        <div class="fold-arrow">▶</div>证明      </div>      <div class="fold-collapse collapse" id="collapse-b02d0016">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403131639082.jpg" alt="证明"></p>        </div>      </div>    </div></li></ul><h4 id="1-3-2-充分必要条件">1.3.2 充分必要条件</h4><p>设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo>:</mo><mi>D</mi><mo>⊂</mo><msup><mi>R</mi><mi>n</mi></msup><mo>→</mo><msup><mi>R</mi><mn>1</mn></msup></mrow><annotation encoding="application/x-tex">f:D \subset R^n \to R^1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span> 是定义在开集 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 上的函数，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup><mo>∈</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">x^* \in D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0391em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 是该函数的局部极小点，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 的一阶导数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(x)=\nabla f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>，二阶导数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi mathvariant="normal">∇</mi><mn>2</mn></msup><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G(x)=\nabla ^2f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">G</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord">∇</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>。</p><ul><li><p><strong>一阶必要条件</strong>。若目标函数在定义域 D 上连续可微，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">x^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span> 是局部极小点的一阶必要条件为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><msup><mi>x</mi><mo>∗</mo></msup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">g(x^*)=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7387em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p></li><li><p><strong>二阶必要条件</strong>。若目标函数在定义域 D 上二阶连续可微，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">x^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span> 是局部极小点的二阶必要条件为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><msup><mi>x</mi><mo>∗</mo></msup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mspace width="1em"></mspace><mi>G</mi><mo stretchy="false">(</mo><msup><mi>x</mi><mo>∗</mo></msup><mo stretchy="false">)</mo><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">g(x^*)=0,\quad G(x^*)\ge 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7387em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">G</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7387em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p></li><li><p><strong>二阶充分条件</strong>。若目标函数在定义域 D 上二阶连续可微，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">x^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span> 是<strong>严格</strong>局部极小点的二阶充分条件为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><msup><mi>x</mi><mo>∗</mo></msup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mspace width="1em"></mspace><mi>G</mi><mo stretchy="false">(</mo><msup><mi>x</mi><mo>∗</mo></msup><mo stretchy="false">)</mo><mtext>是正定的</mtext></mrow><annotation encoding="application/x-tex">g(x^*)=0,\quad G(x^*) \text{是正定的}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7387em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">G</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7387em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord text"><span class="mord cjk_fallback">是正定的</span></span></span></span></span></span></p></li><li><p><strong>充分必要条件（凸最优性定理）</strong>。在无约束最优化问题中，如果目标函数是凸的，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">x^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span> 是全局极小点的充分必要条件是：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><msup><mi>x</mi><mo>∗</mo></msup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">g(x^*)=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7387em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p></li></ul><h4 id="1-3-3-拉格朗日乘子法">1.3.3 拉格朗日乘子法</h4><p>如果现在最优化问题不是单纯的<strong>无约束最优化问题</strong>，而是增设了等式约束的<strong>等式约束最优化问题</strong>，如何求解呢？我们引入 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Lagrange</mtext></mrow><annotation encoding="application/x-tex">\text{Lagrange}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">Lagrange</span></span></span></span></span> 乘子法，将所有的等式约束都转化到目标函数中得到一个等价的无约束最优化问题，即：</p><p>对于这样的等式约束最优化问题：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>min</mi><mo>⁡</mo><mspace width="1em"></mspace></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mtext>s.t.</mtext><mspace width="1em"></mspace></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msub><mi>c</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mi>m</mi><mi mathvariant="normal">.</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\min\quad&amp; f(x)\\\text{s.t.}\quad&amp;c_i(x)=0,i=1,2,\cdots,m.\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop">min</span><span class="mspace" style="margin-right:1em;"></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord">s.t.</span></span><span class="mspace" style="margin-right:1em;"></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mord">.</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>引入拉格朗日乘子将其转化为无约束最优化问题，进而利用上述无约束最优化问题的求解策略进行求解：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>λ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><msub><mi>λ</mi><mi>i</mi></msub><msub><mi>c</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(x,\lambda) = f(x) - \sum_{i=1}^m\lambda_ic_i(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">λ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></p><h3 id="1-4-最优化方法概述">1.4 最优化方法概述</h3><p>现实生活中，对于常见问题的建模往往是极其复杂的，为了求得最优解，我们需要对建立的模型进行微分算子的求解。但是问题是，尽管我们知道最优解一定存在于微分算子的数值解中，但是我们往往不能很快的计算出其数值解，因此我们需要采用别的方法进行计算。最常用的方法就是本书介绍的迭代法。</p><p>大概就两步：确定初值开始判断，如果符合条件的约束则停止迭代得到最终的解；如果不符合约束则对当前迭代值赋予修正量并继续迭代判断。直到找到最终的解。接下来将对 5 个专有名词进行解释，分别为：初始点的选取、迭代点好坏的判定、收敛速度、迭代的终止条件、修正量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>s</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">s^{(k)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.888em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span> 的确定。</p><h4 id="1-4-1-初始点的选取">1.4.1 初始点的选取</h4><p>初始点的选取取决于算法的收敛性能。</p><ul><li>如果一个算法可以做到全局收敛，则初始点的选取是任意的</li><li>如果一个算法只能局部收敛，则初始点的选取往往需要尽可能的接近最优解，而这往往是困难的，因为我们并不知道的最优解是什么。此时可以采用借鉴之前的经验来选择初始点</li></ul><h4 id="1-4-2-迭代点好坏的判定">1.4.2 迭代点好坏的判定</h4><p>判定一个迭代点的好坏时，我们往往会设计一个评价函数，而评价函数的设计取决于约束问题的种类。</p><ul><li>对于无约束最优化问题。我们可以直接将目标函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 设计为评价函数。很显然，如果一个迭代点对应的的函数值比之前点对应的函数值更小，则可以说明当前迭代点由于之前的点</li><li>对于有约束最优化问题。此时我们不仅仅需要考虑函数值的大小，也要考虑当前迭代点的可行程度（离可行域的距离）。因此此类最优化问题往往既包含目标函数，也包含约束函数。</li></ul><h4 id="1-4-3-收敛速度">1.4.3 收敛速度</h4><p>若当前算法一定收敛，我们还需要判断收敛的速度，接下来介绍收敛的速度。</p><p>假设</p><ul><li>设向量序列 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup><mo>⊂</mo><msup><mi>R</mi><mi>n</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{ x^{(k)} \subset R^n \}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.138em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span> 收敛于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">x^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span>​</li></ul><p>定义</p><ol><li>误差序列：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>e</mi><mi>k</mi></msub><mo>=</mo><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup><mo>−</mo><msup><mi>x</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">e_k = x^{(k)} - x^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9713em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span></li><li>收敛率表达式：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>k</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow></munder><mfrac><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><msub><mi>e</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi></mrow><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><msub><mi>e</mi><mi>k</mi></msub><mi mathvariant="normal">∣</mi><msup><mi mathvariant="normal">∣</mi><mi>r</mi></msup></mrow></mfrac><mo>=</mo><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \lim_{k \to \infty} \frac{|| e_{k+1} ||}{||e_k||^r} = C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.3479em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">→</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7521em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣∣</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord">∣</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5904em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣∣</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mord">∣∣</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>，并称序列 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{x^{(k)}\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.138em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mclose">}</span></span></span></span> 的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span> 阶以 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> 为因子收敛于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">x^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span></li></ol><p>线性收敛</p><ul><li>定义：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>0</mn><mo>&lt;</mo><mi>C</mi><mo>&lt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">r=1,0 &lt; C&lt;1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></li><li>性质：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> 越小，算法收敛越快</li></ul><p>超线性收敛</p><ul><li><p>定义：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mi>C</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">r=1,C=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></p></li><li><p>定理：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mo>&gt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">r&gt;1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>​ 时的算法均为超线性收敛算法</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-7f0bbd51" role="button" aria-expanded="false" aria-controls="collapse-7f0bbd51">        <div class="fold-arrow">▶</div>证明      </div>      <div class="fold-collapse collapse" id="collapse-7f0bbd51">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403201331819.png" alt="证明"></p>        </div>      </div>    </div></li></ul><h4 id="1-4-4-迭代的终止条件">1.4.4 迭代的终止条件</h4><ul><li><p>对于收敛速度比较慢的算法：</p><p>根据一阶必要条件，我们知道最优解一定取在微分算子为 0 的向量上，因此我们以下式为终止条件是一种可行的选择</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mo>≤</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">||\nabla f(x^{(k)}|| \le \epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.188em;vertical-align:-0.25em;"></span><span class="mord">∣∣∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mord">∣∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ϵ</span></span></span></span></span></p><p>但是问题在于这对收敛速度很快的算法不使用，如下图的无约束最优化问题。已知两个局部最优解分别为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>x</mi><mn>1</mn><mo>∗</mo></msubsup><mo separator="true">,</mo><msubsup><mi>x</mi><mn>2</mn><mo>∗</mo></msubsup></mrow><annotation encoding="application/x-tex">x_1^*,x_2^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9368em;vertical-align:-0.2481em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-2.4519em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-2.4519em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span></span></span></span>，迭代的两个解分别为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>x</mi><mo stretchy="true">‾</mo></mover><mn>1</mn></msub><mo separator="true">,</mo><msub><mover accent="true"><mi>x</mi><mo stretchy="true">‾</mo></mover><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\overline{x}_1,\overline{x}_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.825em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6306em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span></span></span><span style="top:-3.5506em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6306em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span></span></span><span style="top:-3.5506em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，可以看出：尽管 2 号点相对于 1 号点更靠近局部最优解，但是由于 2 号点的梯度更大，明显不如 1 号点更加局部最优，因此利用微分算子作为迭代的终止条件不适用于收敛速度快的算法</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403201330908.png" alt="无约束最优化问题"></p></li><li><p>对于超线性收敛的算法：</p><p>一个理想的收敛终止条件为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup><mo>−</mo><msup><mi>x</mi><mo>∗</mo></msup><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mo>≤</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">||x^{(k)} - x^*|| \le \epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.138em;vertical-align:-0.25em;"></span><span class="mord">∣∣</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mord">∣∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ϵ</span></span></span></span>，但是由于最优解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">x^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span> 是未知的，因此该方案不可行，同理 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msup><mi>x</mi><mo>∗</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x^*)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 也是未知的，因此 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mi>f</mi><mo stretchy="false">(</mo><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><msup><mi>x</mi><mo>∗</mo></msup><mo stretchy="false">)</mo><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mo>≤</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">||f(x^{(k)}) - f(x^*)|| \le \epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.138em;vertical-align:-0.25em;"></span><span class="mord">∣∣</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord">∣∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ϵ</span></span></span></span> 也是不可行的。那么有没有什么替代的方案呢？答案是有的。</p><ol><li><p>方案一：利用<strong>函数解序列</strong>进行替代。即以下式作为迭代的终止条件</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mo>−</mo><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mo>≤</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">|| x^{(k+1)} - x^{(k)} || \le \epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.188em;vertical-align:-0.25em;"></span><span class="mord">∣∣</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.188em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mord">∣∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ϵ</span></span></span></span></span></p>     <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-f8eb5676" role="button" aria-expanded="false" aria-controls="collapse-f8eb5676">        <div class="fold-arrow">▶</div>证明      </div>      <div class="fold-collapse collapse" id="collapse-f8eb5676">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403201425856.jpg" alt="证明"></p>        </div>      </div>    </div></li><li><p>方案二：利用<strong>函数值序列</strong>进行替代。即以下式作为迭代的终止条件</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mi>f</mi><mo stretchy="false">(</mo><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">)</mo><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mo>≤</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">|| f(x^{(k+1)}) - f(x^{(k)}) || \le \epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.188em;vertical-align:-0.25em;"></span><span class="mord">∣∣</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.188em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord">∣∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ϵ</span></span></span></span></span></p>     <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-1f855ea6" role="button" aria-expanded="false" aria-controls="collapse-1f855ea6">        <div class="fold-arrow">▶</div>证明      </div>      <div class="fold-collapse collapse" id="collapse-1f855ea6">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403201425473.jpg" alt="证明"></p>        </div>      </div>    </div></li></ol></li><li><p>一般情况下，对于上述超线性算法的判断收敛的方法，只用其中一种往往不适当。此时一般使用两种方法集成的思路进行判断。</p></li></ul><h4 id="1-4-5-修正量的确定">1.4.5 修正量的确定</h4><p>本书介绍的迭代修正值都是使得当前迭代后的值小于上一个状态的函数值，我们称这类使评价函数值下降的算法为<strong>单调下降算法</strong>。至于如何得出修正量，我们往往通过求解一个相对简单易于求解的最优化问题（通常成为子问题），来计算获得修正量。</p><h2 id="第二章「约束最优化」线性规划">第二章「约束最优化」线性规划</h2><div class="note note-light">            <p>目标函数和约束函数都是线性函数。</p>          </div><h3 id="2-1-线性规划问题和基本性质">2.1 线性规划问题和基本性质</h3><h4 id="2-1-1-线性规划问题">2.1.1 线性规划问题</h4><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403211918859.png" alt="线性规划问题的一般形式"></p><h4 id="2-1-2-图解法">2.1.2 图解法</h4><p><strong>仅适用于二元变量</strong>的线性规划问题。我们将所有的约束条件全部画到平面直角坐标系中构成一个可行域，然后将目标函数作为一条直线进行平移，直到与可行域初次有交点，则该交点就是最优解对应的点。当然不一定会有交点，一共分为四种情况：</p>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-0f7ad755" role="button" aria-expanded="false" aria-controls="collapse-0f7ad755">        <div class="fold-arrow">▶</div>刚好只有一个最优解      </div>      <div class="fold-collapse collapse" id="collapse-0f7ad755">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403211917063.png" alt="刚好只有一个最优解"></p>        </div>      </div>    </div>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-2097dcc0" role="button" aria-expanded="false" aria-controls="collapse-2097dcc0">        <div class="fold-arrow">▶</div>有无穷多最优解      </div>      <div class="fold-collapse collapse" id="collapse-2097dcc0">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403211920849.png" alt="有无穷多最优解"></p>        </div>      </div>    </div>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-c681e611" role="button" aria-expanded="false" aria-controls="collapse-c681e611">        <div class="fold-arrow">▶</div>有无界解      </div>      <div class="fold-collapse collapse" id="collapse-c681e611">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403211921495.png" alt="有无界解"></p>        </div>      </div>    </div>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-75bcc499" role="button" aria-expanded="false" aria-controls="collapse-75bcc499">        <div class="fold-arrow">▶</div>无可行解 - 可行域为空集      </div>      <div class="fold-collapse collapse" id="collapse-75bcc499">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403211922747.png" alt="无可行解 - 可行域为空集"></p>        </div>      </div>    </div><h4 id="2-1-3-基本性质">2.1.3 基本性质</h4><ol><li><p>线性规划问题的可行域如果非空，则是一个凸集</p>     <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-da25d73d" role="button" aria-expanded="false" aria-controls="collapse-da25d73d">        <div class="fold-arrow">▶</div>证明      </div>      <div class="fold-collapse collapse" id="collapse-da25d73d">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403212109584.jpg" alt="证明"></p>        </div>      </div>    </div></li><li><p>如果线性规划问题有最优解，那么最优解可在可行域的顶点中确定</p></li><li><p>如果可行域有界且可行域只有有限个顶点，则问题的最优解必存在，并在这有限个顶点中确定</p></li><li><p>最优解可由最优顶点处的<strong>有效约束</strong>形成的方程组的解确定</p></li></ol><h4 id="2-1-4-线性规划的标准形">2.1.4 线性规划的标准形</h4><p>为什么要学习线性规划的标准形？</p><ul><li>我们要学习单纯形法来计算多维线性规划的最优解</li><li>单纯形法需要线性规划的具有所谓的标准形</li></ul><p>线性规划的标准形的定义：</p><ul><li>只含有线性等式约束和对变量的非负约束</li></ul><p>一般线性规划转化为标准形的方法：</p><ol><li>对于目标函数：需要将取 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>max</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\max</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mop">max</span></span></span></span> 通过<strong>取相反数</strong>转化为取 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>min</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\min</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6679em;"></span><span class="mop">min</span></span></span></span></li><li>对于约束条件：需要将不等式约束<strong>松弛</strong>转化为等式约束<ul><li>对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≤</mo></mrow><annotation encoding="application/x-tex">\le</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≤</span></span></span></span> 的不等式约束，等式左边加上非负新变量，从而转化为等式</li><li>对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≥</mo></mrow><annotation encoding="application/x-tex">\ge</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≥</span></span></span></span> 的不等式约数，等式左边减去非负新变量，从而转化为等式</li></ul></li><li>对于无约束的变量：需要将其转化为<strong>两个新变量之差</strong>（可正可负），产生了一个新的等式，如果该无约束变量存在于目标函数中，还需要将目标函数中的该变量表示为两个新变量之差</li></ol>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-53f9dab5" role="button" aria-expanded="false" aria-controls="collapse-53f9dab5">        <div class="fold-arrow">▶</div>举个例子      </div>      <div class="fold-collapse collapse" id="collapse-53f9dab5">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403212111330.jpg" alt="一般线性规划转化为标准形 - 演示"></p>        </div>      </div>    </div><h4 id="2-1-5-基本可行解">2.1.5 基本可行解</h4><p>直接说结论：对于标准形的线性规划问题，其基本可行解就是凸集的顶点。</p><p>假设系数矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mrow><mi>m</mi><mo separator="true">,</mo><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">A_{m, n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>。基本可行解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>⊂</mo></mrow><annotation encoding="application/x-tex">\subset</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">⊂</span></span></span></span> 基本解，而基本解的个数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≤</mo><msubsup><mi>C</mi><mi>n</mi><mi>m</mi></msubsup></mrow><annotation encoding="application/x-tex">\le C_{n}^{m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9303em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span>（因为分块矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>B</mi><mrow><mi>m</mi><mo>×</mo><mi>m</mi></mrow></msub></mrow><annotation encoding="application/x-tex">B_{m \times m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2583em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">×</span><span class="mord mathnormal mtight">m</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span> 不一定可逆）。其中基本可行解需要满足解序列元素非负。如果<strong>基本变量</strong>向量对应的解中含有 0 元素，称其为退化的基本可行解，产生退化的基本可行解的原因是存在可删除的约数条件。</p><h4 id="2-1-6-最优解的性质">2.1.6 最优解的性质</h4><ul><li><p>若可行域有界，则线性规划问题的目标函数一定可以在其可行域的顶点上达到最优。</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-072cac81" role="button" aria-expanded="false" aria-controls="collapse-072cac81">        <div class="fold-arrow">▶</div>证明 - 反证法      </div>      <div class="fold-collapse collapse" id="collapse-072cac81">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403222103333.jpg" alt="证明"></p>        </div>      </div>    </div></li><li><p>有时，目标函数可能在多个顶点处达到最大，这时在这些顶点的<strong>凸组合</strong>上也达到最大值，这时线性规划问题有无限多个最优解。</p></li></ul><h3 id="2-2-单纯形法">2.2 单纯形法</h3><div class="note note-light">            <p>本目主要介绍一种常用的计算线性规划问题可行解的方法：单纯形法。其中，<strong>前三条</strong>分别介绍单纯性方法计算步骤的理论可行性，<strong>第四条</strong>具体介绍单纯形法的计算步骤与过程。</p>          </div><h4 id="2-2-1-初始基可行解的确定">2.2.1 初始基可行解的确定</h4><ol><li><strong>直接观察</strong>：直接看出系数矩阵中含有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>×</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">m \times m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 的单位矩阵，则选择该单位矩阵为初始基</li><li><strong>加松弛变量</strong>：当所有的约束条件都是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≤</mo></mrow><annotation encoding="application/x-tex">\le</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≤</span></span></span></span> 时，每一个约束条件都加一个非负的松弛变量，则这 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 个松弛变量的系数就对应了一个 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>×</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">m \times m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 的单位矩阵，选择其为初始基即可</li><li><strong>加非负的人工变量</strong>：求解方法与常规的线性规划问题不一样，见下述 2.2.3 条。<ul><li>对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo></mrow><annotation encoding="application/x-tex">=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">=</span></span></span></span> 约束，我们在约束条件的左边加上非负人工变量</li><li>对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≥</mo></mrow><annotation encoding="application/x-tex">\ge</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≥</span></span></span></span> 约束，我们先在约束条件的左边减去非负松弛变量，再在约束条件的左边加上非负的人工变量</li></ul></li></ol><h4 id="2-2-2-最优性检验">2.2.2 最优性检验</h4>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-73e17d47" role="button" aria-expanded="false" aria-controls="collapse-73e17d47">        <div class="fold-arrow">▶</div>符号说明      </div>      <div class="fold-collapse collapse" id="collapse-73e17d47">        <div class="fold-content">          <p>一、当前局面：已经计算出一个基本可行解</p><p>1.1 确定线性规划的目标</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403281940243.png" alt="确定线性规划的目标"></p><p>1.2 添加松弛变量</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403281942808.png" alt="添加松弛变量"></p><p>1.3 用非基变量线性表示基变量</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403281937210.png" alt="用非基变量线性表示基变量"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403281939336.png" alt="向量表示"></p><p>二、最优性检验</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403282010283.jpg" alt="最优性检验"></p>        </div>      </div>    </div><ol><li><p><strong>唯一最优解</strong>判别定理：对于<strong>目标函数求解最大值</strong>的情形。若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∀</mi><mi>j</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>m</mi><mo>+</mo><mn>1</mn><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\forall j \in [m+1, n]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">∀</span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">]</span></span></span></span> 有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>σ</mi><mi>j</mi></msub><mo>≤</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\sigma_j \le 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9221em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 则当前基本可行解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">x^{(k)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.888em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span> 为最优解。</p></li><li><p><strong>无穷多最优解</strong>判别定理：在满足第一条所有的检验数非正的情况下，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∃</mi><mi>j</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>m</mi><mo>+</mo><mn>1</mn><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\exist j \in [m+1, n]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">∃</span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">]</span></span></span></span>，有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>σ</mi><mi>j</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\sigma_j=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，则该线性规划问题有无穷多最优解。</p>     <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-0c8e4c38" role="button" aria-expanded="false" aria-controls="collapse-0c8e4c38">        <div class="fold-arrow">▶</div>证明      </div>      <div class="fold-collapse collapse" id="collapse-0c8e4c38">        <div class="fold-content">          <p>我们可以将任意一个检验数为 0 的非基变量与基变量进行置换得到新的一个基本可行解对应的目标函数值保持不变。于是下述凸组合内的可行解都是最优解</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>s</mi><mi mathvariant="normal">.</mi><mi>t</mi><mi mathvariant="normal">.</mi><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>k</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">∈</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo stretchy="false">[</mo><mi>m</mi><mo>+</mo><mn>1</mn><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">]</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>σ</mi><mi>k</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">=</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr></mtable></mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}&amp; x^{(k)} \\ &amp; s.t. \begin{cases} k &amp;\in&amp; [m+1, n] \\\sigma_k &amp;=&amp; 0\end{cases}\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.898em;vertical-align:-2.199em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.699em;"><span style="top:-5.511em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"></span></span><span style="top:-3.101em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.199em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.699em;"><span style="top:-5.511em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.101em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal">s</span><span class="mord">.</span><span class="mord mathnormal">t</span><span class="mord">.</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mrel">∈</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mrel">=</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mopen">[</span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">]</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.199em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>        </div>      </div>    </div></li><li><p><strong>无界解</strong>判别定理：pass</p></li><li><p><strong>无解</strong>判别定理：可行域为空集</p></li></ol><h4 id="2-2-3-计算新的基本可行解">2.2.3 计算新的基本可行解</h4><p>对于<strong>常规</strong>的线性规划问题（初始基本可行解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">x^{(0)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.888em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span> 可以直接找到）</p><ol><li>向量方程法<ul><li>确定入基变量：选择目标函数中<strong>系数最大的变量</strong>作为入基变量，即将其对应的系数列向量与出基变量对应的系数列向量进行置换</li><li>确定出基变量：线性表示基方程组以后，利用变量非负的特性找到入基变量刚好取 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 时对应的变量即为出基变量（示例的第二轮迭代中出基变量即为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>5</mn></msub></mrow><annotation encoding="application/x-tex">x_5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>）</li></ul></li><li>系数矩阵法<ul><li>pass</li></ul></li></ol><p>对于<strong>添加人工变量</strong>的线性规划问题（初始基本可行解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">x^{(0)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.888em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span> 不能直接找到，需要手动构造 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">x^{(0)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.888em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span>）</p><ol><li>大 M 法：pass</li><li>两阶段法<ul><li>第一阶段：pass</li><li>第二阶段：pass</li></ul></li></ol><h4 id="2-2-4-单纯形表法">2.2.4 单纯形表法</h4><p>向量方程法：</p>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-8c53e26b" role="button" aria-expanded="false" aria-controls="collapse-8c53e26b">        <div class="fold-arrow">▶</div>实战演练      </div>      <div class="fold-collapse collapse" id="collapse-8c53e26b">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403272056146.jpg" alt="单纯形法的求解步骤 - 1"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403272056440.jpg" alt="单纯形法的求解步骤 - 2"></p>        </div>      </div>    </div><p>单纯形表法：</p>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-766b3028" role="button" aria-expanded="false" aria-controls="collapse-766b3028">        <div class="fold-arrow">▶</div>实战演练      </div>      <div class="fold-collapse collapse" id="collapse-766b3028">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404171407440.jpg" alt="实战演练"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404171410846.jpg" alt="代码验算"></p>        </div>      </div>    </div><h3 id="2-3-对偶与对偶单纯形法">2.3 对偶与对偶单纯形法</h3><h4 id="2-3-1-确定对偶问题">2.3.1 确定对偶问题</h4><p>已知原问题的表达式，如何求解对偶问题的表达式？我们需掌握以下对偶转换规则：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404071144213.png" alt="转化规则"></p><p>从上图不难发现，其实就是对线性规划的三个部分进行转换：目标函数、m个约束条件、n个自变量，下面分别进行文字介绍：</p><ol><li>约束条件：从 m 个变为 n 个<ul><li><strong>常量矩阵</strong>中的元素为原问题中自变量的系数</li><li><strong>二元关系</strong>的变化取决于转化方向<ul><li>max to min：对偶问题约束条件的符号和原问题自变量的符号相同</li><li>min to max：对偶问题约束条件的符号和原问题自变量的符号相反</li></ul></li><li><strong>线性约束</strong>为原问题线性约束的转置</li></ul></li><li>自变量：从 n 个变为 m 个<ul><li><strong>二元关系</strong>的变化同样取决于转化方向<ul><li>max to min：对偶问题自变量的符号和原问题约束条件的符号相反</li><li>min to max：对偶问题自变量的符号和原问题约束条件的符号相同</li></ul></li></ul></li><li>目标函数：自元个数从 n 个变为 m 个<ul><li>变元的系数就是原问题中约束条件的常量矩阵对应的元素值</li></ul></li></ol>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-d8f9745d" role="button" aria-expanded="false" aria-controls="collapse-d8f9745d">        <div class="fold-arrow">▶</div>实战演练      </div>      <div class="fold-collapse collapse" id="collapse-d8f9745d">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202407081127115.png" alt="实例"></p>        </div>      </div>    </div><h4 id="2-3-2-对偶定理">2.3.2 对偶定理</h4><ul><li>对称性：对偶问题的对偶问题是原问题</li><li>无界性：若原问题是无界解，则对偶问题无可行解</li><li>对偶定理：若原问题有最优解，那么对偶问题也有最优解，且目标函数值相等</li><li>互补松弛性：若原问题最优解为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">x^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span>，对偶问题的最优解为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>y</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">y^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8831em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span>，则有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup><msub><mi>y</mi><mi>s</mi></msub><mo>=</mo><mn>0</mn><mo separator="true">,</mo><msup><mi>y</mi><mo>∗</mo></msup><msub><mi>x</mi><mi>s</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x^* y_s=0,y^*x_s=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8831em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8831em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>s</mi></msub><mo separator="true">,</mo><msub><mi>y</mi><mi>s</mi></msub></mrow><annotation encoding="application/x-tex">x_s,y_s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 分别为原问题的松弛变量和对偶问题的松弛变量</li></ul>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-59805afb" role="button" aria-expanded="false" aria-controls="collapse-59805afb">        <div class="fold-arrow">▶</div>实战演练      </div>      <div class="fold-collapse collapse" id="collapse-59805afb">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202407081259988.png" alt="问题"></p><p>对于上述问题，我们可以先写出其对偶问题：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202407081300732.png" alt="对偶问题"></p><p>显然的对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>y</mi><mo>∗</mo></msup><mo>=</mo><mo stretchy="false">(</mo><mfrac><mn>4</mn><mn>5</mn></mfrac><mo separator="true">,</mo><mfrac><mn>3</mn><mn>5</mn></mfrac><msup><mo stretchy="false">)</mo><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">y^*=(\frac{4}{5},\frac{3}{5})^T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8831em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mopen">(</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">5</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span></span></span>，上述 2,3,4 是取不到等号的，对应的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup><mo>=</mo><mo stretchy="false">(</mo><mo>−</mo><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mo>−</mo><msup><mo stretchy="false">)</mo><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">x^*=(-,0,0,0,-)^T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0913em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span></span></span>。还剩两个变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>5</mn></msub></mrow><annotation encoding="application/x-tex">x_1,x_5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 通过 1,5 两个不等式取等进行计算：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mn>3</mn><msub><mi>x</mi><mn>2</mn></msub><mo>=</mo><mn>4</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mn>2</mn><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>x</mi><mn>2</mn></msub><mo>=</mo><mn>3</mn></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}x_1+3x_2=4\\2x_1+x_2=3\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">3</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">4</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>计算可得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn><mo separator="true">,</mo><msub><mi>x</mi><mn>5</mn></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x_1=1,x_5=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，于是原问题的最优解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">x^*=(1, 0, 0, 0, 1)^T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0913em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span></span></span></p>        </div>      </div>    </div><h4 id="2-3-3-对偶单纯形法">2.3.3 对偶单纯形法</h4>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-c81b1c05" role="button" aria-expanded="false" aria-controls="collapse-c81b1c05">        <div class="fold-arrow">▶</div>实战演练      </div>      <div class="fold-collapse collapse" id="collapse-c81b1c05">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404171409291.jpg" alt="实战演练"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404171411832.jpg" alt="代码验算"></p>        </div>      </div>    </div><h2 id="第三章-线性搜索">第三章 线性搜索</h2><div class="note note-light">            <p>本章介绍最优化问题中迭代解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><msub><mi>α</mi><mi>k</mi></msub><msub><mi>d</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">x_{k+1} = x_k + \alpha_kd_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 基于「线性搜索」的迭代方式。</p><p>假设搜索方向 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">d_{k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是一个定值且一定是下降方向，我们讨论步长因子 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>α</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_{k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的计算选择策略。分为两步：</p><ol><li>确定步长的搜索区间</li><li>通过精确 or 不精确算法来搜索合适的步长</li></ol><p>本章的内容分布：</p><ul><li>3.1 介绍<strong>确定初始搜索区间</strong>的「进退法」</li><li>3.2 介绍<strong>缩小搜索区间</strong>的「精确线性搜索算法」：0.618 法、斐波那契法、二分法、插值法</li><li>3.3 介绍<strong>缩小搜索区间</strong>的「不精确线性搜索算法」：Armijo 准则、Goldstein 准则、Wolfe 准则</li></ul>          </div><h3 id="3-1-确定初始搜索区间">3.1 确定初始搜索区间</h3><p>确定初始搜索区间 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[a,b]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mclose">]</span></span></span></span>，我们利用 <a href="https://zhuanlan.zhihu.com/p/109701879">进退法</a></p><h3 id="3-2-精确线性搜索算法">3.2 精确线性搜索算法</h3>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-d825d270" role="button" aria-expanded="false" aria-controls="collapse-d825d270">        <div class="fold-arrow">▶</div>预备知识：单峰函数定理      </div>      <div class="fold-collapse collapse" id="collapse-d825d270">        <div class="fold-content">          <p>在正式开始介绍之前，我们先了解<strong>单峰函数定理</strong>：</p><ul><li><p>定理：对于在区间 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[a,b]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mclose">]</span></span></span></span> 上的一个单峰函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup><mo>∈</mo><mo stretchy="false">[</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">x^* \in [a,b]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0391em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mclose">]</span></span></span></span> 是其极小点，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[a,b]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mclose">]</span></span></span></span> 上的任意两点，且 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>&lt;</mo><msub><mi>x</mi><mn>1</mn></msub><mo>&lt;</mo><msub><mi>x</mi><mn>2</mn></msub><mo>&lt;</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a&lt;x_1&lt;x_2&lt;b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span>，可以通过比较 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x_1),f(x_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 的值来确定点的保留和舍弃</p></li><li><p>迭代：</p><ol><li><p>若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>≥</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x_1) \ge f(x_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup><mo>∈</mo><mo stretchy="false">[</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">x^* \in [x_1,b]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0391em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mclose">]</span></span></span></span></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404121125038.png" alt="保留右区间"></p></li><li><p>若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>&lt;</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x_1) &lt; f(x_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup><mo>∈</mo><mo stretchy="false">[</mo><mi>a</mi><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">x^* \in [a,x_2]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0391em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span>​</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404121125961.png" alt="保留左区间"></p></li><li><p>若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x_1) = f(x_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup><mo>∈</mo><mo stretchy="false">[</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">x^* \in [x_1,x_2]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0391em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></p></li></ol></li></ul><p>于是迭代的关键就在于如何取点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，下面开始介绍三种取点方法。</p>        </div>      </div>    </div>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-649e6489" role="button" aria-expanded="false" aria-controls="collapse-649e6489">        <div class="fold-arrow">▶</div>计算代码：基于 python 实现 0.618 法、斐波那契法、二分法      </div>      <div class="fold-collapse collapse" id="collapse-649e6489">        <div class="fold-content">          <p>迭代计算代码：</p><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br><span class="line">82</span><br><span class="line">83</span><br><span class="line">84</span><br><span class="line">85</span><br><span class="line">86</span><br><span class="line">87</span><br><span class="line">88</span><br><span class="line">89</span><br><span class="line">90</span><br><span class="line">91</span><br><span class="line">92</span><br><span class="line">93</span><br><span class="line">94</span><br><span class="line">95</span><br><span class="line">96</span><br><span class="line">97</span><br><span class="line">98</span><br><span class="line">99</span><br><span class="line">100</span><br><span class="line">101</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">class</span> <span class="hljs-title class_">ExactLinearSearch</span>:<br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">__init__</span>(<span class="hljs-params">self, </span><br><span class="hljs-params">                 a: <span class="hljs-built_in">float</span>, b: <span class="hljs-built_in">float</span>, delta: <span class="hljs-built_in">float</span>, </span><br><span class="hljs-params">                 f: <span class="hljs-type">Callable</span>[[<span class="hljs-built_in">float</span>], <span class="hljs-built_in">float</span>], </span><br><span class="hljs-params">                 criterion: <span class="hljs-built_in">str</span>=<span class="hljs-string">"0.618"</span>, max_iter=<span class="hljs-number">100</span></span>) -&gt; <span class="hljs-literal">None</span>:<br>        <br>        <span class="hljs-variable language_">self</span>.a = a<br>        <span class="hljs-variable language_">self</span>.b = b<br>        <span class="hljs-variable language_">self</span>.delta = delta<br>        <span class="hljs-variable language_">self</span>.f = f<br>        <span class="hljs-variable language_">self</span>.max_iter = max_iter<br>        <br>        <span class="hljs-keyword">if</span> criterion == <span class="hljs-string">"0.618"</span>:<br>            new_a, new_b, x_star, count = <span class="hljs-variable language_">self</span>._f_goad()<br>        <span class="hljs-keyword">elif</span> criterion == <span class="hljs-string">"fibonacci"</span>:<br>            new_a, new_b, x_star, count = <span class="hljs-variable language_">self</span>._f_fibo()<br>        <span class="hljs-keyword">else</span>: <span class="hljs-comment"># criterion == "binary"</span><br>            new_a, new_b, x_star, count = <span class="hljs-variable language_">self</span>._f_binary()<br><br>        fixed = <span class="hljs-keyword">lambda</span> x, acc: <span class="hljs-built_in">round</span>(x, acc)<br>        <span class="hljs-built_in">print</span>(<span class="hljs-string">f"算法为：“<span class="hljs-subst">{criterion}</span>” 法"</span>)<br>        <span class="hljs-built_in">print</span>(<span class="hljs-string">f"共迭代：<span class="hljs-subst">{count}</span> 次"</span>)<br>        <span class="hljs-built_in">print</span>(<span class="hljs-string">f"左边界: <span class="hljs-subst">{fixed(new_a, <span class="hljs-number">4</span>)}</span>"</span>)<br>        <span class="hljs-built_in">print</span>(<span class="hljs-string">f"右边界: <span class="hljs-subst">{fixed(new_b, <span class="hljs-number">4</span>)}</span>"</span>)<br>        <span class="hljs-built_in">print</span>(<span class="hljs-string">f"最优解: <span class="hljs-subst">{fixed(x_star, <span class="hljs-number">4</span>)}</span>"</span>)<br>        <span class="hljs-built_in">print</span>(<span class="hljs-string">f"最优值: <span class="hljs-subst">{fixed(f(x_star), <span class="hljs-number">6</span>)}</span>\n"</span>)<br>        <br>    <br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">_f_goad</span>(<span class="hljs-params">self</span>) -&gt; <span class="hljs-type">Tuple</span>[<span class="hljs-built_in">float</span>, <span class="hljs-built_in">float</span>, <span class="hljs-built_in">float</span>, <span class="hljs-built_in">int</span>]:<br>        a, b, delta, f = <span class="hljs-variable language_">self</span>.a, <span class="hljs-variable language_">self</span>.b, <span class="hljs-variable language_">self</span>.delta, <span class="hljs-variable language_">self</span>.f<br>        count = <span class="hljs-number">0</span><br><br>        lam = a + <span class="hljs-number">0.382</span> * (b - a)<br>        mu = b - <span class="hljs-number">0.382</span> * (b - a)<br><br>        <span class="hljs-keyword">while</span> count &lt;= <span class="hljs-variable language_">self</span>.max_iter:<br>            phi_lam = f(lam)<br>            phi_mu = f(mu)<br><br>            <span class="hljs-keyword">if</span> phi_lam &lt;= phi_mu:<br>                b = mu<br>            <span class="hljs-keyword">else</span>:<br>                a = lam<br>            <br>            lam = a + <span class="hljs-number">0.382</span> * (b - a)<br>            mu = b - <span class="hljs-number">0.382</span> * (b - a)<br><br>            <span class="hljs-keyword">if</span> b - lam &lt;= delta:<br>                <span class="hljs-keyword">return</span> a, b, lam, count<br>            <span class="hljs-keyword">if</span> mu - a &lt;= delta:<br>                <span class="hljs-keyword">return</span> a, b, mu, count<br>            <br>            count += <span class="hljs-number">1</span><br>        <br>        <span class="hljs-keyword">return</span> a, b, lam <span class="hljs-keyword">if</span> f(lam) &lt;= f(mu) <span class="hljs-keyword">else</span> mu, count<br><br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">_f_fibo</span>(<span class="hljs-params">self</span>) -&gt; <span class="hljs-type">Tuple</span>[<span class="hljs-built_in">float</span>, <span class="hljs-built_in">float</span>, <span class="hljs-built_in">float</span>, <span class="hljs-built_in">int</span>]:<br>        a, b, delta, f, max_iter = <span class="hljs-variable language_">self</span>.a, <span class="hljs-variable language_">self</span>.b, <span class="hljs-variable language_">self</span>.delta, <span class="hljs-variable language_">self</span>.f, <span class="hljs-variable language_">self</span>.max_iter<br>        count = <span class="hljs-literal">None</span><br><br>        F = [<span class="hljs-number">0.0</span>] * max_iter<br>        F[<span class="hljs-number">1</span>] = F[<span class="hljs-number">2</span>] = <span class="hljs-number">1</span><br>        <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">3</span>, max_iter):<br>            F[i] = F[i - <span class="hljs-number">1</span>] + F[i - <span class="hljs-number">2</span>]<br>            <span class="hljs-keyword">if</span> F[i] &gt;= (b - a) / delta:<br>                count = i - <span class="hljs-number">2</span><br>                <span class="hljs-keyword">break</span><br>        <br>        <span class="hljs-keyword">if</span> count == <span class="hljs-literal">None</span>:<br>            ValueError(<span class="hljs-string">"区间过大或精度过高导致，找不到合适的迭代次数"</span>)<br><br>        lam, mu = a, b<br>        <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">3</span>, count + <span class="hljs-number">1</span>):<br><br>            lam = a + (<span class="hljs-number">1</span> - F[i - <span class="hljs-number">1</span>] / F[i]) * (b - a)<br>            mu = b - (<span class="hljs-number">1</span> - F[i - <span class="hljs-number">1</span>] / F[i]) * (b - a)<br><br>            <span class="hljs-keyword">if</span> f(lam) &lt;= f(mu):<br>                b = mu<br>            <span class="hljs-keyword">else</span>:<br>                a = lam<br><br>        <span class="hljs-keyword">return</span> a, b, lam <span class="hljs-keyword">if</span> f(lam) &lt;= f(mu) <span class="hljs-keyword">else</span> mu, count<br>    <br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">_f_binary</span>(<span class="hljs-params">self</span>) -&gt; <span class="hljs-type">Tuple</span>[<span class="hljs-built_in">float</span>, <span class="hljs-built_in">float</span>, <span class="hljs-built_in">float</span>, <span class="hljs-built_in">int</span>]:<br>        a, b, delta, f, max_iter = <span class="hljs-variable language_">self</span>.a, <span class="hljs-variable language_">self</span>.b, <span class="hljs-variable language_">self</span>.delta, <span class="hljs-variable language_">self</span>.f, <span class="hljs-variable language_">self</span>.max_iter<br>        count = <span class="hljs-literal">None</span><br>        <br>        count = np.ceil(np.log2((b - a) / delta)).astype(<span class="hljs-built_in">int</span>)<br><br>        <span class="hljs-keyword">if</span> count &gt; max_iter:<br>            ValueError(<span class="hljs-string">"区间过大或精度过高导致迭代次数过高"</span>)<br>    <br>        <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(count):<br>            c = (a + b) / <span class="hljs-number">2.0</span><br>            <span class="hljs-keyword">if</span> f(c) &gt;= <span class="hljs-number">0.0</span>:<br>                b = c<br>            <span class="hljs-keyword">else</span>:<br>                a = c<br><br>        <span class="hljs-keyword">return</span> a, b, c, count<br></code></pre></td></tr></tbody></table></figure><p>分别调用 0.618 法、斐波那契法、二分法进行迭代计算：</p><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br></pre></td><td class="code"><pre><code class="hljs python">a, b, delta = -<span class="hljs-number">1</span>, <span class="hljs-number">1</span>, <span class="hljs-number">0.01</span><br>f = <span class="hljs-keyword">lambda</span> x: np.exp(-x) + np.exp(x)  <span class="hljs-comment"># 原函数</span><br>calc_goad = ExactLinearSearch(a, b, delta, f, criterion=<span class="hljs-string">"0.618"</span>)<br>calc_fibo = ExactLinearSearch(a, b, delta, f, criterion=<span class="hljs-string">"fibonacci"</span>)<br><br>a, b, delta = -<span class="hljs-number">3</span>, <span class="hljs-number">6</span>, <span class="hljs-number">0.1</span><br>f = <span class="hljs-keyword">lambda</span> x: <span class="hljs-number">2</span> * x + <span class="hljs-number">2</span>               <span class="hljs-comment"># 导函数</span><br>calc_bina = ExactLinearSearch(a, b, delta, f, criterion=<span class="hljs-string">"binary"</span>)<br></code></pre></td></tr></tbody></table></figure><p>计算结果为：</p><figure class="highlight makefile"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br></pre></td><td class="code"><pre><code class="hljs makefile">算法为：“0.618” 法<br>共迭代：10 次<br><span class="hljs-section">左边界: -0.0031</span><br><span class="hljs-section">右边界: 0.0069</span><br><span class="hljs-section">最优解: 0.0007</span><br><span class="hljs-section">最优值: 2.000001</span><br><br>算法为：“fibonacci” 法<br>共迭代：11 次<br><span class="hljs-section">左边界: -0.0225</span><br><span class="hljs-section">右边界: 0.0</span><br><span class="hljs-section">最优解: -0.0139</span><br><span class="hljs-section">最优值: 2.000193</span><br><br>算法为：“binary” 法<br>共迭代：7 次<br><span class="hljs-section">左边界: -1.0312</span><br><span class="hljs-section">右边界: -0.9609</span><br><span class="hljs-section">最优解: -0.9609</span><br><span class="hljs-section">最优值: 0.078125</span><br></code></pre></td></tr></tbody></table></figure>        </div>      </div>    </div><h4 id="3-2-1-0-618-法">3.2.1 0.618 法</h4><p>基于「函数值」进行选点。</p><p>按照黄金分割的比例取点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，不断迭代判断 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x_1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 的值，直到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>b</mi><mi>k</mi></msub><mo>−</mo><msub><mi>λ</mi><mi>k</mi></msub><mo>&lt;</mo><mi>δ</mi></mrow><annotation encoding="application/x-tex">b_k-\lambda_k&lt;\delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03785em;">δ</span></span></span></span> 或 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>μ</mi><mi>k</mi></msub><mo>−</mo><msub><mi>a</mi><mi>k</mi></msub><mo>&lt;</mo><mi>δ</mi></mrow><annotation encoding="application/x-tex">\mu_k-a_k&lt;\delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03785em;">δ</span></span></span></span> 则结束迭代，取最优解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">x^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span> 为对应的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\lambda_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 或 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>μ</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\mu_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 即可。</p><h4 id="3-2-2-Fibonacci-法">3.2.2 Fibonacci 法</h4><p>基于「函数值」进行选点。</p><p>第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 次迭代的区间长度是上一个区间长度的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><msub><mi>F</mi><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msub><msub><mi>F</mi><mrow><mi>n</mi><mo>−</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mfrac></mrow><annotation encoding="application/x-tex">\frac{F_{n-k}}{F_{n-k+1}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4284em;vertical-align:-0.4925em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9359em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:-0.1389em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2107em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4575em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:-0.1389em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2107em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4925em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>​，即：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>b</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>a</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mfrac><msub><mi>F</mi><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msub><msub><mi>F</mi><mrow><mi>n</mi><mo>−</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mfrac><mo stretchy="false">(</mo><msub><mi>b</mi><mi>k</mi></msub><mo>−</mo><msub><mi>a</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">b_{k+1} - a_{k+1} = \frac{F_{n-k}}{F_{n-k+1}} (b_{k} - a_{k})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9028em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2547em;vertical-align:-0.8943em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8943em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p><p>经过 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 次迭代后，得到区间 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>a</mi><mi>n</mi></msub><mo separator="true">,</mo><msub><mi>b</mi><mi>n</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[a_n,b_n]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span>，且 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>b</mi><mi>n</mi></msub><mo>−</mo><msub><mi>a</mi><mi>n</mi></msub><mo>≤</mo><mi>δ</mi></mrow><annotation encoding="application/x-tex">b_n-a_n \le \delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03785em;">δ</span></span></span></span>，于是可得：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>b</mi><mi>n</mi></msub><mo>−</mo><msub><mi>a</mi><mi>n</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><msub><mi>F</mi><mn>1</mn></msub><msub><mi>F</mi><mn>2</mn></msub></mfrac><mo stretchy="false">(</mo><msub><mi>b</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><msub><mi>F</mi><mn>1</mn></msub><msub><mi>F</mi><mn>2</mn></msub></mfrac><mfrac><msub><mi>F</mi><mn>2</mn></msub><msub><mi>F</mi><mn>3</mn></msub></mfrac><mo stretchy="false">(</mo><msub><mi>b</mi><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>−</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>⋯</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><msub><mi>F</mi><mn>1</mn></msub><msub><mi>F</mi><mn>2</mn></msub></mfrac><mfrac><msub><mi>F</mi><mn>2</mn></msub><msub><mi>F</mi><mn>3</mn></msub></mfrac><mo>⋯</mo><mfrac><msub><mi>F</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><msub><mi>F</mi><mi>n</mi></msub></mfrac><mo stretchy="false">(</mo><msub><mi>b</mi><mn>1</mn></msub><mo>−</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><msub><mi>F</mi><mn>1</mn></msub><msub><mi>F</mi><mi>n</mi></msub></mfrac><mo stretchy="false">(</mo><msub><mi>b</mi><mn>1</mn></msub><mo>−</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>≤</mo><mi>δ</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}b_n-a_n &amp;= \frac{F_1}{F_2}(b_{n-1}-a_{n-1}) \\&amp;= \frac{F_1}{F_2} \frac{F_2}{F_3}(b_{n-2}-a_{n-2}) \\&amp;= \cdots \\&amp;= \frac{F_1}{F_2} \frac{F_2}{F_3} \cdots \frac{F_{n-1}}{F_n}(b_{1}-a_{1}) \\&amp;= \frac{F_1}{F_n}(b_1-a_1) \le \delta\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:11.4853em;vertical-align:-5.4927em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.9927em;"><span style="top:-7.9927em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-5.4963em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"></span></span><span style="top:-3.5203em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"></span></span><span style="top:-1.5em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"></span></span><span style="top:0.9963em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.4927em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.9927em;"><span style="top:-7.9927em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.836em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-5.4963em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.836em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.836em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-3.5203em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="minner">⋯</span></span></span><span style="top:-1.5em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.836em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.836em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.836em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:0.9963em;"><span class="pstrut" style="height:3.3603em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.836em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03785em;">δ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.4927em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>在已知上界 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>δ</mi></mrow><annotation encoding="application/x-tex">\delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03785em;">δ</span></span></span></span> 的情况下可以直接计算出 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 的值，于是迭代 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 次即可得到最终的步长值 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\lambda_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 或 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>μ</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\mu_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p><h4 id="3-2-3-二分法">3.2.3 二分法</h4><p>基于「一阶导数」进行选点。</p><p>单调性存在于导数上。极值点左边导数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">&lt;0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，极值点右边导数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">&gt;0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，于是可以进行二分搜索。</p><h4 id="3-2-4-插值法">3.2.4 插值法</h4><p>基于「函数值、一阶导数」进行选点。</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404170915756.png" alt="三点二次插值法"></p><p><strong>三点二次插值法</strong>。给定初始迭代区间 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>a</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>b</mi><mn>0</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[a_0,b_0]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span> 和初始迭代解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>t</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">t_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7651em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>。每次在目标函数上取三个点来拟合一个二次函数，通过拟合出来的二次函数的最小值，来更新三个点为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>b</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>t</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">a_1,b_1,t_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，直到区间小于上界长度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>δ</mi></mrow><annotation encoding="application/x-tex">\delta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03785em;">δ</span></span></span></span>，终止迭代，极小值就是当前状态二次函数的最小值。</p><p>那么我们如何求解当前状态二次函数的最小值呢？假定三个已知点为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>&lt;</mo><msub><mi>x</mi><mn>2</mn></msub><mo>&lt;</mo><msub><mi>x</mi><mn>3</mn></msub></mrow><annotation encoding="application/x-tex">x_1&lt;x_2&lt;x_3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，且 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>&gt;</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>&lt;</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>3</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x_1)&gt;f(x_2)&lt;f(x_3)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>，我们需要知道二次函数的三个参数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo separator="true">,</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">a,b,c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">c</span></span></span></span>，刚好三个已知点可以得到三个方程，从而解出二次函数的三个未知数。</p><p><strong>两点二次插值法</strong>。只不过少取了一个点，多利用了一个点的导数值来确定二次函数的三个参数罢了，其余迭代过程与三点二次插值完全一致。</p><h3 id="3-3-不精确线性搜索算法">3.3 不精确线性搜索算法</h3><div class="note note-light">            <p>精确搜索有时会导致搜索时间过长，尤其是当最优解离当前点还很远时。接下来我们介绍不精确线性搜索方法，在确保每一步的函数值都有充分下降的必要条件下，确保收敛并提升计算效率。真的有这么厉害的算法吗？其实根本逻辑很简单，<strong>Armijo 准则</strong>限定了搜索方向的上界、<strong>Goldstein 准则</strong>在前者的基础上又限定了搜索方向的下界（防止过小导致收敛过慢）、<strong>Wolfe 准则</strong>在前者的基础上又完善了下界的约束（确保不会把可行解排除在搜索区间外）</p>          </div><p>符号定义：我们在已知上一步解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">x_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 和下次迭代的下降方向 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">d_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的基础上，需要寻找合适的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span>。于是唯一变量就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span>，我们直接定义关于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 的一元函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><mi>α</mi><msub><mi>d</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(\alpha) = f(x_k + \alpha d_k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">ϕ</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>g</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">g_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 表示梯度。</p><h4 id="3-3-1-Armijo-准则">3.3.1 Armijo 准则</h4><p>只限定搜索上界，给定初始点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>、系数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span></span></span></span>：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><mi>α</mi><msub><mi>d</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mo>≤</mo><mi>ρ</mi><mo>⋅</mo><msubsup><mi>g</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>d</mi><mi>k</mi></msub><mo>⋅</mo><mi>α</mi><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x_k + \alpha d_k) \le \rho \cdot g_k^Td_k \cdot \alpha + f(x_k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1383em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p><h4 id="3-3-2-Goldstein-准则">3.3.2 Goldstein 准则</h4><p>又限定了搜索下界，同样给定初始点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>、系数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span></span></span></span>：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><mi>α</mi><msub><mi>d</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mo>≤</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>ρ</mi><mo>⋅</mo><msubsup><mi>g</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>d</mi><mi>k</mi></msub><mo>⋅</mo><mi>α</mi><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><mi>α</mi><msub><mi>d</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mo>≥</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>⋅</mo><msubsup><mi>g</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>d</mi><mi>k</mi></msub><mo>⋅</mo><mi>α</mi><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}f(x_k + \alpha d_k) \le &amp;\rho \cdot g_k^Td_k \cdot \alpha + f(x_k)\\f(x_k + \alpha d_k) \ge &amp;(1-\rho) \cdot g_k^Td_k \cdot \alpha + f(x_k)\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.1027em;vertical-align:-1.3013em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8013em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span></span></span><span style="top:-2.3587em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3013em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8013em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal">ρ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-2.3587em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">ρ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3013em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><h4 id="3-3-3-Wolfe-准则">3.3.3 Wolfe 准则</h4><p>完善了搜索下界的约束，即保证新点的梯度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>g</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mi>T</mi></msubsup><msub><mi>d</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">g_{k+1}^Td_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1828em;vertical-align:-0.3414em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3414em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 不低于老点梯度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>g</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>d</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">g_{k}^Td_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1244em;vertical-align:-0.2831em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> 倍。给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub><mo separator="true">,</mo><mi>ρ</mi><mo separator="true">,</mo><mi>σ</mi></mrow><annotation encoding="application/x-tex">x_0,\rho,\sigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">ρ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span>：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><mi>α</mi><msub><mi>d</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mo>≤</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>ρ</mi><mo>⋅</mo><msubsup><mi>g</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>d</mi><mi>k</mi></msub><mo>⋅</mo><mi>α</mi><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>g</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><msub><mi>α</mi><mi>k</mi></msub><msub><mi>d</mi><mi>k</mi></msub><msup><mo stretchy="false">)</mo><mi>T</mi></msup><msub><mi>d</mi><mi>k</mi></msub><mo>≥</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>σ</mi><msubsup><mi>g</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>d</mi><mi>k</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mn>0</mn><mo>&lt;</mo><mi>ρ</mi><mo>&lt;</mo><mi>σ</mi><mo>&lt;</mo><mn>1</mn></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}f(x_k + \alpha d_k) \le &amp;\rho \cdot g_k^Td_k \cdot \alpha + f(x_k)\\g(x_k + \alpha_kd_k)^Td_k \ge&amp; \sigma g_{k}^Td_k\\0 &lt; \rho &lt; \sigma &lt; 1\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.6027em;vertical-align:-2.0513em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5513em;"><span style="top:-4.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span></span></span><span style="top:-3.1087em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span></span></span><span style="top:-1.6087em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">ρ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.0513em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5513em;"><span style="top:-4.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal">ρ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-3.1087em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.5513em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><blockquote><p>参考：</p><ul><li><a href="https://www.bilibili.com/video/BV11N411m75b/">路人 - 视频 - 更通俗的解释</a></li><li><a href="https://blog.csdn.net/luzhanbo207/article/details/121559905">路人 - 博客 - 更通俗的解释</a></li></ul></blockquote><h2 id="第四章「无约束最优化」">第四章「无约束最优化」</h2><div class="note note-light">            <p>本章介绍无约束函数的最优化算法。其中：</p><ul><li>最速下降法基于「一阶梯度」。最基本的方法</li><li>牛顿法基于「二阶梯度」。最主要的方法</li><li>共轭梯度法基于「一阶梯度」。解大型最优化问题的首选</li><li>拟牛顿法基于「函数值和一阶梯度」。尤其是其中的 BFGS 是目前最成功的方法</li></ul>          </div>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-6f5deacf" role="button" aria-expanded="false" aria-controls="collapse-6f5deacf">        <div class="fold-arrow">▶</div>极小化 Rosenbrock 函数的 python 代码      </div>      <div class="fold-collapse collapse" id="collapse-6f5deacf">        <div class="fold-content">          <p>目标函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span>、一阶梯度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span></span></span></span>、二阶梯度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">G</span></span></span></span>、初始点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">def</span> <span class="hljs-title function_">f</span>(<span class="hljs-params">x: np.ndarray</span>) -&gt; <span class="hljs-built_in">float</span>:<br>    <span class="hljs-keyword">return</span> (<span class="hljs-number">1</span> - x[<span class="hljs-number">0</span>])**<span class="hljs-number">2</span> + <span class="hljs-number">100</span> * (x[<span class="hljs-number">1</span>] - x[<span class="hljs-number">0</span>]**<span class="hljs-number">2</span>)**<span class="hljs-number">2</span><br><br><br><span class="hljs-keyword">def</span> <span class="hljs-title function_">g</span>(<span class="hljs-params">x: np.ndarray</span>) -&gt; np.ndarray:<br>    grad_x = -<span class="hljs-number">2</span> * (<span class="hljs-number">1</span> - x[<span class="hljs-number">0</span>]) - <span class="hljs-number">400</span> * x[<span class="hljs-number">0</span>] * (x[<span class="hljs-number">1</span>] - x[<span class="hljs-number">0</span>]**<span class="hljs-number">2</span>)<br>    grad_y = <span class="hljs-number">200</span> * (x[<span class="hljs-number">1</span>] - x[<span class="hljs-number">0</span>]**<span class="hljs-number">2</span>)<br>    <span class="hljs-keyword">return</span> np.array([grad_x, grad_y])<br><br><br><span class="hljs-keyword">def</span> <span class="hljs-title function_">G</span>(<span class="hljs-params">x: np.ndarray</span>) -&gt; np.ndarray:<br>    grad_xx = <span class="hljs-number">2</span> - <span class="hljs-number">400</span> * x[<span class="hljs-number">1</span>] + <span class="hljs-number">1200</span> * x[<span class="hljs-number">0</span>]**<span class="hljs-number">2</span><br>    grad_xy = -<span class="hljs-number">400</span> * x[<span class="hljs-number">0</span>]<br>    grad_yx = -<span class="hljs-number">400</span> * x[<span class="hljs-number">0</span>]<br>    grad_yy = <span class="hljs-number">200</span><br>    <span class="hljs-keyword">return</span> np.array([<br>        [grad_xx, grad_xy],<br>        [grad_yx, grad_yy]<br>    ])<br><br><br>initial_point = [-<span class="hljs-number">1.2</span>, <span class="hljs-number">1</span>]<br></code></pre></td></tr></tbody></table></figure><p>已知最优点为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">x^*=(1, 1)^T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0913em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span></span></span>，最优解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msup><mi>x</mi><mo>∗</mo></msup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f(x^*)=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，以书中例题初始点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1.2</mn><mo separator="true">,</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">(-1.2,1)^T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0913em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1.2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span></span></span> 为例开始迭代。</p><p><strong>一、最速下降法</strong></p><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">def</span> <span class="hljs-title function_">gradient_descent</span>(<span class="hljs-params">initial_point, max_iter=<span class="hljs-number">5</span>, eps=<span class="hljs-number">1e-6</span></span>):<br>    x = np.array(initial_point)<br>    points = [x]<br>    gradients = [g(x)]<br>    alphas = []<br><br>    <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(max_iter):<br>        grad = gradients[-<span class="hljs-number">1</span>]<br><br>        <span class="hljs-comment"># 搜索方向</span><br>        direction = -grad<br><br>        <span class="hljs-comment"># 步长因子：无法确定准确的步长最小化函数，因此此处采用二分法搜索最优步长</span><br>        alpha = <span class="hljs-number">1</span><br>        <span class="hljs-keyword">while</span> f(x + alpha * direction) &gt; f(x):<br>            alpha /= <span class="hljs-number">2</span><br><br>        x = x + alpha * direction<br>        points.append(x)<br>        gradients.append(g(x))<br>        alphas.append(alpha)<br><br>        <span class="hljs-keyword">if</span> np.linalg.norm(grad) &lt; eps:<br>            <span class="hljs-keyword">break</span><br><br>    <span class="hljs-keyword">return</span> points, gradients, alphas<br><br><br>points, gradients, alphas = gradient_descent(initial_point, max_iter=<span class="hljs-number">100</span>, eps=<span class="hljs-number">1e-6</span>)<br><br><span class="hljs-keyword">for</span> i, (point, grad, alpha) <span class="hljs-keyword">in</span> <span class="hljs-built_in">enumerate</span>(<span class="hljs-built_in">zip</span>(points, gradients, [<span class="hljs-number">1</span>] + alphas)):<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"Iteration <span class="hljs-subst">{i}</span>:"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Point       = <span class="hljs-subst">{point}</span>"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Gradient    = <span class="hljs-subst">{grad}</span>"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Step Size   = <span class="hljs-subst">{alpha}</span>"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Direction   = <span class="hljs-subst">{-grad}</span>"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Function Val= <span class="hljs-subst">{f(point)}</span>\n"</span>)<br></code></pre></td></tr></tbody></table></figure><p>迭代100次后输出为：</p><figure class="highlight makefile"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br></pre></td><td class="code"><pre><code class="hljs makefile">Iteration 98:<br>  Point       = [0.93432802 0.87236513]<br>  Gradient    = [ 0.0942865  -0.12074477]<br>  Step Size   = 0.00390625<br>  Direction   = [-0.0942865   0.12074477]<br>  Function Val= 0.004349256784446673<br><br>Iteration 99:<br>  Point       = [0.93414387 0.87260096]<br>  Gradient    = [-0.12281587 -0.00476179]<br>  Step Size   = 0.001953125<br>  Direction   = [0.12281587 0.00476179]<br>  Function Val= 0.004337086557103718<br><br>Iteration 100:<br>  Point       = [0.93438374 0.87261026]<br>  Gradient    = [ 0.04171114 -0.09254423]<br>  Step Size   = 0.001953125<br>  Direction   = [-0.04171114  0.09254423]<br>  Function Val= 0.004326904052586884<br></code></pre></td></tr></tbody></table></figure><p><strong>二、牛顿法</strong></p><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">def</span> <span class="hljs-title function_">newton_method</span>(<span class="hljs-params">initial_point, max_iter=<span class="hljs-number">5</span>, eps=<span class="hljs-number">1e-6</span></span>):<br>    x = np.array(initial_point)<br>    points = [x]<br>    gradients = [g(x)]<br>    Hessians = [G(x)]<br><br>    <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(max_iter):<br>        grad = gradients[-<span class="hljs-number">1</span>]<br>        Hessian = Hessians[-<span class="hljs-number">1</span>]<br><br>        <span class="hljs-comment"># 搜索方向</span><br>        direction = np.linalg.inv(Hessian) @ grad<br><br>        <span class="hljs-comment"># 步长因子：假定使用固定步长的牛顿法</span><br>        alpha = <span class="hljs-number">1</span><br><br>        x = x - alpha * direction<br>        points.append(x)<br>        gradients.append(g(x))<br>        Hessians.append(G(x))<br><br>        <span class="hljs-keyword">if</span> np.linalg.norm(grad) &lt; eps:<br>            <span class="hljs-keyword">break</span><br><br>    <span class="hljs-keyword">return</span> points, gradients, Hessians<br><br><br>points, gradients, Hessians = newton_method(initial_point, max_iter=<span class="hljs-number">50</span>, eps=<span class="hljs-number">1e-6</span>)<br><br><span class="hljs-keyword">for</span> i, (point, grad, Hessian) <span class="hljs-keyword">in</span> <span class="hljs-built_in">enumerate</span>(<span class="hljs-built_in">zip</span>(points, gradients, Hessians)):<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"Iteration <span class="hljs-subst">{i}</span>:"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Point       = <span class="hljs-subst">{point}</span>"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Gradient    = <span class="hljs-subst">{grad}</span>"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Hessian     = <span class="hljs-subst">{Hessian}</span>"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Function Val= <span class="hljs-subst">{f(point)}</span>\n"</span>)<br></code></pre></td></tr></tbody></table></figure><p>迭代7次即收敛：</p><figure class="highlight makefile"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br></pre></td><td class="code"><pre><code class="hljs makefile">Iteration 5:<br>  Point       = [0.9999957  0.99999139]<br>  Gradient    = [-8.60863343e-06 -2.95985458e-11]<br>  Hessian     = [[ 801.99311306 -399.99827826]<br> [-399.99827826  200.        ]]<br>  Function Val= 1.8527397192178054e-11<br><br>Iteration 6:<br>  Point       = [1. 1.]<br>  Gradient    = [ 7.41096051e-09 -3.70548037e-09]<br>  Hessian     = [[ 802.00000001 -400.        ]<br> [-400.          200.        ]]<br>  Function Val= 3.4326461875363225e-20<br><br>Iteration 7:<br>  Point       = [1. 1.]<br>  Gradient    = [-0.  0.]<br>  Hessian     = [[ 802. -400.]<br> [-400.  200.]]<br>  Function Val= 0.0<br></code></pre></td></tr></tbody></table></figure><p><strong>三、共轭梯度法</strong></p><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">def</span> <span class="hljs-title function_">conjugate_gradient</span>(<span class="hljs-params">initial_point, max_iter=<span class="hljs-number">5</span>, eps=<span class="hljs-number">1e-6</span></span>):<br>    x = np.array(initial_point)<br>    points = [x]<br>    gradients = [g(x)]<br>    directions = [-g(x)]<br>    alphas = []<br><br>    <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(max_iter):<br>        grad = gradients[-<span class="hljs-number">1</span>]<br><br>        <span class="hljs-comment"># 搜索方向：FR公式</span><br>        <span class="hljs-keyword">if</span> i == <span class="hljs-number">0</span>:<br>            direction = -grad<br>        <span class="hljs-keyword">else</span>:<br>            beta = np.dot(g(x), g(x)) / np.dot(g(points[-<span class="hljs-number">2</span>]), g(points[-<span class="hljs-number">2</span>]))<br>            direction = -g(x) + beta * direction<br><br>        <span class="hljs-comment"># 步长因子：精确线搜索直接得到闭式解</span><br>        alpha = -np.dot(grad, direction) / np.dot(direction, G(x) @ direction)<br>        <br><br>        x = x + alpha * direction<br><br>        directions.append(direction)<br>        alphas.append(alpha)<br>        points.append(x)<br>        gradients.append(g(x))<br><br>        <span class="hljs-keyword">if</span> np.linalg.norm(grad) &lt; eps:<br>            <span class="hljs-keyword">break</span><br><br>    <span class="hljs-keyword">return</span> points, gradients, alphas<br><br><br>points, gradients, alphas = conjugate_gradient(initial_point, max_iter=<span class="hljs-number">1000</span>, eps=<span class="hljs-number">1e-6</span>)<br><br><span class="hljs-keyword">for</span> i, (point, grad, alpha) <span class="hljs-keyword">in</span> <span class="hljs-built_in">enumerate</span>(<span class="hljs-built_in">zip</span>(points, gradients, alphas)):<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"Iteration <span class="hljs-subst">{i}</span>:"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Point       = <span class="hljs-subst">{point}</span>"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Gradient    = <span class="hljs-subst">{grad}</span>"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Step Size   = <span class="hljs-subst">{alpha}</span>"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Direction   = <span class="hljs-subst">{-grad}</span>"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Function Val= <span class="hljs-subst">{f(point)}</span>\n"</span>)<br></code></pre></td></tr></tbody></table></figure><p>迭代1000次后输出为：</p><figure class="highlight makefile"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br></pre></td><td class="code"><pre><code class="hljs makefile">Iteration 997:<br>  Point       = [0.9999994  0.99999875]<br>  Gradient    = [ 1.90794906e-05 -1.01414007e-05]<br>  Step Size   = 0.0005161468619784313<br>  Direction   = [-1.90794906e-05  1.01414007e-05]<br>  Function Val= 6.191018745155016e-13<br><br>Iteration 998:<br>  Point       = [0.99999931 0.99999861]<br>  Gradient    = [ 5.43686111e-06 -3.40374227e-06]<br>  Step Size   = 0.0005078748917694624<br>  Direction   = [-5.43686111e-06  3.40374227e-06]<br>  Function Val= 4.986125950068217e-13<br><br>Iteration 999:<br>  Point       = [0.9999993 0.9999986]<br>  Gradient    = [ 1.34784246e-06 -1.36924747e-06]<br>  Step Size   = 0.0005356250138048412<br>  Direction   = [-1.34784246e-06  1.36924747e-06]<br>  Function Val= 4.881643528976312e-13<br></code></pre></td></tr></tbody></table></figure><p><strong>四、拟牛顿法</strong></p><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">def</span> <span class="hljs-title function_">bfgs</span>(<span class="hljs-params">initial_point, max_iter=<span class="hljs-number">5</span>, eps=<span class="hljs-number">1e-6</span></span>):<br>    x = np.array(initial_point)<br>    points = [x]<br>    gradients = [g(x)]<br>    B = G(x)<br><br>    <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(max_iter):<br>        grad = gradients[-<span class="hljs-number">1</span>]<br><br>        <span class="hljs-comment"># 迭代公式</span><br>        x = x - np.linalg.inv(B) @ grad<br><br>        <span class="hljs-comment"># 更新 B 矩阵</span><br>        s = x - points[-<span class="hljs-number">1</span>]<br>        y = g(x) - gradients[-<span class="hljs-number">1</span>]<br>        B = B + np.outer(y, y) / (s @ y) - (B @ np.outer(s, s) @ B) / (s @ B @ s)<br><br>        points.append(x)<br>        gradients.append(g(x))<br><br>        <span class="hljs-keyword">if</span> np.linalg.norm(grad) &lt; eps:<br>            <span class="hljs-keyword">break</span><br><br>    <span class="hljs-keyword">return</span> points, gradients<br><br><br>points, gradients = bfgs(initial_point, max_iter=<span class="hljs-number">1000</span>, eps=<span class="hljs-number">1e-6</span>)<br><br><span class="hljs-keyword">for</span> i, (point, grad) <span class="hljs-keyword">in</span> <span class="hljs-built_in">enumerate</span>(<span class="hljs-built_in">zip</span>(points, gradients)):<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"Iteration <span class="hljs-subst">{i}</span>:"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Point       = <span class="hljs-subst">{point}</span>"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Gradient    = <span class="hljs-subst">{grad}</span>"</span>)<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">f"  Function Val= <span class="hljs-subst">{f(point)}</span>\n"</span>)<br></code></pre></td></tr></tbody></table></figure><p>迭代 78 次收敛：</p><figure class="highlight makefile"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br></pre></td><td class="code"><pre><code class="hljs makefile">Iteration 76:<br>  Point       = [1.00000075 0.99999921]<br>  Gradient    = [ 0.000918   -0.00045825]<br>  Function Val= 5.255482679080297e-10<br><br>Iteration 77:<br>  Point       = [1.00000006 1.00000012]<br>  Gradient    = [ 6.46055099e-07 -2.63592925e-07]<br>  Function Val= 3.7061757712619374e-15<br><br>Iteration 78:<br>  Point       = [1. 1.]<br>  Gradient    = [6.75185684e-10 4.68913797e-10]<br>  Function Val= 6.510026595267928e-19<br></code></pre></td></tr></tbody></table></figure>        </div>      </div>    </div><h3 id="4-1-最速下降法">4.1 最速下降法</h3><p>放一张生动的图：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406132016327.png" alt="最速下降法 - 迭代示意图"></p><p>迭代公式：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>x</mi><mi>k</mi></msub><mo>−</mo><msub><mi>α</mi><mi>k</mi></msub><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x_{k+1} = x_k - \alpha_k \nabla f(x_k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p><p>每次迭代时，<strong>下降方向</strong> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">d_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 采用当前解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">x_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 处的负梯度方向 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">- \nabla f(x_k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>，<strong>步长因子</strong> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>α</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 采用精确线性搜索算法的计算结果。</p><p>易证相邻迭代解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>k</mi></msub><mo separator="true">,</mo><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">x_k,x_{k+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span> 的方向 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mi>k</mi></msub><mo separator="true">,</mo><msub><mi>d</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">d_k,d_{k+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9028em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span> 是正交的：由于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><mi>α</mi><msub><mi>d</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(\alpha) = f(x_k + \alpha d_k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">ϕ</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>，在采用线搜索找最优步长时，步长的搜索结果 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>α</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 即为使得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>ϕ</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\phi'(\alpha)=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">ϕ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 的解，于是可得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>=</mo><msup><mi>ϕ</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>ϕ</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><msub><mi>α</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><msub><mi>α</mi><mi>k</mi></msub><msub><mi>d</mi><mi>k</mi></msub><mo stretchy="false">)</mo><msub><mi>d</mi><mi>k</mi></msub><mo>=</mo><mo>−</mo><msubsup><mi>d</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mi>T</mi></msubsup><mo>⋅</mo><msub><mi>d</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">0=\phi'(\alpha) = \phi'(\alpha_k) = \nabla f(x_k+\alpha_k d_k)d_k = -d_{k+1}^T \cdot d_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">ϕ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">ϕ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1828em;vertical-align:-0.3414em;"></span><span class="mord">−</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3414em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>d</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mi>T</mi></msubsup><mo>⋅</mo><msub><mi>d</mi><mi>k</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d_{k+1}^T \cdot d_k = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1828em;vertical-align:-0.3414em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3414em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>。如图：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406132046068.png" alt="相邻迭代解的方向正交"></p><p>也正因为搜索方向正交的特性导致最速下降法的收敛速度往往不尽如人意。优点在于程序设计简单并且计算和存储量都不大，以及对初始点的要求不高。</p><h3 id="4-2-牛顿法">4.2 牛顿法</h3><p>放一张生动的图：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406140902324.gif" alt="牛顿法 - 迭代示意图"></p><div class="note note-warning">            <p>注：本图演示的是求解一维函数零点的迭代过程，需要使得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f(x)=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，因此比值式为原函数除以导函数。后续介绍的是极小化函数的过程，需要使得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f'(x)=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，因此比值式为一阶导数除以二阶导数，高维就是二阶梯度的逆乘以一阶梯度。</p>          </div><p>迭代公式：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>x</mi><mi>k</mi></msub><mo>−</mo><msup><mi mathvariant="normal">∇</mi><mn>2</mn></msup><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><msup><mo stretchy="false">)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x_{k+1} = x_k - \nabla^2f(x_k)^{-1}\nabla f(x_k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord">∇</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p><p>牛顿法相较于最速下降法有了更快的收敛速度，但是由于需要计算和存储海塞矩阵导致计算量增加并且有些目标函数可能根本求不出二阶梯度。同时牛顿法对于初始迭代点的选择比最速下降法要苛刻的多。</p><h3 id="4-3-共轭梯度法">4.3 共轭梯度法</h3><p>我们利用共轭梯度法解决「正定二次函数」的极小化问题。由于最速下降法中相邻迭代点的方向是正交的导致搜索效率下降，牛顿法又由于需要计算和存储海塞矩阵导致存储开销过大，共轭梯度法的核心思想是<strong>相邻两个迭代点的搜索方向是关于正定二次型的正定阵正交的</strong>。这样既保证了迭代收敛的速度也避免了计算存储海塞矩阵的开销。美中不足的是当共轭梯度法解决其他问题是往往会出现对线搜索的过度依赖，一旦线搜索变差会导致整个迭代过程精度变差。</p><p>概念补充：</p><ol><li>共轭：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 与 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> 共轭当且仅当 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mi>T</mi></msup><mi>G</mi><mi>y</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x^TGy=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0358em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，其中 G 为对称正定阵</li><li>正定二次：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mi>T</mi></msup><mi>G</mi><mi>x</mi><mo>−</mo><msup><mi>b</mi><mi>T</mi></msup><mi>x</mi><mo>+</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">f(x) =\frac{1}{2} x^TGx - b^Tx + c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.9247em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">c</span></span></span></span></li></ol><p>公式推导：</p><ol><li><p>首先给定初始迭代点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，收敛阈值 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ϵ</span></span></span></span>，迭代公式是不变的：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><msub><mi>α</mi><mi>k</mi></msub><msub><mi>d</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">x_{k+1} = x_k + \alpha_k d_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，关键在于计算每一次迭代过程中的步长因子 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>α</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>​ 和搜索方向 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">d_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>​</p></li><li><p>确定步长因子 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>α</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mi>α</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mrow><mi>min</mi><mo>⁡</mo></mrow><mi>α</mi></munder><mi>ϕ</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mrow><mi>min</mi><mo>⁡</mo></mrow><mi>α</mi></munder><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><mi>α</mi><msub><mi>d</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mrow><mi>min</mi><mo>⁡</mo></mrow><mi>α</mi></munder><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><mi>α</mi><msub><mi>d</mi><mi>k</mi></msub><msup><mo stretchy="false">)</mo><mi>T</mi></msup><mi>G</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><mi>α</mi><msub><mi>d</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mo>−</mo><msup><mi>b</mi><mi>T</mi></msup><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><mi>α</mi><msub><mi>d</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mo>+</mo><mi>c</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mrow><mi>min</mi><mo>⁡</mo></mrow><mi>α</mi></munder><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><msubsup><mi>x</mi><mi>k</mi><mi>T</mi></msubsup><mi>G</mi><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><mn>2</mn><mi>α</mi><msubsup><mi>x</mi><mi>k</mi><mi>T</mi></msubsup><mi>G</mi><msub><mi>d</mi><mi>k</mi></msub><mo>+</mo><msup><mi>α</mi><mn>2</mn></msup><msubsup><mi>d</mi><mi>K</mi><mi>T</mi></msubsup><msub><mi>d</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mo>−</mo><msup><mi>b</mi><mi>T</mi></msup><msub><mi>x</mi><mi>k</mi></msub><mo>−</mo><msup><mi>b</mi><mi>T</mi></msup><mi>α</mi><msub><mi>d</mi><mi>k</mi></msub><mo>+</mo><mi>c</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mrow><mi>min</mi><mo>⁡</mo></mrow><mi>α</mi></munder><mfrac><mn>1</mn><mn>2</mn></mfrac><msubsup><mi>x</mi><mi>k</mi><mi>T</mi></msubsup><mi>G</mi><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><mi>α</mi><msubsup><mi>x</mi><mi>k</mi><mi>T</mi></msubsup><mi>G</mi><msub><mi>d</mi><mi>k</mi></msub><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>α</mi><mn>2</mn></msup><msubsup><mi>d</mi><mi>k</mi><mi>T</mi></msubsup><mi>G</mi><msub><mi>d</mi><mi>k</mi></msub><mo>−</mo><msup><mi>b</mi><mi>T</mi></msup><mi>α</mi><msub><mi>d</mi><mi>k</mi></msub><mo>+</mo><mi>c</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\alpha &amp;= \min_{\alpha} \phi(\alpha) \\&amp;= \min_{\alpha} f(x_k+\alpha d_k) \\&amp;= \min_{\alpha} \frac{1}{2}(x_k+\alpha d_k)^TG(x_k+\alpha d_k) - b^T(x_k+\alpha d_k) + c \\&amp;= \min_{\alpha} \frac{1}{2} (x_k^TGx_k + 2\alpha x_k^T G d_k + \alpha^2d_K^Td_k) - b^Tx_k - b^T \alpha d_k + c\\&amp;= \min_{\alpha} \frac{1}{2} x_k^TGx_k + \alpha x_k^T G d_k + \frac{1}{2}\alpha^2d_k^T G d_k - b^T \alpha d_k + c\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:10.6443em;vertical-align:-5.0722em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.5722em;"><span style="top:-8.0536em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span><span style="top:-6.2136em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"></span></span><span style="top:-3.8922em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"></span></span><span style="top:-1.5707em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"></span></span><span style="top:0.7507em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.0722em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.5722em;"><span style="top:-8.0536em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">min</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">ϕ</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mclose">)</span></span></span><span style="top:-6.2136em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">min</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-3.8922em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">min</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">c</span></span></span><span style="top:-1.5707em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">min</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">K</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">c</span></span></span><span style="top:0.7507em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">min</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.0722em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>由于目标函数是正定二次型，显然可以直接求出步长因子的闭式解：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi>d</mi><mi>ϕ</mi><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><mi>α</mi></mrow></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msubsup><mi>x</mi><mi>k</mi><mi>T</mi></msubsup><mi>G</mi><msub><mi>d</mi><mi>k</mi></msub><mo>+</mo><mi>α</mi><msubsup><mi>d</mi><mi>k</mi><mi>T</mi></msubsup><mi>G</mi><msub><mi>d</mi><mi>k</mi></msub><mo>−</mo><msup><mi>b</mi><mi>T</mi></msup><msub><mi>d</mi><mi>k</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>0</mn></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\frac{d \phi(\alpha)}{d\alpha} &amp;= x_k^T G d_k + \alpha d_k^T G d_k - b^Td_k \\&amp;=0\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.913em;vertical-align:-1.7065em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.2065em;"><span style="top:-4.2065em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">ϕ</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.3805em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.7065em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.2065em;"><span style="top:-4.2065em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.3805em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.7065em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>于是可以导出当前的步长因子 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>α</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的闭式解为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><msub><mi>α</mi><mi>k</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><msup><mi>b</mi><mi>T</mi></msup><mo>−</mo><msup><mi>x</mi><mi>T</mi></msup><mi>G</mi><mo stretchy="false">)</mo><msub><mi>d</mi><mi>k</mi></msub></mrow><mrow><msubsup><mi>d</mi><mi>k</mi><mi>T</mi></msubsup><mi>G</mi><msub><mi>d</mi><mi>k</mi></msub></mrow></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>−</mo><mfrac><mrow><msubsup><mi>g</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>d</mi><mi>k</mi></msub></mrow><mrow><msubsup><mi>d</mi><mi>k</mi><mi>T</mi></msubsup><mi>G</mi><msub><mi>d</mi><mi>k</mi></msub></mrow></mfrac></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\alpha_k &amp;= \frac{(b^T - x^TG)d_k}{d_k^TGd_k} \\&amp;= -\frac{g_k^T d_k}{d_k^TGd_k}\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.6655em;vertical-align:-2.5828em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.0828em;"><span style="top:-5.0828em;"><span class="pstrut" style="height:3.5183em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.5183em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.5828em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.0828em;"><span style="top:-5.0828em;"><span class="pstrut" style="height:3.5183em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5183em;"><span style="top:-2.2869em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8231em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0144em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.5183em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5183em;"><span style="top:-2.2869em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8231em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0144em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.5828em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li><li><p>确定搜索方向 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">d_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>d</mi><mi>k</mi></msub><mo>=</mo><mo>−</mo><msub><mi>g</mi><mi>k</mi></msub><mo>+</mo><mi>β</mi><msub><mi>d</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">d_k = -g_k + \beta d_{k-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord">−</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.9028em;vertical-align:-0.2083em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span></span></p><p>可见只需要确定组合系数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span>。由于共轭梯度法遵循相邻迭代点的搜索方向共轭，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>d</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mi>T</mi></msubsup><mi>G</mi><msub><mi>d</mi><mi>k</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d_{k-1}^TGd_k=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1828em;vertical-align:-0.3414em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3414em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，因此对上式两侧同时左乘 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>d</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mi>T</mi></msubsup><mi>G</mi></mrow><annotation encoding="application/x-tex">d_{k-1}^TG</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1828em;vertical-align:-0.3414em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3414em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span></span></span></span>，有：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msubsup><mi>d</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mi>T</mi></msubsup><mi>G</mi><msub><mi>d</mi><mi>k</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>−</mo><msubsup><mi>d</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mi>T</mi></msubsup><mi>G</mi><msub><mi>g</mi><mi>k</mi></msub><mo>+</mo><mi>β</mi><msubsup><mi>d</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mi>T</mi></msubsup><mi>G</mi><msub><mi>d</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>0</mn></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}d_{k-1}^TGd_k &amp;= -d_{k-1}^TGg_k + \beta d_{k-1}^TGd_{k-1} \\&amp;= 0\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0513em;vertical-align:-1.2757em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7757em;"><span style="top:-3.8843em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.3843em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2757em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7757em;"><span style="top:-3.8843em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">−</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.3843em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2757em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>于是可得当前的组合系数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span> 为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>β</mi><mo>=</mo><mfrac><mrow><msubsup><mi>d</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mi>T</mi></msubsup><mi>G</mi><msub><mi>g</mi><mi>k</mi></msub></mrow><mrow><msubsup><mi>d</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mi>T</mi></msubsup><mi>G</mi><msub><mi>d</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">\beta = \frac{d_{k-1}^TGg_k}{d_{k-1}^TGd_{k-1}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.6455em;vertical-align:-1.0728em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5728em;"><span style="top:-2.2869em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8231em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3596em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.7314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3414em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0728em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p> <div class="note note-warning">            <p>上述组合系数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span> 的结果是共轭梯度最原始的表达式，后人又进行了变形，<s>没证出来，难崩，直接背吧</s>，给出 FR 的组合系数表达式：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>β</mi><mo>=</mo><mfrac><mrow><msubsup><mi>g</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>g</mi><mi>k</mi></msub></mrow><mrow><msubsup><mi>g</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mi>T</mi></msubsup><msub><mi>g</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mfrac></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\beta = \frac{g_k^Tg_k}{g_{k-1}^T g_{k-1}}\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.8911em;vertical-align:-1.1956em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6956em;"><span style="top:-3.6956em;"><span class="pstrut" style="height:3.5183em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5183em;"><span style="top:-2.2869em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8231em;"><span style="top:-2.3987em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3596em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0728em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.1956em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>          </div><p>当然了由于初始迭代时没有前一个搜索方向，因此直接用初始点的梯度作为搜索方向，即：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>d</mi><mn>0</mn></msub><mo>=</mo><mo>−</mo><msub><mi>g</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">d_0=-g_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord">−</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p><p>于是可以导出当前的搜索方向 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">d_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的闭式解为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>d</mi><mi>k</mi></msub><mo>=</mo><mo>−</mo><msub><mi>g</mi><mi>k</mi></msub><mo>+</mo><mfrac><mrow><msubsup><mi>g</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>g</mi><mi>k</mi></msub></mrow><mrow><msubsup><mi>g</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mi>T</mi></msubsup><msub><mi>g</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mfrac><msub><mi>d</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">d_k = -g_k + \frac{g_k^Tg_k}{g_{k-1}^T g_{k-1}} d_{k-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.1944em;"></span><span class="mord">−</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.5911em;vertical-align:-1.0728em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5183em;"><span style="top:-2.2869em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8231em;"><span style="top:-2.3987em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3596em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0728em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span></span></p></li></ol><p>迭代公式：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><msub><mi>α</mi><mi>k</mi></msub><msub><mi>d</mi><mi>k</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mo lspace="0em" rspace="0em">=</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msub><mi>x</mi><mi>k</mi></msub><mo>+</mo><mo stretchy="false">(</mo><mo>−</mo><mfrac><mrow><msubsup><mi>g</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>d</mi><mi>k</mi></msub></mrow><mrow><msubsup><mi>d</mi><mi>k</mi><mi>T</mi></msubsup><mi>G</mi><msub><mi>d</mi><mi>k</mi></msub></mrow></mfrac><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo>−</mo><msub><mi>g</mi><mi>k</mi></msub><mo>+</mo><mfrac><mrow><msubsup><mi>g</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>g</mi><mi>k</mi></msub></mrow><mrow><msubsup><mi>g</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><mi>T</mi></msubsup><msub><mi>g</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></mfrac><msub><mi>d</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}x_{k+1} =&amp; x_k + \alpha_k d_k\\=&amp; x_k + (-\frac{g_k^T d_k}{d_k^TGd_k}) (-g_k + \frac{g_k^Tg_k}{g_{k-1}^T g_{k-1}} d_{k-1})\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.3911em;vertical-align:-1.9456em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.4456em;"><span style="top:-5.1239em;"><span class="pstrut" style="height:3.5183em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span></span><span style="top:-2.9456em;"><span class="pstrut" style="height:3.5183em;"></span><span class="mord"><span class="mrel">=</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.9456em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.4456em;"><span style="top:-5.1239em;"><span class="pstrut" style="height:3.5183em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.9456em;"><span class="pstrut" style="height:3.5183em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5183em;"><span style="top:-2.2869em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8231em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0144em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span><span class="mopen">(</span><span class="mord">−</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5183em;"><span style="top:-2.2869em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8231em;"><span style="top:-2.3987em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3596em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0728em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.9456em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><h3 id="4-4-拟牛顿法">4.4 拟牛顿法</h3><p>4.1 和 4.2 介绍的基于一阶梯度和二阶梯度的下降法都可以统一成下面的表达式：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>x</mi><mi>k</mi></msub><mo>−</mo><msub><mi>α</mi><mi>k</mi></msub><msub><mi>B</mi><mi>k</mi></msub><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x_{k+1} = x_k - \alpha_k B_k \nabla f(x_k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p><ul><li>4.1 的最速下降法的步长因子通过精确线搜索获得，海塞矩阵的逆 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>B</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">B_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 不存在，可以看做为单位阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span></span></span></span></li><li>4.2 的牛顿法的步长因子同样可以通过精确线搜索获得，当然也可以设置为定值，海塞矩阵的逆 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>B</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">B_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 对应二阶梯度的逆 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msup><mi mathvariant="normal">∇</mi><mn>2</mn></msup><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">(\nabla^2 f(x_k))^{-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord">∇</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span></li></ul><p>前者收敛速度差、后者计算量和存储量大，我们尝试构造一个对称正定阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>B</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">B_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 来近似代替二阶梯度的逆，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>B</mi><mi>k</mi></msub><mo>≈</mo><mo stretchy="false">(</mo><msup><mi mathvariant="normal">∇</mi><mn>2</mn></msup><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">B_k \approx (\nabla^2 f(x_k))^{-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord">∇</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span>，使得该法具备较快的收敛速度与较少的内存开销。</p><p>介绍最著名的 <strong>BFGS</strong> 拟牛顿法，它的核心思想是每次迭代过程中对其进行快速校正，从而在确保收敛速度的情况下提升计算效率。迭代公式如下：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>x</mi><mi>k</mi></msub><mo>−</mo><msubsup><mi>B</mi><mi>k</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><msub><mi>g</mi><mi>k</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mtext>记:&nbsp;</mtext><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>s</mi><mi>k</mi></msub><mo>=</mo><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>x</mi><mi>k</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>y</mi><mi>k</mi></msub><mo>=</mo><msub><mi>g</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>g</mi><mi>k</mi></msub></mrow></mstyle></mtd></mtr></mtable></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mtext>则:&nbsp;</mtext><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>B</mi><mn>0</mn></msub><mo>=</mo><msup><mi mathvariant="normal">∇</mi><mn>2</mn></msup><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><msub><mi>B</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>B</mi><mi>k</mi></msub><mo>+</mo><mfrac><mrow><msub><mi>y</mi><mi>k</mi></msub><msubsup><mi>y</mi><mi>k</mi><mi>T</mi></msubsup></mrow><mrow><msubsup><mi>s</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>y</mi><mi>k</mi></msub></mrow></mfrac><mo>−</mo><mfrac><mrow><msub><mi>B</mi><mi>k</mi></msub><msub><mi>s</mi><mi>k</mi></msub><msubsup><mi>s</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>B</mi><mi>k</mi></msub></mrow><mrow><msubsup><mi>s</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>B</mi><mi>k</mi></msub><msub><mi>s</mi><mi>k</mi></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr></mtable></mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}&amp;x_{k+1} = x_k - B_k^{-1}g_k \\&amp;\text{记: }\begin{cases}s_k= x_{k+1} - x_k\\y_k=g_{k+1} - g_k\end{cases}\\&amp;\text{则: }\begin{cases}B_0 = \nabla^2f(x_0)\\\displaystyle B_{k+1} = B_k + \frac{y_ky_k^T}{s_k^Ty_k} - \frac{B_ks_ks_k^TB_k}{s_k^TB_ks_k}\end{cases}\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:9.0969em;vertical-align:-4.2985em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.7985em;"><span style="top:-8.1707em;"><span class="pstrut" style="height:4.2364em;"></span><span class="mord"></span></span><span style="top:-5.7607em;"><span class="pstrut" style="height:4.2364em;"></span><span class="mord"></span></span><span style="top:-1.9743em;"><span class="pstrut" style="height:4.2364em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.2985em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.7985em;"><span style="top:-8.1707em;"><span class="pstrut" style="height:4.2364em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.4086em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2914em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-5.7607em;"><span class="pstrut" style="height:4.2364em;"></span><span class="mord"><span class="mord"></span><span class="mord text"><span class="mord cjk_fallback">记</span><span class="mord">:&nbsp;</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-1.9743em;"><span class="pstrut" style="height:4.2364em;"></span><span class="mord"><span class="mord"></span><span class="mord text"><span class="mord cjk_fallback">则</span><span class="mord">:&nbsp;</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-2.5em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.492em;"><span class="pstrut" style="height:3.15em;"></span><span style="height:0.016em;width:0.8889em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.8889em" height="0.016em" style="width:0.8889em" viewBox="0 0 888.89 16" preserveAspectRatio="xMinYMin"><path d="M384 0 H504 V16 H384z M384 0 H504 V16 H384z"></path></svg></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.292em;"><span class="pstrut" style="height:3.15em;"></span><span style="height:0.016em;width:0.8889em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.8889em" height="0.016em" style="width:0.8889em" viewBox="0 0 888.89 16" preserveAspectRatio="xMinYMin"><path d="M384 0 H504 V16 H384z M384 0 H504 V16 H384z"></path></svg></span></span><span style="top:-4.3em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.2364em;"><span style="top:-4.7467em;"><span class="pstrut" style="height:3.5183em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord">∇</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-2.7964em;"><span class="pstrut" style="height:3.5183em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5183em;"><span style="top:-2.2869em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8231em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0144em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5183em;"><span style="top:-2.2869em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8231em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0144em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.7364em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.2985em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><blockquote><p>参考：</p><ul><li><a href="https://max.book118.com/html/2017/0606/111956214.shtm">西北工业大学 - 课件 - 含例题与解析</a></li><li><a href="https://www.math.pku.edu.cn/teachers/lidf/docs/statcomp/html/_statcompbook/opt-uncons.html#opt-uncons">北京大学 - 教材 - 含例题与解析</a></li><li><a href="https://blog.csdn.net/Serendipity_zyx/article/details/120515338">路人 - 博客 - 更通俗的解释</a></li><li><a href="https://zh.wikipedia.org/wiki/%E7%89%9B%E9%A1%BF%E6%B3%95">牛顿法 - wiki - 更权威的解释</a></li></ul></blockquote><h2 id="第五章「无约束最优化」最小二乘">第五章「无约束最优化」最小二乘</h2><div class="note note-light">            <p>本章继续介绍无约束函数的最优化算法。不过和第四章的区别在于现在的目标函数是二次函数，称为「最小二乘问题」。</p><p>所谓的无约束最小二乘问题，本质上是第四章介绍的无约束问题的一个子集，只不过因为使用场景很多所以单独拿出来进行讨论。也正因为使用场景多，学者们针对此类问题设计出了更加高效的最优化算法。</p><p>无约束最小二乘问题的形式定义为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><munder><mrow><mi>min</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>∈</mo><msup><mi>R</mi><mi>n</mi></msup></mrow></munder><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><mo stretchy="false">[</mo><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">]</mo><mn>2</mn></msup><mo separator="true">,</mo><mspace width="1em"></mspace><mi>m</mi><mo>≥</mo><mi>n</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\min_{x\in R^n}f(x)=\frac{1}{2}\sum_{i=1}^m[r_{i}(x)]^2,\quad m\ge n\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.2291em;vertical-align:-1.3645em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8645em;"><span style="top:-3.8645em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6679em;"><span style="top:-2.3557em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.5935em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">min</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7717em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3645em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r_i(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 称为「残量函数」。本质上最小二乘问题就是寻找一个函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><msub><mi>α</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x,\alpha_i),(i=1,2,\cdots,m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span> 来拟合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span>，于是问题就转化为了</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>min</mi><mo>⁡</mo><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><mo stretchy="false">[</mo><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">]</mo><mn>2</mn></msup></mrow><mo>=</mo><mi>min</mi><mo>⁡</mo><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><msub><mi>α</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>−</mo><msub><mi>b</mi><mi>i</mi></msub><msup><mo stretchy="false">]</mo><mn>2</mn></msup></mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\min { \sum_{i=1}^m [r_i(x)]^2 }=\min { \sum_{i=1}^m [f(x,\alpha_i) - b_i]^2 }\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.2291em;vertical-align:-1.3645em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8645em;"><span style="top:-3.8645em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mop">min</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop">min</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3645em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>当 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r_i(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 为线性函数时，当前问题为线性最小二乘问题；当 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r_i(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 为非线性函数时，当前问题为非线性最小二乘问题。本章将分别讨论这两种最小二乘问题的优化求解策略。</p>          </div><h3 id="5-1-线性最小二乘">5.1 线性最小二乘</h3><p>此时可以直接将目标函数写成：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>min</mi><mo>⁡</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mi>A</mi><mi>x</mi><mo>−</mo><mi>b</mi><mi mathvariant="normal">∣</mi><msup><mi mathvariant="normal">∣</mi><mn>2</mn></msup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mi>T</mi></msup><msup><mi>A</mi><mi>T</mi></msup><mi>A</mi><mi>x</mi><mo>−</mo><msup><mi>b</mi><mi>T</mi></msup><mi>A</mi><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>b</mi><mi>T</mi></msup><mi>b</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\min f(x)&amp;=\frac{1}{2}|| Ax-b ||^2\\&amp;= \frac{1}{2}x^TA^TAx-b^TAx+\frac{1}{2}b^Tb\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.6149em;vertical-align:-2.0574em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5574em;"><span style="top:-4.5574em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mop">min</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.0574em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5574em;"><span style="top:-4.5574em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord">∣∣</span><span class="mord mathnormal">A</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">b</span><span class="mord">∣</span><span class="mord"><span class="mord">∣</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.3214em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">A</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">A</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.0574em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>利用一阶必要条件可得：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msup><mi>A</mi><mi>T</mi></msup><mi>A</mi><mi>x</mi><mo>−</mo><msup><mi>A</mi><mi>T</mi></msup><mi>b</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>0</mn></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\nabla f(x)&amp;=A^TAx - A^Tb\\&amp;=0\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0513em;vertical-align:-1.2757em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7757em;"><span style="top:-3.8843em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span><span style="top:-2.3843em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2757em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7757em;"><span style="top:-3.8843em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">A</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">b</span></span></span><span style="top:-2.3843em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2757em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>于是可得最优闭式解：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup><mo>=</mo><mo stretchy="false">(</mo><msup><mi>A</mi><mi>T</mi></msup><mi>A</mi><msup><mo stretchy="false">)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mi>A</mi><mi>T</mi></msup><mi>b</mi></mrow><annotation encoding="application/x-tex">x^*=(A^T A)^{-1}A^Tb</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7387em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7387em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1413em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">A</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">b</span></span></span></span></span></p><p>当然 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>A</mi><mi>T</mi></msup><mi>A</mi></mrow><annotation encoding="application/x-tex">A^TA</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8413em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">A</span></span></span></span> 并不都是可逆的，并且在数据量足够大时，即使可逆也会让求逆操作即为耗时。针对此问题，提出了线性最小二乘的 QR 正交分解算法。</p><h3 id="5-2-非线性最小二乘">5.2 非线性最小二乘</h3><p>同样可以采用第四章学到的各种下降迭代算法，这里引入高斯牛顿法，推导的解的迭代公式为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>=</mo><msup><mi>x</mi><mi>k</mi></msup><mo>−</mo><mo stretchy="false">(</mo><msubsup><mi>A</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>A</mi><mi>k</mi></msub><msup><mo stretchy="false">)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><msubsup><mi>A</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>r</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">x^{k+1}=x^k - (A_k^TA_k)^{-1}A_k^Tr_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8991em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9824em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1413em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p><p>其中：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>A</mi><mi>k</mi></msub><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">∇</mi><msub><mi>r</mi><mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">∇</mi><msub><mi>r</mi><mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi mathvariant="normal">∇</mi><msub><mi>r</mi><mi>m</mi></msub><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo separator="true">,</mo><mspace width="1em"></mspace><msub><mi>r</mi><mi>k</mi></msub><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>r</mi><mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>r</mi><mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi><mi mathvariant="normal">⋮</mi><mpadded height="0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>r</mi><mi>m</mi></msub><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}A_k =  \begin{bmatrix}\nabla r_1(x_k)\\\nabla r_2(x_k)\\\vdots \\\nabla r_m(x_k)\end{bmatrix},\quadr_k = \begin{bmatrix}r_1(x_k)\\r_2(x_k)\\\vdots \\r_m(x_k)\end{bmatrix}\\\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.76em;vertical-align:-2.63em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.13em;"><span style="top:-5.13em;"><span class="pstrut" style="height:4.98em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.95em;"><span style="top:-4.95em;"><span class="pstrut" style="height:7.4em;"></span><span style="width:0.667em;height:5.400em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="5.400em" viewBox="0 0 667 5400"><path d="M403 1759 V84 H666 V0 H319 V1759 v1800 v1759 h347 v-84H403z M403 1759 V0 H319 V1759 v1800 v1759 h84z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.45em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.98em;"><span style="top:-5.8275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">∇</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-4.6275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">∇</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-2.7675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-1.5675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">∇</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.48em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.95em;"><span style="top:-4.95em;"><span class="pstrut" style="height:7.4em;"></span><span style="width:0.667em;height:5.400em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="5.400em" viewBox="0 0 667 5400"><path d="M347 1759 V0 H0 V84 H263 V1759 v1800 v1759 H0 v84 H347zM347 1759 V0 H263 V1759 v1800 v1759 h84z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.45em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.95em;"><span style="top:-4.95em;"><span class="pstrut" style="height:7.4em;"></span><span style="width:0.667em;height:5.400em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="5.400em" viewBox="0 0 667 5400"><path d="M403 1759 V84 H666 V0 H319 V1759 v1800 v1759 h347 v-84H403z M403 1759 V0 H319 V1759 v1800 v1759 h84z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.45em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.98em;"><span style="top:-5.8275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-4.6275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-2.7675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-1.5675em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.48em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.95em;"><span style="top:-4.95em;"><span class="pstrut" style="height:7.4em;"></span><span style="width:0.667em;height:5.400em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.667em" height="5.400em" viewBox="0 0 667 5400"><path d="M347 1759 V0 H0 V84 H263 V1759 v1800 v1759 H0 v84 H347zM347 1759 V0 H263 V1759 v1800 v1759 h84z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.45em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.63em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><blockquote><p>参考：</p><ul><li><a href="http://faculty.bicmr.pku.edu.cn/~wenzw/optbook/lect/14-lsp-new-zxx.pdf">北京大学 - 课件 - 非线性最小二乘</a></li><li><a href="https://zhuanlan.zhihu.com/p/124934931">路人 - 博客 - 通俗解释非线性最小二乘</a></li></ul></blockquote><h2 id="第六章「约束最优化」二次规划">第六章「约束最优化」二次规划</h2><div class="note note-light">            <p>目标函数是二次函数，约束函数是线性函数。一般形式为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>min</mi><mo>⁡</mo><mspace width="1em"></mspace><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mi>T</mi></msup><mi>G</mi><mi>x</mi><mo>+</mo><msup><mi>g</mi><mi>T</mi></msup><mi>x</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>s</mi><mi mathvariant="normal">.</mi><mi>t</mi><mi mathvariant="normal">.</mi><mspace width="1em"></mspace><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msubsup><mi>a</mi><mi>i</mi><mi>T</mi></msubsup><mi>x</mi><mo>=</mo><msub><mi>b</mi><mi>i</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo separator="true">,</mo><mi>i</mi><mo>∈</mo><mi>S</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msubsup><mi>a</mi><mi>i</mi><mi>T</mi></msubsup><mi>x</mi><mo>≥</mo><msub><mi>b</mi><mi>i</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo separator="true">,</mo><mi>i</mi><mo>∈</mo><mi>M</mi></mrow></mstyle></mtd></mtr></mtable></mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}&amp;\min \quad q(x) = \frac{1}{2}x^TGx + g^Tx \\&amp;s.t.\quad \begin{cases}a_i^Tx = b_i&amp; ,i\in S\\a_i^Tx \ge b_i&amp; ,i \in M\\\end{cases}\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.6075em;vertical-align:-2.5537em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.0537em;"><span style="top:-5.4823em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"></span></span><span style="top:-2.7463em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.5537em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.0537em;"><span style="top:-5.4823em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">min</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">x</span></span></span><span style="top:-2.7463em;"><span class="pstrut" style="height:3.75em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal">s</span><span class="mord">.</span><span class="mord mathnormal">t</span><span class="mord">.</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.5537em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>本章将分别讨论约束函数「含有等式」和「含有不等式」两类二次规划问题。</p>          </div><h3 id="6-1-等式约束二次规划">6.1 等式约束二次规划</h3><p>我们引入<strong>拉格朗日方法 (Lagrange Method, 简称 LM)</strong>。此时可以用矩阵表示问题和约束条件，并且不加证明的给出最优解和对应乘子就是满足 KKT 条件下的解。</p><p>拉格朗日函数：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>λ</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mi>T</mi></msup><mi>G</mi><mi>x</mi><mo>+</mo><msup><mi>g</mi><mi>T</mi></msup><mi>x</mi><mo>−</mo><msup><mi>λ</mi><mi>T</mi></msup><mo stretchy="false">(</mo><msup><mi>A</mi><mi>T</mi></msup><mi>x</mi><mo>−</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(x,\lambda) = \frac{1}{2}x^TGx+g^Tx -\lambda^T(A^Tx - b)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">λ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">G</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0858em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1413em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">b</span><span class="mclose">)</span></span></span></span></span></p><p>一阶必要条件：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>L</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>G</mi><mi>x</mi><mo>+</mo><mi>g</mi><mo>−</mo><mi>A</mi><mi>λ</mi><mo>=</mo><mn>0</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>L</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">∂</mi><mi>λ</mi></mrow></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msup><mi>A</mi><mi>x</mi></msup><mo>−</mo><mi>b</mi><mo>=</mo><mn>0</mn></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\frac{\partial L(x, \lambda) }{\partial x} &amp;= Gx+g-A\lambda = 0 \\\frac{\partial L(x, \lambda) }{\partial \lambda} &amp;= A^x - b = 0\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.826em;vertical-align:-2.163em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.663em;"><span style="top:-4.663em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">λ</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">λ</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">λ</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.163em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.663em;"><span style="top:-4.663em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">G</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal">λ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.163em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>最优解的矩阵形式：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>G</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mi>A</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><msup><mi>A</mi><mi>T</mi></msup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msup><mi>x</mi><mo>∗</mo></msup></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msup><mi>λ</mi><mo>∗</mo></msup></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mo>−</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.16em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>g</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\begin{bmatrix}G &amp; -A \\-A^T &amp; 0\end{bmatrix}\begin{bmatrix}x^* \\\lambda^*\end{bmatrix}=-\begin{bmatrix}g \\b\end{bmatrix}\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.7013em;vertical-align:-1.1007em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6007em;"><span style="top:-3.6007em;"><span class="pstrut" style="height:3.4507em;"></span><span class="mord"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4507em;"><span style="top:-3.6107em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">G</span></span></span><span style="top:-2.4093em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9507em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4507em;"><span style="top:-3.6107em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord mathnormal">A</span></span></span><span style="top:-2.4093em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9507em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.1007em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><h3 id="6-2-不等式约束二次规划">6.2 不等式约束二次规划</h3><p>我们引入<strong>有效集方法 (Active Set Method, 简称 ASM)</strong>。首先显然的最优解一定成立于等式约束，或成立于不等式取等。我们可以直接枚举每一种约束条件的组合（所有等式+枚举的不等式，并将不等式看做等式约束），然后判定当前的解是否满足没有选择的不等式约束。这种方法是可行的但是计算量极大。于是有效集方法应运而生。</p><p>算法流程：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406191553692.png" alt="Primal ASM 算法"></p><p>算法解析：</p><ul><li><strong>计算初始可行点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">x^{(0)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.888em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">0</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span></strong>，可用线性规划中的大 M 法计算获得</li><li><strong>计算前进方向 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">p_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></strong>，通过求解等式约束二次规划子问题获得</li><li>若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>k</mi></msub><mo>=</mo><mn mathvariant="bold">0</mn></mrow><annotation encoding="application/x-tex">p_k = \bf{0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">0</span></span></span></span></span></span>​，则<strong>计算工作集约束 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>W</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">W_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 对应的所有拉格朗日乘子 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\lambda_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></strong><ul><li>若所有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mi>i</mi></msub><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\lambda_i \ge 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，则停止迭代，得到最优解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup><mo>=</mo><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">x^*=x^{(k)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.888em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span></span></li><li>若存在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mi>i</mi></msub><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\lambda_i &lt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，则在工作集中剔除 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\lambda_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 最小时对应的约束 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">a_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>，得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>W</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>W</mi><mi>k</mi></msub><mo>∖</mo><mo stretchy="false">{</mo><mtext> </mtext><msub><mi>a</mi><mi>j</mi></msub><mtext> </mtext><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">W_{k+1}=W_k \setminus \set{a_j}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∖</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mopen">{</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">}</span></span></span></span></li></ul></li><li>若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>k</mi></msub><mo mathvariant="normal">≠</mo><mn mathvariant="bold">0</mn></mrow><annotation encoding="application/x-tex">p_k \ne \bf{0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">0</span></span></span></span></span></span>，则<strong>计算步长因子 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>α</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></strong>，并置 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mo>=</mo><msup><mi>x</mi><mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup><mo>+</mo><msub><mi>α</mi><mi>k</mi></msub><msub><mi>p</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">x^{(k+1)} = x^{(k)} + \alpha_kp_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.888em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9713em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.888em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><ul><li>若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>α</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\alpha_k=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，则工作集不变，得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>W</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>W</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">W_{k+1}=W_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></li><li>若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>α</mi><mi>k</mi></msub><mo>&lt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\alpha_k &lt; 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，则在工作集中添加 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>α</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\alpha_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 最小时对应的约束 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">a_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>，得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>W</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>W</mi><mi>k</mi></msub><mo>⋃</mo><mo stretchy="false">{</mo><mtext> </mtext><msub><mi>a</mi><mi>j</mi></msub><mtext> </mtext><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">W_{k+1}=W_k \bigcup \set {a_j}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">⋃</span><span class="mopen">{</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">}</span></span></span></span></li></ul></li></ul><blockquote><p>参考：</p><ul><li><a href="https://blog.csdn.net/dymodi/article/details/50358278">路人 - 博客 - 引入+原理+示例</a></li></ul></blockquote><h2 id="第七章「约束最优化」">第七章「约束最优化」</h2><div class="note note-light">            <p>本章我们简单讨论<strong>约束最优化问题</strong>中，针对<strong>等式约束</strong>的「二次罚函数」算法，以及拉格朗日乘子法。</p>          </div><h3 id="二次罚函数法">二次罚函数法</h3><p>对于等式约束问题：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>min</mi><mo>⁡</mo><mspace width="1em"></mspace><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>s</mi><mi mathvariant="normal">.</mi><mi>t</mi><mi mathvariant="normal">.</mi><mspace width="1em"></mspace><msub><mi>a</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mtext>&nbsp;</mtext><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><mi>p</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}&amp;\min\quad f(x) \\&amp;s.t.\quad a_i(x) = 0,\ i=1,2,...,p\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.75em;"><span class="pstrut" style="height:2.84em;"></span><span class="mord"></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:2.84em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">min</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal">s</span><span class="mord">.</span><span class="mord mathnormal">t</span><span class="mord">.</span><span class="mspace" style="margin-right:1em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace">&nbsp;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">...</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>我们定义二次罚函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>E</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P_E(x, \sigma)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mclose">)</span></span></span></span>，其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> 为惩罚因子：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>P</mi><mi>E</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>σ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>σ</mi><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>p</mi></munderover><msubsup><mi>a</mi><mi>i</mi><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P_E(x, \sigma) = f(x) + \frac{1}{2} \sigma\sum_{i=1}^p a_i^2(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.9762em;vertical-align:-1.2777em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6985em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3471em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></p><p>不加证明的给出结论：当惩罚因子 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">\to \infty</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord">∞</span></span></span></span> 时，目标函数趋于最小值。下面给出算法流程：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406120820032.png" alt="等式约束的二次罚函数法"></p><h3 id="拉格朗日乘子法">拉格朗日乘子法</h3><p>目标函数：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406281100567.png" alt="目标函数"></p><p>拉格朗日函数：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406281102611.png" alt="拉格朗日函数"></p><p>KKT 条件：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406281102383.png" alt="KKT 条件"></p><p>在实际求解时，我们只需要罗列上述 KKT 条件中的 (1) 和 (5)，另外三个显然不需要再罗列了。我们只需要对 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 个不等式约束条件对应的乘子进行是否为零的讨论即可，显然需要讨论 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>m</mi></msup></mrow><annotation encoding="application/x-tex">2^m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6644em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span></span></span></span></span></span></span></span> 次。</p><h2 id="后记">后记</h2><p>下面罗列一下考试题型（考前就透露完了🤣）：</p><h3 id="一、选择-6-2"><strong>一、选择 <code>6 * 2'</code></strong></h3><ul><li><p>严格凸函数的定义（ch1.2）（判断<a href="https://www.cnblogs.com/bigmonkey/p/11913079.html">海塞矩阵是否正定</a>即可）</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-a7ec1960" role="button" aria-expanded="false" aria-controls="collapse-a7ec1960">        <div class="fold-arrow">▶</div>例题      </div>      <div class="fold-collapse collapse" id="collapse-a7ec1960">        <div class="fold-content">          <p>考虑函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">f(x, y) = x^2 + y^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>。我们计算它的 Hessian 矩阵：</p><ol><li><p>计算一阶导数：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>f</mi><mi>x</mi></msub><mo>=</mo><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mn>2</mn><mi>x</mi></mrow><annotation encoding="application/x-tex">f_x = \frac{\partial f}{\partial x} = 2x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span><span class="mord mathnormal">x</span></span></span></span></span></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>f</mi><mi>y</mi></msub><mo>=</mo><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mn>2</mn><mi>y</mi></mrow><annotation encoding="application/x-tex">f_y = \frac{\partial f}{\partial y} = 2y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span></span></p></li><li><p>计算二阶导数：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>f</mi><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>=</mo><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">f_{xx} = \frac{\partial^2 f}{\partial x^2} = 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">xx</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span></span></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>f</mi><mrow><mi>y</mi><mi>y</mi></mrow></msub><mo>=</mo><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">f_{yy} = \frac{\partial^2 f}{\partial y^2} = 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">yy</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3715em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span></span></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>f</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub><mo>=</mo><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f_{xy} = \frac{\partial^2 f}{\partial x \partial y} = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3715em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">x</span><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>f</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub><mo>=</mo><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f_{yx} = \frac{\partial^2 f}{\partial y \partial x} = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3715em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p></li><li><p>构造 Hessian 矩阵：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>H</mi><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>f</mi><mrow><mi>x</mi><mi>x</mi></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>f</mi><mrow><mi>x</mi><mi>y</mi></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>f</mi><mrow><mi>y</mi><mi>x</mi></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>f</mi><mrow><mi>y</mi><mi>y</mi></mrow></msub></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">H = \begin{pmatrix}f_{xx} &amp; f_{xy} \\f_{yx} &amp; f_{yy}\end{pmatrix} = \begin{pmatrix}2 &amp; 0 \\0 &amp; 2\end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">xx</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">yy</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p></li><li><p>检查 Hessian 矩阵是否为正定矩阵。对于一个 2x2 的对称矩阵：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>H</mi><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">H = \begin{pmatrix}a &amp; b \\b &amp; c\end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">b</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p><p>要判断它是否正定，可以使用以下条件：</p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">a &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></li><li>矩阵的行列式 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mi>c</mi><mo>−</mo><msup><mi>b</mi><mn>2</mn></msup><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">ac - b^2 &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">a</span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8532em;vertical-align:-0.0391em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></li></ul></li></ol><p>对于我们的 Hessian 矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span>：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>a</mi><mo>=</mo><mn>2</mn><mo separator="true">,</mo><mspace width="1em"></mspace><mi>b</mi><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mspace width="1em"></mspace><mi>c</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">   a = 2, \quad b = 0, \quad c = 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span></span></p><p>显然：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>2</mn><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">   2 &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6835em;vertical-align:-0.0391em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>行列式</mtext><mo>=</mo><mn>2</mn><mo>⋅</mo><mn>2</mn><mo>−</mo><msup><mn>0</mn><mn>2</mn></msup><mo>=</mo><mn>4</mn><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">   \text{行列式} = 2 \cdot 2 - 0^2 = 4 &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord cjk_fallback">行列式</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6835em;vertical-align:-0.0391em;"></span><span class="mord">4</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p><p>因此，Hessian 矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> 是正定的，所以函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">f(x, y) = x^2 + y^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 是严格凸函数。</p>        </div>      </div>    </div></li><li><p>海塞矩阵正定负定和极值的关系（ch1.3）（显然的）</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-6b88c869" role="button" aria-expanded="false" aria-controls="collapse-6b88c869">        <div class="fold-arrow">▶</div>讲解      </div>      <div class="fold-collapse collapse" id="collapse-6b88c869">        <div class="fold-content">          <p>驻点的性质与海塞矩阵的行列式（determinant，det）和特征值（eigenvalues）的符号密切相关。以下是具体的关系：</p><ol><li><p><strong>正定矩阵</strong>：如果海塞矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> 是正定矩阵（即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>det</mtext><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\text{det}(H) &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">det</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 且 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f_{xx} &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">xx</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>），则驻点是局部极小值点。这是因为此时所有的特征值都是正的，函数在该点附近是向上的碗形结构。</p></li><li><p><strong>负定矩阵</strong>：如果海塞矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> 是负定矩阵（即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>det</mtext><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\text{det}(H) &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">det</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 且 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f_{xx} &lt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">xx</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>），则驻点是局部极大值点。这是因为此时所有的特征值都是负的，函数在该点附近是向下的碗形结构。</p></li><li><p><strong>不定矩阵</strong>：如果海塞矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> 的行列式小于零（即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>det</mtext><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\text{det}(H) &lt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">det</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>），则驻点是鞍点。这是因为此时特征值的符号不同，函数在该点附近的形状像马鞍。</p></li><li><p><strong>退化矩阵</strong>：如果海塞矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> 的行列式等于零（即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>det</mtext><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\text{det}(H) = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">det</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>），则无法通过二阶导数判断驻点的性质，需要进一步分析。这是因为此时海塞矩阵是奇异的，不能提供足够的信息来确定驻点的性质。</p></li></ol><p>总结以上内容可以得出如下的关系：</p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>det</mtext><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\text{det}(H) &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">det</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 且 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f_{xx} &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">xx</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>：局部极小值点。</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>det</mtext><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\text{det}(H) &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">det</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 且 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>f</mi><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f_{xx} &lt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">xx</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>：局部极大值点。</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>det</mtext><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\text{det}(H) &lt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">det</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>：鞍点。</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>det</mtext><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\text{det}(H) = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">det</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>：需要进一步分析。</li></ul>        </div>      </div>    </div></li><li><p>求二元函数的极值（ch1.3）（先利用一阶必要条件求出所有驻点，然后利用海塞矩阵的正定性判定极值点。答案：一个极大一个极小，另两个什么都不是）</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-cbf53915" role="button" aria-expanded="false" aria-controls="collapse-cbf53915">        <div class="fold-arrow">▶</div>例题      </div>      <div class="fold-collapse collapse" id="collapse-cbf53915">        <div class="fold-content">          <p><strong>问题：</strong></p><p>给定函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>3</mn><mi>x</mi><mo>+</mo><msup><mi>y</mi><mn>3</mn></msup><mo>−</mo><mn>3</mn><mi>y</mi></mrow><annotation encoding="application/x-tex">f(x, y) = x^3 - 3x + y^3 - 3y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">3</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">3</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>，求其所有极值点。</p><p><strong>解题步骤：</strong></p><ol><li><p><strong>计算一阶梯度以求驻点</strong>：</p><p>计算函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x, y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> 对 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> 的一阶偏导数：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">\frac{\partial f}{\partial x} = 3x^2 - 3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9474em;vertical-align:-0.0833em;"></span><span class="mord">3</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span></span></span></span></span></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">\frac{\partial f}{\partial y} = 3y^2 - 3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0585em;vertical-align:-0.1944em;"></span><span class="mord">3</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span></span></span></span></span></p><p>解方程组 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\frac{\partial f}{\partial x} = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2772em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.10764em;">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\frac{\partial f}{\partial y} = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.4133em;vertical-align:-0.4811em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.10764em;">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">3x^2 - 3 = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9474em;vertical-align:-0.0833em;"></span><span class="mord">3</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">3y^2 - 3 = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0585em;vertical-align:-0.1944em;"></span><span class="mord">3</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p><p>化简方程组得到：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><mi>x</mi><mo>=</mo><mo>±</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x^2 = 1 \implies x = \pm 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6684em;vertical-align:-0.024em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">±</span><span class="mord">1</span></span></span></span></span></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn><mtext>  </mtext><mo>⟹</mo><mtext>  </mtext><mi>y</mi><mo>=</mo><mo>±</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">y^2 = 1 \implies y = \pm 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0585em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6684em;vertical-align:-0.024em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟹</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">±</span><span class="mord">1</span></span></span></span></span></p><p>所以，驻点为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1, 1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1, -1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-1, 1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-1, -1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span></span></span></span>。</p></li><li><p><strong>计算二阶梯度以检验驻点的性质</strong>：</p><p>计算二阶偏导数：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>6</mn><mi>x</mi></mrow><annotation encoding="application/x-tex">\frac{\partial^2 f}{\partial x^2} = 6x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">6</span><span class="mord mathnormal">x</span></span></span></span></span></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>6</mn><mi>y</mi></mrow><annotation encoding="application/x-tex">\frac{\partial^2 f}{\partial y^2} = 6y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.3715em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">6</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span></span></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\frac{\partial^2 f}{\partial x \partial y} = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.3715em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">x</span><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p><p>对于每个驻点，计算 Hessian 矩阵：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>H</mi><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>6</mn><mi>x</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>6</mn><mi>y</mi></mrow></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">H = \begin{pmatrix}6x &amp; 0 \\0 &amp; 6y\end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span><span class="mord mathnormal">x</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p><p><strong>驻点 (1, 1)：</strong></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>6</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>6</mn></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">H(1,1) = \begin{pmatrix}6 &amp; 0 \\0 &amp; 6\end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p><p>行列式为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>det</mtext><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mn>6</mn><mo>×</mo><mn>6</mn><mo>−</mo><mn>0</mn><mo>×</mo><mn>0</mn><mo>=</mo><mn>36</mn><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\text{det}(H(1,1)) = 6 \times 6 - 0 \times 0 = 36 &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">det</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">6</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">6</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6835em;vertical-align:-0.0391em;"></span><span class="mord">36</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p><p>主对角线元素 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>6</mn><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">6 &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6835em;vertical-align:-0.0391em;"></span><span class="mord">6</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，所以 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1, 1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 是<strong>局部极小值点</strong>。</p><p><strong>驻点 (-1, -1)：</strong></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>6</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>6</mn></mrow></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">H(-1,-1) = \begin{pmatrix}-6 &amp; 0 \\0 &amp; -6\end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">6</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">6</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p><p>行列式为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>det</mtext><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>6</mn><mo>×</mo><mo>−</mo><mn>6</mn><mo>−</mo><mn>0</mn><mo>×</mo><mn>0</mn><mo>=</mo><mn>36</mn><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\text{det}(H(-1,-1)) = -6 \times -6 - 0 \times 0 = 36 &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">det</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">6</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">6</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6835em;vertical-align:-0.0391em;"></span><span class="mord">36</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p><p>主对角线元素 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>6</mn><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">-6 &lt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">6</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，所以 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-1, -1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 是<strong>局部极大值点</strong>。</p><p><strong>驻点 (1, -1)：</strong></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>6</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>6</mn></mrow></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">H(1,-1) = \begin{pmatrix}6 &amp; 0 \\0 &amp; -6\end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">6</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p><p>行列式为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>det</mtext><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mn>6</mn><mo>×</mo><mo>−</mo><mn>6</mn><mo>−</mo><mn>0</mn><mo>×</mo><mn>0</mn><mo>=</mo><mo>−</mo><mn>36</mn><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\text{det}(H(1,-1)) = 6 \times -6 - 0 \times 0 = -36 &lt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">det</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">6</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">6</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">36</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p><p>由于行列式为负，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1, -1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 是<strong>鞍点</strong>。</p><p><strong>驻点 (-1, 1)：</strong></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>6</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>6</mn></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">H(-1,1) = \begin{pmatrix}-6 &amp; 0 \\0 &amp; 6\end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">6</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p><p>行列式为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>det</mtext><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>6</mn><mo>×</mo><mn>6</mn><mo>−</mo><mn>0</mn><mo>×</mo><mn>0</mn><mo>=</mo><mo>−</mo><mn>36</mn><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\text{det}(H(-1,1)) = -6 \times 6 - 0 \times 0 = -36 &lt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">det</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">6</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">6</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">36</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p><p>由于行列式为负，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-1, 1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 是<strong>鞍点</strong>。</p></li></ol><p><strong>结论：</strong></p><p>函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>3</mn><mi>x</mi><mo>+</mo><msup><mi>y</mi><mn>3</mn></msup><mo>−</mo><mn>3</mn><mi>y</mi></mrow><annotation encoding="application/x-tex">f(x, y) = x^3 - 3x + y^3 - 3y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">3</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">3</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> 在点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1, 1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 处有一个局部极小值点，在点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-1, -1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 处有一个局部极大值点，而在点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1, -1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-1, 1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 处为鞍点。</p>        </div>      </div>    </div></li><li><p>拉格朗日函数中等式乘子的性质（ch1.3）（等式约束的乘子 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\lambda_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是任意实数，不等式约束的乘子 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>μ</mi><mi>i</mi></msub><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\mu_i\ge 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>）</p></li></ul><h3 id="二、填空-5-2"><strong>二、填空 <code>5 * 2'</code></strong></h3><ul><li><p>求解给定点的下降方向（ch1.3）</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-49bb6dd1" role="button" aria-expanded="false" aria-controls="collapse-49bb6dd1">        <div class="fold-arrow">▶</div>例题      </div>      <div class="fold-collapse collapse" id="collapse-49bb6dd1">        <div class="fold-content">          <p>求解一个函数的下降方向（Descent Direction），可以使用其梯度（Gradient）。梯度的反方向是函数值下降最快的方向。具体来说：</p><ol><li><p><strong>一元函数：</strong></p><ul><li>对于一元函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>，其梯度是导数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f'(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>。</li><li>下降方向是梯度的负方向，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">-f'(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>。</li></ul><p><strong>例子：</strong><br>设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">f(x) = x^2 + 2x + 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>。</p><ul><li>首先计算导数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">f'(x) = 2x + 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span>。</li><li>在某点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 处，下降方向是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mo stretchy="false">(</mo><mn>2</mn><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">-f'(x_0) = -(2x_0 + 2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mopen">(</span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mclose">)</span></span></span></span>。</li></ul></li><li><p><strong>二元函数：</strong></p><ul><li>对于二元函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x, y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span>，其梯度是偏导数的向量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">(</mo><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac><mo separator="true">,</mo><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">\nabla f(x, y) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.8em;vertical-align:-0.65em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.10764em;">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.10764em;">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">)</span></span></span></span></span></span>。</li><li>下降方向是梯度的负方向，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">(</mo><mo>−</mo><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac><mo separator="true">,</mo><mo>−</mo><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">-\nabla f(x, y) = \left( -\frac{\partial f}{\partial x}, -\frac{\partial f}{\partial y} \right)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.8em;vertical-align:-0.65em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size2">(</span></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.10764em;">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.10764em;">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size2">)</span></span></span></span></span></span>。</li></ul><p><strong>例子：</strong><br>设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi></mrow><annotation encoding="application/x-tex">f(x, y) = x^2 + y^2 + 2x + 2y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>。</p><ul><li>首先计算偏导数：<p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\frac{\partial f}{\partial x} = 2x + 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span></span></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac><mo>=</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\frac{\partial f}{\partial y} = 2y + 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span></span></p></li><li>在某点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x_0, y_0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 处，梯度是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>2</mn><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><mn>2</mn><mo separator="true">,</mo><mn>2</mn><msub><mi>y</mi><mn>0</mn></msub><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\nabla f(x_0, y_0) = (2x_0 + 2, 2y_0 + 2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mclose">)</span></span></span></span>。</li><li>下降方向是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mi mathvariant="normal">∇</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mo>−</mo><mo stretchy="false">(</mo><mn>2</mn><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mo>−</mo><mo stretchy="false">(</mo><mn>2</mn><msub><mi>y</mi><mn>0</mn></msub><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">-\nabla f(x_0, y_0) = (-(2x_0 + 2), -(2y_0 + 2))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mord">∇</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mopen">(</span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mopen">(</span><span class="mord">2</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mclose">))</span></span></span></span>。</li></ul></li></ol>        </div>      </div>    </div></li><li><p>线性规划的基本概念（ch2.1）</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-9f553363" role="button" aria-expanded="false" aria-controls="collapse-9f553363">        <div class="fold-arrow">▶</div>讲解      </div>      <div class="fold-collapse collapse" id="collapse-9f553363">        <div class="fold-content">          <p>这个不知道要怎么考，大致罗列一下所有的基本概念。</p><ul><li>定义：目标函数和约束条件都是线性的</li><li>图解法：对于二元函数，可以使用平面图法进行求解。那么共有 4 种可能的结果，分别为：有唯一解、有无穷多解、有无界解、无可行解</li><li>性质：分别从可行域和最优解两个角度展开：<ul><li>对于可行域，如果可行域是非空则可行域一定是凸集，这个很显然，用 ch1.2 的可行域凸集定理证明即可</li><li>对于最优解，首先最优解如果存在则一定存在于所有约束条件中取等时，其次最优解如果存在一定是在可行域的顶点上。</li></ul></li></ul>        </div>      </div>    </div></li><li><p>拟牛顿法的近似海塞矩阵公式（ch4.4）（见书 4.4.9和 4.4.10）</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-4c49716f" role="button" aria-expanded="false" aria-controls="collapse-4c49716f">        <div class="fold-arrow">▶</div>讲解      </div>      <div class="fold-collapse collapse" id="collapse-4c49716f">        <div class="fold-content">          <p>对于原函数，有：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>G</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mo>≈</mo><msub><mi>g</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>g</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">G_{k+1}(x_{k+1}-x_k) \approx g_{k+1} - g_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">G</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p><p>我们希望构造出的对称矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>B</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">B_{k+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span> 满足上式中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>G</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">G_{k+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">G</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span> 的条件，于是就有下面两种拟牛顿条件，其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>H</mi><mi>k</mi></msub><mo>=</mo><msubsup><mi>B</mi><mi>k</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">H_k=B_k^{-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1555em;vertical-align:-0.3013em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8542em;"><span style="top:-2.3987em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.1031em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span></span></span></span>：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>B</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>g</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>g</mi><mi>k</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>H</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>g</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>g</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>x</mi><mi>k</mi></msub></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}B_{k+1}(x_{k+1}-x_k)= g_{k+1} - g_k\\H_{k+1}(g_{k+1} - g_k) = x_{k+1}-x_k\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>我们记 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>s</mi><mi>k</mi></msub><mo>=</mo><msub><mi>x</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>x</mi><mi>k</mi></msub><mo separator="true">,</mo><msub><mi>y</mi><mi>k</mi></msub><mo>=</mo><msub><mi>g</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>g</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">s_k=x_{k+1}-x_k,y_k=g_{k+1} - g_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>。则关于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>B</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">B_{k+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8917em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span> 的 BFGS 校正迭代公式如下：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>B</mi><mn>0</mn></msub><mo>=</mo><msup><mi mathvariant="normal">∇</mi><mn>2</mn></msup><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mstyle scriptlevel="0" displaystyle="true"><msub><mi>B</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>B</mi><mi>k</mi></msub><mo>+</mo><mfrac><mrow><msub><mi>y</mi><mi>k</mi></msub><msubsup><mi>y</mi><mi>k</mi><mi>T</mi></msubsup></mrow><mrow><msubsup><mi>s</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>y</mi><mi>k</mi></msub></mrow></mfrac><mo>−</mo><mfrac><mrow><msub><mi>B</mi><mi>k</mi></msub><msub><mi>s</mi><mi>k</mi></msub><msubsup><mi>s</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>B</mi><mi>k</mi></msub></mrow><mrow><msubsup><mi>s</mi><mi>k</mi><mi>T</mi></msubsup><msub><mi>B</mi><mi>k</mi></msub><msub><mi>s</mi><mi>k</mi></msub></mrow></mfrac></mstyle></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases}B_0 = \nabla^2f(x_0)\\\displaystyle B_{k+1} = B_k + \frac{y_ky_k^T}{s_k^Ty_k} - \frac{B_ks_ks_k^TB_k}{s_k^TB_ks_k}\end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.9728em;vertical-align:-1.7364em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-2.5em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.492em;"><span class="pstrut" style="height:3.15em;"></span><span style="height:0.016em;width:0.8889em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.8889em" height="0.016em" style="width:0.8889em" viewBox="0 0 888.89 16" preserveAspectRatio="xMinYMin"><path d="M384 0 H504 V16 H384z M384 0 H504 V16 H384z"></path></svg></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.292em;"><span class="pstrut" style="height:3.15em;"></span><span style="height:0.016em;width:0.8889em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.8889em" height="0.016em" style="width:0.8889em" viewBox="0 0 888.89 16" preserveAspectRatio="xMinYMin"><path d="M384 0 H504 V16 H384z M384 0 H504 V16 H384z"></path></svg></span></span><span style="top:-4.3em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.2364em;"><span style="top:-4.7467em;"><span class="pstrut" style="height:3.5183em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord">∇</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-2.7964em;"><span class="pstrut" style="height:3.5183em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5183em;"><span style="top:-2.2869em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8231em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0144em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5183em;"><span style="top:-2.2869em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8231em;"><span style="top:-2.3987em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3013em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-2.4169em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2831em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0144em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.7364em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>        </div>      </div>    </div></li></ul><h3 id="三、证明-1-10"><strong>三、证明 <code>1 * 10'</code></strong></h3><ul><li><p>证明高维凸规划问题（ch1.2）</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-0f7ac154" role="button" aria-expanded="false" aria-controls="collapse-0f7ac154">        <div class="fold-arrow">▶</div>例题      </div>      <div class="fold-collapse collapse" id="collapse-0f7ac154">        <div class="fold-content">          <p>关键在于证明可行域是凸集，目标函数是凸函数。证明可行域是凸集比较简单，书 ch1.2 中的「定理1.2.18」给出了详细的可行域凸性判定定理。证明目标函数是凸函数有三种方法，书 ch1.2 中「定理1.2.19-1.2.21」分别从函数值、一阶导数、二阶导数三个角度进行了凸函数判定定理的介绍，下面仅从二阶导数的角度给出凸函数判定的示例。</p><p><strong>二元凸函数</strong></p><p>选取函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>4</mn><msup><mi>y</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">f(x, y) = 3x^2 + 2xy + 4y^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord">3</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord">4</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>，计算 Hessian 矩阵：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>f</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>6</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>8</mn></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">H(f) = \begin{pmatrix}\frac{\partial^2 f}{\partial x^2} &amp; \frac{\partial^2 f}{\partial x \partial y} \\\frac{\partial^2 f}{\partial y \partial x} &amp; \frac{\partial^2 f}{\partial y^2}\end{pmatrix} = \begin{pmatrix}6 &amp; 2 \\2 &amp; 8\end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.1023em;vertical-align:-1.3011em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8011em;"><span style="top:-3.8011em;"><span class="pstrut" style="height:3.07em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.07em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.10764em;">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.07em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.07em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.10764em;">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3011em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8011em;"><span style="top:-3.8011em;"><span class="pstrut" style="height:3.07em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.07em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight">x</span><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.10764em;">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.07em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.07em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.10764em;">f</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3011em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size4">)</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">8</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p><p>Hessian 矩阵的特征值均为正数，因此海塞矩阵是正定的，因此 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>4</mn><msup><mi>y</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">f(x, y) = 3x^2 + 2xy + 4y^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord">3</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord">4</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 是凸函数并且是严格凸的。</p><p><strong>三元凸函数</strong></p><p>选取函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mi>z</mi><mo>+</mo><msup><mi>z</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">g(x, y, z) = 2x^2 + 2xy + 3y^2 + 2xz + z^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord">3</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>，计算 Hessian 矩阵：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>g</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>g</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>g</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>x</mi><mi mathvariant="normal">∂</mi><mi>z</mi></mrow></mfrac></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>g</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>g</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>g</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>y</mi><mi mathvariant="normal">∂</mi><mi>z</mi></mrow></mfrac></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>g</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>z</mi><mi mathvariant="normal">∂</mi><mi>x</mi></mrow></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>g</mi></mrow><mrow><mi mathvariant="normal">∂</mi><mi>z</mi><mi mathvariant="normal">∂</mi><mi>y</mi></mrow></mfrac></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><msup><mi mathvariant="normal">∂</mi><mn>2</mn></msup><mi>g</mi></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>z</mi><mn>2</mn></msup></mrow></mfrac></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>4</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>6</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">H(g) = \begin{pmatrix}\frac{\partial^2 g}{\partial x^2} &amp; \frac{\partial^2 g}{\partial x \partial y} &amp; \frac{\partial^2 g}{\partial x \partial z} \\\frac{\partial^2 g}{\partial y \partial x} &amp; \frac{\partial^2 g}{\partial y^2} &amp; \frac{\partial^2 g}{\partial y \partial z} \\\frac{\partial^2 g}{\partial z \partial x} &amp; \frac{\partial^2 g}{\partial z \partial y} &amp; \frac{\partial^2 g}{\partial z^2}\end{pmatrix} = \begin{pmatrix}4 &amp; 2 &amp; 2 \\2 &amp; 6 &amp; 0 \\2 &amp; 0 &amp; 2\end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:4.6534em;vertical-align:-2.0767em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35em;"><span style="top:-4.35em;"><span class="pstrut" style="height:6.2em;"></span><span style="width:0.875em;height:4.200em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.875em" height="4.200em" viewBox="0 0 875 4200"><path d="M863,9c0,-2,-2,-5,-6,-9c0,0,-17,0,-17,0c-12.7,0,-19.3,0.3,-20,1c-5.3,5.3,-10.3,11,-15,17c-242.7,294.7,-395.3,682,-458,1162c-21.3,163.3,-33.3,349,-36,557 l0,684c0.2,6,0,26,0,60c2,159.3,10,310.7,24,454c53.3,528,210,949.7,470,1265c4.7,6,9.7,11.7,15,17c0.7,0.7,7,1,19,1c0,0,18,0,18,0c4,-4,6,-7,6,-9c0,-2.7,-3.3,-8.7,-10,-18c-135.3,-192.7,-235.5,-414.3,-300.5,-665c-65,-250.7,-102.5,-544.7,-112.5,-882c-2,-104,-3,-167,-3,-189l0,-692c0,-162.7,5.7,-314,17,-454c20.7,-272,63.7,-513,129,-723c65.3,-210,155.3,-396.3,270,-559c6.7,-9.3,10,-15.3,10,-18z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.85em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5767em;"><span style="top:-4.5767em;"><span class="pstrut" style="height:3.07em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.07em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.0256em;"><span class="pstrut" style="height:3.07em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.07em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-1.4744em;"><span class="pstrut" style="height:3.07em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.07em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.0767em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5767em;"><span style="top:-4.5767em;"><span class="pstrut" style="height:3.07em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.07em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight">x</span><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.0256em;"><span class="pstrut" style="height:3.07em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.07em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-1.4744em;"><span class="pstrut" style="height:3.07em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.07em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.0767em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5767em;"><span style="top:-4.5767em;"><span class="pstrut" style="height:3.07em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.07em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight">x</span><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.0256em;"><span class="pstrut" style="height:3.07em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.07em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4811em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-1.4744em;"><span class="pstrut" style="height:3.07em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.07em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7463em;"><span style="top:-2.786em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight" style="margin-right:0.05556em;">∂</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.0767em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35em;"><span style="top:-4.35em;"><span class="pstrut" style="height:6.2em;"></span><span style="width:0.875em;height:4.200em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.875em" height="4.200em" viewBox="0 0 875 4200"><path d="M76,0c-16.7,0,-25,3,-25,9c0,2,2,6.3,6,13c21.3,28.7,42.3,60.3,63,95c96.7,156.7,172.8,332.5,228.5,527.5c55.7,195,92.8,416.5,111.5,664.5c11.3,139.3,17,290.7,17,454c0,28,1.7,43,3.3,45l0,609c-3,4,-3.3,16.7,-3.3,38c0,162,-5.7,313.7,-17,455c-18.7,248,-55.8,469.3,-111.5,664c-55.7,194.7,-131.8,370.3,-228.5,527c-20.7,34.7,-41.7,66.3,-63,95c-2,3.3,-4,7,-6,11c0,7.3,5.7,11,17,11c0,0,11,0,11,0c9.3,0,14.3,-0.3,15,-1c5.3,-5.3,10.3,-11,15,-17c242.7,-294.7,395.3,-681.7,458,-1161c21.3,-164.7,33.3,-350.7,36,-558l0,-744c-2,-159.3,-10,-310.7,-24,-454c-53.3,-528,-210,-949.7,-470,-1265c-4.7,-6,-9.7,-11.7,-15,-17c-0.7,-0.7,-6.7,-1,-18,-1z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.85em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.6em;vertical-align:-1.55em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.875em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.875em" height="3.600em" viewBox="0 0 875 3600"><path d="M863,9c0,-2,-2,-5,-6,-9c0,0,-17,0,-17,0c-12.7,0,-19.3,0.3,-20,1c-5.3,5.3,-10.3,11,-15,17c-242.7,294.7,-395.3,682,-458,1162c-21.3,163.3,-33.3,349,-36,557 l0,84c0.2,6,0,26,0,60c2,159.3,10,310.7,24,454c53.3,528,210,949.7,470,1265c4.7,6,9.7,11.7,15,17c0.7,0.7,7,1,19,1c0,0,18,0,18,0c4,-4,6,-7,6,-9c0,-2.7,-3.3,-8.7,-10,-18c-135.3,-192.7,-235.5,-414.3,-300.5,-665c-65,-250.7,-102.5,-544.7,-112.5,-882c-2,-104,-3,-167,-3,-189l0,-92c0,-162.7,5.7,-314,17,-454c20.7,-272,63.7,-513,129,-723c65.3,-210,155.3,-396.3,270,-559c6.7,-9.3,10,-15.3,10,-18z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.875em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.875em" height="3.600em" viewBox="0 0 875 3600"><path d="M76,0c-16.7,0,-25,3,-25,9c0,2,2,6.3,6,13c21.3,28.7,42.3,60.3,63,95c96.7,156.7,172.8,332.5,228.5,527.5c55.7,195,92.8,416.5,111.5,664.5c11.3,139.3,17,290.7,17,454c0,28,1.7,43,3.3,45l0,9c-3,4,-3.3,16.7,-3.3,38c0,162,-5.7,313.7,-17,455c-18.7,248,-55.8,469.3,-111.5,664c-55.7,194.7,-131.8,370.3,-228.5,527c-20.7,34.7,-41.7,66.3,-63,95c-2,3.3,-4,7,-6,11c0,7.3,5.7,11,17,11c0,0,11,0,11,0c9.3,0,14.3,-0.3,15,-1c5.3,-5.3,10.3,-11,15,-17c242.7,-294.7,395.3,-681.7,458,-1161c21.3,-164.7,33.3,-350.7,36,-558l0,-144c-2,-159.3,-10,-310.7,-24,-454c-53.3,-528,-210,-949.7,-470,-1265c-4.7,-6,-9.7,-11.7,-15,-17c-0.7,-0.7,-6.7,-1,-18,-1z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>Hessian 矩阵的特征值均为正数，因此海塞矩阵是正定的，因此 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo separator="true">,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mi>z</mi><mo>+</mo><msup><mi>z</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">g(x, y, z) = 2x^2 + 2xy + 3y^2 + 2xz + z^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord">3</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 是凸函数并且是严格凸的。</p><p>当然我们也可以通过计算主子矩阵的行列式来代替计算矩阵的特征值，上述二元同样也可以。例如对于下述对称矩阵：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>A</mi><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>1</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>1</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>1</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>1</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">A = \begin{pmatrix} 2 &amp; -1 &amp; 0 \\ -1 &amp; 2 &amp; -1 \\ 0 &amp; -1 &amp; 2 \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.6em;vertical-align:-1.55em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.875em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.875em" height="3.600em" viewBox="0 0 875 3600"><path d="M863,9c0,-2,-2,-5,-6,-9c0,0,-17,0,-17,0c-12.7,0,-19.3,0.3,-20,1c-5.3,5.3,-10.3,11,-15,17c-242.7,294.7,-395.3,682,-458,1162c-21.3,163.3,-33.3,349,-36,557 l0,84c0.2,6,0,26,0,60c2,159.3,10,310.7,24,454c53.3,528,210,949.7,470,1265c4.7,6,9.7,11.7,15,17c0.7,0.7,7,1,19,1c0,0,18,0,18,0c4,-4,6,-7,6,-9c0,-2.7,-3.3,-8.7,-10,-18c-135.3,-192.7,-235.5,-414.3,-300.5,-665c-65,-250.7,-102.5,-544.7,-112.5,-882c-2,-104,-3,-167,-3,-189l0,-92c0,-162.7,5.7,-314,17,-454c20.7,-272,63.7,-513,129,-723c65.3,-210,155.3,-396.3,270,-559c6.7,-9.3,10,-15.3,10,-18z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.875em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.875em" height="3.600em" viewBox="0 0 875 3600"><path d="M76,0c-16.7,0,-25,3,-25,9c0,2,2,6.3,6,13c21.3,28.7,42.3,60.3,63,95c96.7,156.7,172.8,332.5,228.5,527.5c55.7,195,92.8,416.5,111.5,664.5c11.3,139.3,17,290.7,17,454c0,28,1.7,43,3.3,45l0,9c-3,4,-3.3,16.7,-3.3,38c0,162,-5.7,313.7,-17,455c-18.7,248,-55.8,469.3,-111.5,664c-55.7,194.7,-131.8,370.3,-228.5,527c-20.7,34.7,-41.7,66.3,-63,95c-2,3.3,-4,7,-6,11c0,7.3,5.7,11,17,11c0,0,11,0,11,0c9.3,0,14.3,-0.3,15,-1c5.3,-5.3,10.3,-11,15,-17c242.7,-294.7,395.3,-681.7,458,-1161c21.3,-164.7,33.3,-350.7,36,-558l0,-144c-2,-159.3,-10,-310.7,-24,-454c-53.3,-528,-210,-949.7,-470,-1265c-4.7,-6,-9.7,-11.7,-15,-17c-0.7,-0.7,-6.7,-1,-18,-1z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>我们计算其所有主子矩阵的行列式：</p><ul><li>一阶主子矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\begin{pmatrix} 2 \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.85em;"><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span></span></span></span> 的行列式为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span>。</li><li>二阶主子矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>1</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>1</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">\begin{pmatrix} 2 &amp; -1 \\ -1 &amp; 2 \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span> 的行列式为：<p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>det</mi><mo>⁡</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>1</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>1</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><mo>=</mo><mn>2</mn><mo>⋅</mo><mn>2</mn><mo>−</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>4</mn><mo>−</mo><mn>1</mn><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">\det \begin{pmatrix} 2 &amp; -1 \\ -1 &amp; 2 \end{pmatrix} = 2 \cdot 2 - (-1) \cdot (-1) = 4 - 1 = 3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mop">det</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">4</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span></span></span></span></span></p></li><li>三阶主子矩阵即矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 本身的行列式：<p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>det</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">∣</mo><mtable rowspacing="0.16em" columnalign="center center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>1</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>1</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>1</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>1</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd></mtr></mtable><mo fence="true">∣</mo></mrow><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">\det(A) = \begin{vmatrix} 2 &amp; -1 &amp; 0 \\ -1 &amp; 2 &amp; -1 \\ 0 &amp; -1 &amp; 2 \end{vmatrix}=4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">det</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.6em;vertical-align:-1.55em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.333em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.333em" height="3.600em" viewBox="0 0 333 3600"><path d="M145 15 v585 v2400 v585 c2.667,10,9.667,15,21,15c10,0,16.667,-5,20,-15 v-585 v-2400 v-585 c-2.667,-10,-9.667,-15,-21,-15c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v2400 v585 h43z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.05em;"><span class="pstrut" style="height:5.6em;"></span><span style="width:0.333em;height:3.600em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.333em" height="3.600em" viewBox="0 0 333 3600"><path d="M145 15 v585 v2400 v585 c2.667,10,9.667,15,21,15c10,0,16.667,-5,20,-15 v-585 v-2400 v-585 c-2.667,-10,-9.667,-15,-21,-15c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v2400 v585 h43z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">4</span></span></span></span></span></p></li></ul><p>由于所有主子矩阵的行列式都大于零，矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 是正定的。假如这是一个函数的海塞矩阵，则该函数是凸函数并且是严格凸的。</p>        </div>      </div>    </div></li></ul><h3 id="四、计算-68"><strong>四、计算 <code>68'</code></strong></h3><ul><li><p>计算点到超平面的距离，转化为等式约束的最优化问题（ch1.3）</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-4236d39e" role="button" aria-expanded="false" aria-controls="collapse-4236d39e">        <div class="fold-arrow">▶</div>讲解      </div>      <div class="fold-collapse collapse" id="collapse-4236d39e">        <div class="fold-content">          <p>这个应该是很显然的一道题，我们定义目标函数为点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> 到超平面上点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 的距离，约束条件为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 在超平面上。求解方法就是构造一个拉格朗日函数，然后利用一阶必要条件求解即可。</p>        </div>      </div>    </div></li><li><p>线性规划问题中求对偶问题表达式（ch2.3.1）</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-e2dc11d3" role="button" aria-expanded="false" aria-controls="collapse-e2dc11d3">        <div class="fold-arrow">▶</div>例题      </div>      <div class="fold-collapse collapse" id="collapse-e2dc11d3">        <div class="fold-content">          <p>直接看 2.3.1 的实战演练即可。</p>        </div>      </div>    </div></li><li><p>线性规划问题中已知原问题最优解，求解对偶问题最优解（ch2.3.2）（利用互补松弛定理、原问题和对偶问题最优解相等）</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-17bf9d6f" role="button" aria-expanded="false" aria-controls="collapse-17bf9d6f">        <div class="fold-arrow">▶</div>讲解      </div>      <div class="fold-collapse collapse" id="collapse-17bf9d6f">        <div class="fold-content">          <p>见 ch2.3.2 有详细求解步骤。注意可能试卷中给的是原问题的最优解，需要求解对偶问题的最优解，原理是一样的，把原问题看成对偶问题，对偶问题看成原问题，就和 ch2.3.2 的求解逻辑完全一致了。</p>        </div>      </div>    </div></li><li><p>0.618 法求精确步长（ch3.2）</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-ee1d904a" role="button" aria-expanded="false" aria-controls="collapse-ee1d904a">        <div class="fold-arrow">▶</div>讲解      </div>      <div class="fold-collapse collapse" id="collapse-ee1d904a">        <div class="fold-content">          <p>每次缩函数值大的点即可。</p>        </div>      </div>    </div></li><li><p>最速下降法、牛顿法求解方法及其优缺点，共轭梯度法的优缺点（ch4）（直接给步长）</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-774ef8ba" role="button" aria-expanded="false" aria-controls="collapse-774ef8ba">        <div class="fold-arrow">▶</div>讲解      </div>      <div class="fold-collapse collapse" id="collapse-774ef8ba">        <div class="fold-content">          <p>同样很显然的一道题，透露说只要迭代一步？总之就那两个迭代公式，况且步长都给了，记一下方向的迭代即可，最速下降法就是负梯度方向，牛顿法就是二阶梯度的逆乘负梯度作为新的方向。至于优缺点，简单记忆一下即可。最速下降法无非程序简单但因为搜索方向是正交的导致收敛速度差，牛顿法虽然收敛速度快了但是存储的内容太多导致计算量变大，开销增加。</p><p>至于共轭梯度法的优缺点，优点就是该法是最速下降法和牛顿法的结合，每次的搜索方向是共轭的，这样就不用存储海塞矩阵并且收敛速度往往比最速下降法更快。主要用于解决正定二次函数的极小化问题。但在解决其余问题时可能会对搜索步长有极高的依赖，一旦搜索步长不够精准会导致整体的精度下降，收敛速度也会下降。</p>        </div>      </div>    </div></li><li><p><s>有效集方法解不等式约束的二次规划问题（ch6）（小概率考到，考到就gg，因为不会）</s></p></li><li><p>等式约束下的二次罚函数法（ch7）（一个约束函数，一个等式约束条件，共 4 问）</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-fa06789a" role="button" aria-expanded="false" aria-controls="collapse-fa06789a">        <div class="fold-arrow">▶</div>例题      </div>      <div class="fold-collapse collapse" id="collapse-fa06789a">        <div class="fold-content">          <ol><li><p>用拉格朗日乘子法求出最优解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">x^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span> 和拉格朗日乘子 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>λ</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">\lambda^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span></p></li><li><p>写出二次罚函数表达式 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>P</mi><mi>E</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P_E(x,\sigma)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mclose">)</span></span></span></span>，讨论罚因子 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> 在什么取值范围下可以使海塞矩阵 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi mathvariant="normal">∇</mi><mn>2</mn></msup><msub><mi>P</mi><mi>E</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\nabla^2P_E(x,\sigma)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord">∇</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mclose">)</span></span></span></span> 正定</p></li><li><p>求最优解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">x^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span>（直接计算一阶梯度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∇</mi></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">∇</span></span></span></span> 并用 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> 表示 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>x</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">x^*</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6887em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6887em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span></span></span></span></span></span></span></span> 即可）</p></li><li><p>当 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>σ</mi><mo>→</mo><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">\sigma \to \infty</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord">∞</span></span></span></span>​ 时求最优解，并判断是否和第一问计算结果一致</p></li></ol><p>示例：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202407081507550.jpg" alt="等式约束的二次罚函数法"></p>        </div>      </div>    </div></li><li><p>约束最优化问题，约束条件中只有不等式（ch7）（用拉格朗日乘子法，已知有 3 个不等式，且 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">2^3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span> 个答案中只有一个是合法结果）</p>      <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-f639b72c" role="button" aria-expanded="false" aria-controls="collapse-f639b72c">        <div class="fold-arrow">▶</div>例题      </div>      <div class="fold-collapse collapse" id="collapse-f639b72c">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202407081608997.jpg" alt="拉格朗日乘子法解一般约束优化问题"></p>        </div>      </div>    </div></li></ul><p><s>很遗憾将这门课学成了面向已知考试题型的过拟合形式。我并不觉得我掌握了多少优化理论的知识，最多称得上知道了优化问题的大致分类和一些基本的优化应用。从我的笔记就可以看出，自第三章开始就没怎么涉及到理论的证明，确实是证不明白🤡。但这门课自下届开始就被取消了hh。。。。你我皆是局内人，祝好。</s></p>]]></content>
+    <content type="html"><![CDATA[<h2 id="概率论与数理统计">概率论与数理统计</h2><h2 id="前言">前言</h2><p>学科情况：</p><table><thead><tr><th style="text-align:center">主讲教师</th><th style="text-align:center">学分配额</th><th style="text-align:center">学科类别</th></tr></thead><tbody><tr><td style="text-align:center">周效尧</td><td style="text-align:center">4</td><td style="text-align:center">学科基础课</td></tr></tbody></table><p>成绩组成：</p><table><thead><tr><th style="text-align:center">平时</th><th style="text-align:center">测验（×2）</th><th style="text-align:center">期末</th></tr></thead><tbody><tr><td style="text-align:center">10%</td><td style="text-align:center">30%</td><td style="text-align:center">60%</td></tr></tbody></table><p>教材情况：</p><table><thead><tr><th style="text-align:center">课程名称</th><th style="text-align:center">选用教材</th><th style="text-align:center">版次</th><th style="text-align:center">作者</th><th style="text-align:center">出版社</th><th style="text-align:center">ISBN号</th></tr></thead><tbody><tr><td style="text-align:center">概率论与数理统计Ⅰ</td><td style="text-align:center">《概率论与数理统计》</td><td style="text-align:center">第一版</td><td style="text-align:center">刘国祥王晓谦     等主编</td><td style="text-align:center">科学出版社</td><td style="text-align:center">978-7-03-038317-4</td></tr></tbody></table><p>学习资源：</p><ul><li><span class="github-emoji"><span>📺</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/1f4fa.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span> 视频资源：<a href="https://www.bilibili.com/video/BV1ot411y7mU">《概率论与数理统计》教学视频全集（宋浩）</a></li><li><span class="github-emoji"><span>📖</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/1f4d6.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span>  教材答案：<ul><li>本地路径（自用）：[概率论答案.pdf](D:\华为云盘\2. Score\4. 概率论与数理统计\概率论答案.pdf)</li><li>网盘资源（共享）：<a href="https://pan.baidu.com/s/1yeC0rxatHaLeNHQaW85Kpw?pwd=448w">提取码：448w</a></li></ul></li></ul><h2 id="第1章-事件与概率">第1章 事件与概率</h2><h3 id="1-1-随机事件与样本空间">1.1 随机事件与样本空间</h3><h4 id="1-1-1-样本空间">1.1.1 样本空间</h4><p>随机事件发生的总集合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Ω</span></span></span></span></p><h4 id="1-1-2-随机事件">1.1.2 随机事件</h4><p>事件是否发生取决于观察结果的事件</p><h4 id="1-1-3-事件之间的关系与运算">1.1.3 事件之间的关系与运算</h4><ol><li>包含：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>⊂</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \subset B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> or <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mo>⊂</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">B \subset A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span></li><li>相等：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>=</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A=B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></li><li>并（和）：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>∪</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \cup B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></li><li>交（积）：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>∩</mo><mi>B</mi><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi>A</mi><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \cap B \quad (AB)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:1em;"></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span></li><li><strong>互斥</strong>（互不相容）：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mi>B</mi><mo>=</mo><mi mathvariant="normal">Φ</mi></mrow><annotation encoding="application/x-tex">AB=\Phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Φ</span></span></span></span></li><li><strong>对立事件</strong>（余事件）：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>∩</mo><mi>B</mi><mo>=</mo><mi mathvariant="normal">Φ</mi><mo>∧</mo><mi>A</mi><mo>∪</mo><mi>B</mi><mo>=</mo><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">A \cap B=\Phi \land A \cup B=\Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Φ</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∧</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∪</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Ω</span></span></span></span></li><li>差：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>−</mo><mi>B</mi><mo>=</mo><mi>A</mi><mo>∩</mo><mover accent="true"><mi>B</mi><mo stretchy="true">‾</mo></mover><mo>=</mo><mi>A</mi><mover accent="true"><mi>B</mi><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">A-B=A \cap \overline{B} = A \overline B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8833em;"></span><span class="mord mathnormal">A</span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span></span></li><li>德摩根律</li></ol><p>将事件发生的概率论转化为<strong>集合论</strong>进行计算与分析</p><h3 id="1-2-概率的三种定义及其性质">1.2 概率的三种定义及其性质</h3><h4 id="1-2-1-概率的统计定义">1.2.1 概率的统计定义</h4><p>从频率出发得到。</p><h4 id="1-2-2-概率的古典定义">1.2.2 概率的古典定义</h4><p>特征：</p><ul><li>样本空间是有限集</li><li>等可能性（试验中每个基本事件发生的概率是等可能的）</li></ul><p>内容：</p><ol><li>模型与计算公式</li><li>基本组合分析公式<ul><li>乘法、加法原理</li><li>排列公式</li><li>组合公式</li></ul></li><li>实例<ul><li>超几何概率</li><li>分房问题</li><li>生日问题</li></ul></li><li>古典概率的基本性质</li></ol><h4 id="1-2-3-概率的几何定义">1.2.3 概率的几何定义</h4><p>特征：</p><ul><li>样本空间不可列</li><li>等可能性</li></ul><p>内容：</p><ol><li>模型与计算公式</li><li>实例<ul><li>一维几何图形：公交车乘车问题</li><li>二维几何图形：会面问题、蒲丰（Buffon）投针问题</li></ul></li><li>几何概率的基本性质</li></ol><h4 id="1-2-4-概率的性质">1.2.4 概率的性质</h4><p>pass</p><h3 id="1-3-常用概型公式">1.3 常用概型公式</h3><h4 id="1-3-1-条件概率计算公式">1.3.1 条件概率计算公式</h4><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mi mathvariant="normal">∣</mi><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mi>B</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">P(B|A) = \frac{P(AB)}{P(A)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord">∣</span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><h4 id="1-3-2-乘法原理计算公式">1.3.2 乘法原理计算公式</h4><ul><li><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>基本</mtext><mo>:</mo><mtext>前提</mtext><mo>:</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\text{基本}:\text{前提}:P(A)&gt;0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord cjk_fallback">基本</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord cjk_fallback">前提</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mi mathvariant="normal">∣</mi><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(AB) = P(A)P(B|A)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord">∣</span><span class="mord mathnormal">A</span><span class="mclose">)</span></span></span></span></span></p></li><li><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>推广</mtext><mo>:</mo><mtext>前提</mtext><mo>:</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mn>1</mn></msub><msub><mi>A</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>A</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\text{推广}:\text{前提}:P(A_1A_2,...,A_n)&gt;0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord cjk_fallback">推广</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord cjk_fallback">前提</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">...</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mn>1</mn></msub><msub><mi>A</mi><mn>2</mn></msub><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><msub><mi>A</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mn>2</mn></msub><mi mathvariant="normal">∣</mi><msub><mi>A</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mn>3</mn></msub><mi mathvariant="normal">∣</mi><msub><mi>A</mi><mn>1</mn></msub><msub><mi>A</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>⋯</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>n</mi></msub><mi mathvariant="normal">∣</mi><msub><mi>A</mi><mn>1</mn></msub><msub><mi>A</mi><mn>2</mn></msub><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><msub><mi>A</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(A_1A_2...A_n) = P(A_1)P(A_2|A_1)P(A_3|A_1A_2) \cdots P(A_n|A_1A_2...A_{n-1})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">...</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">...</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p></li></ul><h4 id="1-3-3-全概公式">1.3.3 全概公式</h4><p>我们将样本空间 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Ω</span></span></span></span> 完全划分为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个互斥的区域，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>A</mi><mi>i</mi></msub></mstyle></mrow><annotation encoding="application/x-tex">\Omega = \displaystyle \sum_{i=1}^{n} A_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Ω</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> ，则在样本空间中事件 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> 发生的概率 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(B)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span> 就是在各子样本空间中概率之和，经过上述乘法公式变形，计算公式如下：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.16em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mi mathvariant="normal">Ω</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><msub><mi>A</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><msub><mi>A</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mo>⋯</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><msub><mi>A</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mi mathvariant="normal">∣</mi><msub><mi>A</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mi mathvariant="normal">∣</mi><msub><mi>A</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mo>⋯</mo><mo>+</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mi mathvariant="normal">∣</mi><msub><mi>A</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mi mathvariant="normal">∣</mi><msub><mi>A</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{equation*}\begin{aligned}    P(B) &amp;= P(B \Omega) \\    &amp;= P(BA_1) + P(BA_2) + \cdots + P(BA_n) \\    &amp;= P(A_1)P(B|A_1) + P(A_2)P(B|A_2) + \cdots + P(A_n)P(B|A_n) \\    &amp;= \sum_{i=1}^n P(A_i)P(B|A_i)\end{aligned}\end{equation*}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:7.7291em;vertical-align:-3.6145em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.1145em;"><span style="top:-6.1145em;"><span class="pstrut" style="height:6.1145em;"></span><span class="mord"><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.1145em;"><span style="top:-6.9259em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span><span style="top:-5.4259em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"></span></span><span style="top:-3.9259em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"></span></span><span style="top:-1.6145em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.6145em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.1145em;"><span style="top:-6.9259em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord">Ω</span><span class="mclose">)</span></span></span><span style="top:-5.4259em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-3.9259em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-1.6145em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.6145em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.6145em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><h4 id="1-3-4-贝叶斯公式">1.3.4 贝叶斯公式</h4><p>在上述全概公式的背景之下，现在希望求解事件 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> 在第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> 个子样本空间 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">A_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 中发生的概率，或者说第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> 个子样本空间对于事件 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> 的发生贡献了多少概率，记作 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>j</mi></msub><mi mathvariant="normal">∣</mi><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(A_j|B)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span> ，计算公式如下：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>j</mi></msub><mi mathvariant="normal">∣</mi><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mi mathvariant="normal">∣</mi><msub><mi>A</mi><mi>j</mi></msub><mo stretchy="false">)</mo></mrow><mstyle scriptlevel="0" displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mi mathvariant="normal">∣</mi><msub><mi>A</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mstyle></mfrac></mrow><annotation encoding="application/x-tex">P(A_j|B) = \frac{P(A_j)P(B|A_j)}{\displaystyle \sum_{i=1}^n P(A_i)P(B|A_i)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:4.2461em;vertical-align:-2.8191em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.11em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-3.8814em;"><span class="pstrut" style="height:3.6514em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-4.3284em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.8191em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><div class="note note-light">            <p>可以发现全概公式是计算事件发生的所有子样本空间的概率贡献，而贝叶斯公式是计算事件发生的总概率中某些子样本空间的概率贡献，前者是正向思维，后者是逆向思维</p>          </div><h3 id="1-4-事件的独立性及伯努利概型">1.4 事件的独立性及伯努利概型</h3><h4 id="1-4-1-独立性">1.4.1 独立性</h4><p><strong>定义</strong></p><ul><li><p>基本：若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo separator="true">,</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A,B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">A</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> 相互独立，则满足：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(AB)=P(A)P(B)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span></span></p></li><li><p>推广：若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>A</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>A</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">A_1,A_2,...,A_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">...</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 相互独立，则满足：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi mathvariant="normal">∀</mi><mspace width="1em"></mspace><mn>1</mn><mo>≤</mo><msub><mi>i</mi><mn>1</mn></msub><mo>&lt;</mo><msub><mi>i</mi><mn>2</mn></msub><mo>&lt;</mo><mo>⋯</mo><mo>&lt;</mo><msub><mi>i</mi><mi>k</mi></msub><mo>≤</mo><mi>n</mi><mtext>&nbsp;</mtext><mo stretchy="false">(</mo><mi>k</mi><mo>=</mo><mn>2</mn><mo separator="true">,</mo><mn>3</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>s</mi><mi mathvariant="normal">.</mi><mi>t</mi><mi mathvariant="normal">.</mi><mspace width="1em"></mspace><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><msub><mi>i</mi><mn>1</mn></msub></msub><msub><mi>A</mi><msub><mi>i</mi><mn>2</mn></msub></msub><mo>⋯</mo><msub><mi>A</mi><msub><mi>i</mi><mi>k</mi></msub></msub><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><msub><mi>i</mi><mn>1</mn></msub></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><msub><mi>i</mi><mn>2</mn></msub></msub><mo stretchy="false">)</mo><mo>⋯</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>A</mi><msub><mi>i</mi><mi>k</mi></msub></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\forall \quad 1 \le i_1&lt;i_2&lt;\cdots&lt;i_k \le n\ (k=2,3,\cdots,n) \\s.t. \quad P(A_{i_1}A_{i_2}\cdots A_{i_k}) = P(A_{i_1})P(A_{i_2})\cdots P(A_{i_k})\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∀</span><span class="mspace" style="margin-right:1em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">n</span><span class="mspace">&nbsp;</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="mord">.</span><span class="mord mathnormal">t</span><span class="mord">.</span><span class="mspace" style="margin-right:1em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1512em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2559em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1512em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2559em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ul><p><strong>定理</strong></p><ul><li><p>基本：若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo separator="true">,</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A,B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">A</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> 相互独立，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo separator="true">,</mo><mover accent="true"><mi>B</mi><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">A,\overline{B}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">A</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span></span> 相互独立；<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>A</mi><mo stretchy="true">‾</mo></mover><mo separator="true">,</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\overline{A},B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0778em;vertical-align:-0.1944em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> 相互独立；<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>A</mi><mo stretchy="true">‾</mo></mover><mo separator="true">,</mo><mover accent="true"><mi>B</mi><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\overline{A},\overline{B}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0778em;vertical-align:-0.1944em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span></span> 相互独立</p></li><li><p>推广：若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>A</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>A</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">A_1,A_2,...,A_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">...</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 相互独立，则其中任意 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo stretchy="false">(</mo><mn>2</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">k(2 \le k \le n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mopen">(</span><span class="mord">2</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 个也相互独立，且满足：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>P</mi><mo stretchy="false">(</mo><mover accent="true"><msub><mi>A</mi><msub><mi>i</mi><mn>1</mn></msub></msub><mo>^</mo></mover><mover accent="true"><msub><mi>A</mi><msub><mi>i</mi><mn>2</mn></msub></msub><mo>^</mo></mover><mo>⋯</mo><mover accent="true"><msub><mi>A</mi><msub><mi>i</mi><mi>k</mi></msub></msub><mo>^</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mover accent="true"><msub><mi>A</mi><msub><mi>i</mi><mn>1</mn></msub></msub><mo>^</mo></mover><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mover accent="true"><msub><mi>A</mi><msub><mi>i</mi><mn>2</mn></msub></msub><mo>^</mo></mover><mo stretchy="false">)</mo><mo>⋯</mo><mi>P</mi><mo stretchy="false">(</mo><mover accent="true"><msub><mi>A</mi><msub><mi>i</mi><mi>k</mi></msub></msub><mo>^</mo></mover><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>s</mi><mi mathvariant="normal">.</mi><mi>t</mi><mi mathvariant="normal">.</mi><mspace width="1em"></mspace><mover accent="true"><msub><mi>A</mi><msub><mi>i</mi><mi>j</mi></msub></msub><mo>^</mo></mover><mo>=</mo><mi>A</mi><mtext>&nbsp;</mtext><mi>o</mi><mi>r</mi><mtext>&nbsp;</mtext><mover accent="true"><mi>A</mi><mo stretchy="true">‾</mo></mover><mtext>&nbsp;</mtext><mo stretchy="false">(</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}P(\hat{A_{i_1}}\hat{A_{i_2}}\cdots \hat{A_{i_k}}) = P(\hat{A_{i_1}})P(\hat{A_{i_2}})\cdots P(\hat{A_{i_k}}) \\s.t. \quad \hat{A_{i_j}} = A \ or \ \overline{A}\ (j=1,2,\cdots,k)\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.2135em;vertical-align:-1.3568em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8568em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span></span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">^</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span></span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">^</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1512em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2559em;"><span></span></span></span></span></span></span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">^</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2559em;"><span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span></span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">^</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span></span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">^</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3488em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1512em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2559em;"><span></span></span></span></span></span></span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">^</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2559em;"><span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-2.3032em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="mord">.</span><span class="mord mathnormal">t</span><span class="mord">.</span><span class="mspace" style="margin-right:1em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2819em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3473em;"><span></span></span></span></span></span></span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">^</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3473em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">A</span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right:0.02778em;">or</span><span class="mspace">&nbsp;</span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace">&nbsp;</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3568em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ul><p><strong>概念辨析</strong></p><ul><li><p>两两独立：对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个事件，两两独立，而不考虑三个及以上的关系。</p></li><li><p>相互独立：对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个事件，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><mo>→</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">2 \to n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个事件的独立关系都需要考虑。</p></li><li><p>总结：对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个事件，满足两两独立需要 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>C</mi><mi>n</mi><mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">C_n^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0611em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span> 个等式关系，对于相互独立需要 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>n</mi></msup><mo>−</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">2^n-(n+1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7477em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 个等式关系，因此：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>两两独立</mtext><mo>⊂</mo><mtext>相互独立</mtext></mrow><annotation encoding="application/x-tex">\text{两两独立} \subset \text{相互独立}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord text"><span class="mord cjk_fallback">两两独立</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⊂</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord cjk_fallback">相互独立</span></span></span></span></span></span></p></li></ul><h4 id="1-4-2-伯努利概型">1.4.2 伯努利概型</h4><p>定义：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 重伯努利概型</p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 重：发生 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 次独立试验</li><li>伯努利概型：每次试验只有两种可能的结果</li></ul><p>模型：</p><ul><li><p>二项概率公式：n 次独立重复试验发生 k 次的概率：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mi>C</mi><mi>n</mi><mi>k</mi></msubsup><msup><mi>p</mi><mi>k</mi></msup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><msup><mo stretchy="false">)</mo><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup></mrow><annotation encoding="application/x-tex">C_n^k p^k (1-p)^{n-k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1491em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1491em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span></span></span></span></span></p></li><li><p>几何概率公式：在第 n 次试验首次成功的概率：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><msup><mo stretchy="false">)</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>p</mi></mrow><annotation encoding="application/x-tex">(1-p)^{n-1}p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord mathnormal">p</span></span></span></span></span></p></li></ul><h2 id="第2章-随机事件及其分布">第2章 随机事件及其分布</h2><div class="note note-light">            <p>我们知道，解决事件发生概率的问题，除了事件表示以外，我们还关心每一个事件发生的概率 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(X=k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span>，以及某些事件发生的概率 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mo stretchy="false">[</mo><mi>r</mi><mi>a</mi><mi>n</mi><mi>g</mi><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(X=[range))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">an</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord mathnormal">e</span><span class="mclose">))</span></span></span></span>。接下来我们将：</p><ul><li>首先介绍<strong>随机变量的概念</strong>以及<strong>分布函数的概念</strong></li><li>接着介绍随机变量对应的概率发生情况组成的集合。离散型的叫<strong>分布列</strong>，连续型的叫<strong>概率密度函数</strong>，并在其中贯穿分布函数的应用</li><li>最后介绍<strong>分布函数的复合</strong>。从离散型和连续型随机变量两个方向展开</li></ul>          </div><h3 id="2-1-随机变量及其概率分布">2.1 随机变量及其概率分布</h3><h4 id="2-1-1-随机变量的概念">2.1.1 随机变量的概念</h4><p>总的来说，随机变量就是一个样本空间与实数集的映射。我们定义样本空间 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi><mo>=</mo><mo stretchy="false">{</mo><mi>ω</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\Omega=\{ \omega \}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">Ω</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="mclose">}</span></span></span></span>，其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span></span></span></span> 表示所有可能的事件，实数集 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span></span></span></span>，随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span>，则随机变量满足以下映射关系</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>X</mi><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>=</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">X(\omega)=R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span></span></span></span></span></p><h4 id="2-1-2-随机变量的分布函数">2.1.2 随机变量的分布函数</h4><ol><li>分布函数的定义：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(x)=P(X \le x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></li><li>分布函数的性质：<ul><li>非负有界性：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≤</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">0 \le F(x) \le 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></li><li>单调不减性：若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>&lt;</mo><msub><mi>x</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_1 &lt; x_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>≤</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(x_1) \le F(x_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mi>F</mi><mo stretchy="false">(</mo><mo>−</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><mo>−</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle F(-\infty) = \lim_{x \to -\infty} F(x) = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord">−</span><span class="mord">∞</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.5083em;vertical-align:-0.7583em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mrel mtight">→</span><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7583em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mi>F</mi><mo stretchy="false">(</mo><mo>+</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle F(+\infty) = \lim_{x \to +\infty} F(x) = 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord">+</span><span class="mord">∞</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.5083em;vertical-align:-0.7583em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mrel mtight">→</span><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7583em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></li><li>右连续性：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><msubsup><mi>x</mi><mn>0</mn><mo>+</mo></msubsup></mrow></munder><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mspace width="1em"></mspace><mo stretchy="false">(</mo><mo>−</mo><mi mathvariant="normal">∞</mi><mo>&lt;</mo><msub><mi>x</mi><mn>0</mn></msub><mo>&lt;</mo><mo>+</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \lim_{x\to x_0^+}F(x) = F(x_0)\quad(-\infty &lt; x_0 &lt; +\infty)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.8087em;vertical-align:-1.0587em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.2593em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mrel mtight">→</span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.821em;"><span style="top:-2.1885em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span><span style="top:-2.9043em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mbin mtight">+</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3115em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.0587em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:1em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">∞</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">+</span><span class="mord">∞</span><span class="mclose">)</span></span></span></span></li></ul></li></ol><h3 id="2-2-离散型随机变量及其分布列">2.2 离散型随机变量及其分布列</h3><h4 id="2-2-1-离散性随机变量的分布列">2.2.1 离散性随机变量的分布列</h4><p>随机变量的取值都是整数，有以下三种表示方法</p><ol><li><p>公式法</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>p</mi><mi>k</mi></msub><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mo separator="true">,</mo><mspace width="1em"></mspace><mi>k</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo></mrow><annotation encoding="application/x-tex">p_k = P(X=x_k),\quad k = 1,2,\cdots,</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span></span></span></span></span></p></li><li><p>服从法</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>X</mi><mo>∼</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>1</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>2</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>3</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>p</mi><mn>1</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>p</mi><mn>2</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>p</mi><mn>3</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">X \sim \begin{pmatrix}x_1 &amp; x_2 &amp; x_3 &amp; \cdots \\p_1 &amp; p_2 &amp; p_3 &amp; \cdots\end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="minner">⋯</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p></li><li><p>表格法</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.16em" columnalign="center center center center center" columnlines="solid none none none" columnspacing="1em" rowlines="solid"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>X</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>1</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>2</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>3</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>P</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>p</mi><mn>1</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>p</mi><mn>2</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>p</mi><mn>3</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">⋯</mo></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{array}{c|cccc}X &amp; x_1 &amp; x_2 &amp; x_3 &amp; \cdots \\\hlineP &amp; p_1 &amp; p_2 &amp; p_3 &amp; \cdots\end{array}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.45em;"><span class="pstrut" style="height:3.45em;"></span><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="vertical-separator" style="height:2.4em;border-right-width:0.04em;border-right-style:solid;margin:0 -0.02em;vertical-align:-0.95em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="minner">⋯</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span style="top:-3.7em;"><span class="pstrut" style="height:3.45em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span></span></span></p></li></ol><h4 id="2-2-2-常用离散性随机变量及其分布列">2.2.2 常用离散性随机变量及其分布列</h4><ul><li><p>0-1分布：即一个事件只有两面性，我们称这样的随机变量服从0-1分布或者两点分布，记作</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>X</mi><mo>∼</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><mo>−</mo><mi>p</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>p</mi></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">X \sim \begin{pmatrix}0 &amp; 1 \\1-p &amp; p\end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p></li><li><p>二项分布：其实就是 n 重伯努利试验，我们称这样的随机变量服从二项分布，分布列为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>C</mi><mi>n</mi><mi>k</mi></msubsup><msup><mi>p</mi><mi>k</mi></msup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><msup><mo stretchy="false">)</mo><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup></mrow><annotation encoding="application/x-tex">P(X=k) = C_n^kp^k(1-p)^{n-k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0991em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0991em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span></span></span></span>，记作</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>B</mi><mo stretchy="false">(</mo><mi>n</mi><mo separator="true">,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sim B(n,p)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mclose">)</span></span></span></span></span></p></li><li><p>几何分布：同样是伯努利事件，现在需要求解第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 次事件首次发生的概率，此时分布列为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><msup><mo stretchy="false">)</mo><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>p</mi></mrow><annotation encoding="application/x-tex">P(X=k)=(1-p)^{k-1}p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0991em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mord mathnormal">p</span></span></span></span>，记作</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>G</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sim G(p)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">G</span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mclose">)</span></span></span></span></span></p></li><li><p>超几何分布：就是在 N 件含有 M 件次品的样品中无放回的抽取 n 件，问其中含有次品数量的分布列，为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msubsup><mi>C</mi><mi>M</mi><mi>k</mi></msubsup><msubsup><mi>C</mi><mrow><mi>N</mi><mo>−</mo><mi>M</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msubsup></mrow><msubsup><mi>C</mi><mi>N</mi><mi>n</mi></msubsup></mfrac><mo separator="true">,</mo><mspace width="1em"></mspace><mi>k</mi><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mi>min</mi><mo>⁡</mo><mrow><mo stretchy="false">(</mo><mi>n</mi><mo separator="true">,</mo><mi>M</mi><mo stretchy="false">)</mo></mrow></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle P(X=k)=\frac{C_M^k C_{N-M}^{n-k}}{C_N^n}, \quad k=0,1,2,\cdots,\min{(n, M)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.6106em;vertical-align:-0.9795em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6311em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6462em;"><span style="top:-2.4065em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2935em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.7419em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-2.4247em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2753em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8892em;"><span style="top:-2.4065em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span></span></span></span><span style="top:-3.1031em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3519em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9795em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">min</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mclose">)</span></span></span></span></span>，记作</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>X</mi><mo>∼</mo><mtext>超几何分布</mtext><mo stretchy="false">(</mo><mi>n</mi><mo separator="true">,</mo><mi>N</mi><mo separator="true">,</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sim \text{超几何分布}(n,N,M)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord cjk_fallback">超几何分布</span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mclose">)</span></span></span></span></span></p></li><li><p>泊松分布：当二项分布中，试验次数很大或者概率很小时，可以近似为泊松分布，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>C</mi><mi>n</mi><mi>k</mi></msubsup><msup><mi>p</mi><mi>k</mi></msup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><msup><mo stretchy="false">)</mo><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup><mo>→</mo><mfrac><msup><mi>λ</mi><mi>k</mi></msup><mrow><mi>k</mi><mo stretchy="false">!</mo></mrow></mfrac><msup><mi>e</mi><mrow><mo>−</mo><mi>λ</mi></mrow></msup></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle P(X=k)=C_n^k p^k(1-p)^{n-k} \to \frac{\lambda^k}{k!}e^{-\lambda}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1491em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1491em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2121em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5261em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">!</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">λ</span></span></span></span></span></span></span></span></span></span></span></span>，其中常数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\lambda &gt; 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">λ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，记作</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>P</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sim P(\lambda)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal">λ</span><span class="mclose">)</span></span></span></span></span></p><p>显然，泊松分布含有下面两个性质</p><ol><li><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>&gt;</mo><mn>0</mn><mo separator="true">,</mo><mi>k</mi><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo separator="true">,</mo><mo>⋯</mo></mrow><annotation encoding="application/x-tex">P(X=k) &gt; 0,k=0,1,\cdots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span></span></span></span></p></li><li><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \sum_{k=0}^\infty P(X=k)=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.9535em;vertical-align:-1.3021em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8479em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">∞</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3021em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></p>     <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-60042435" role="button" aria-expanded="false" aria-controls="collapse-60042435">        <div class="fold-arrow">▶</div>泊松分布正规性证明      </div>      <div class="fold-collapse collapse" id="collapse-60042435">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403151012817.png" alt="泊松分布正规性证明"></p>        </div>      </div>    </div></li></ol></li></ul><h3 id="2-3-连续型随机变量及其概率密度函数">2.3 连续型随机变量及其概率密度函数</h3><p>说白了其实就是离散性随机变量的积分加强版。现在随着事件发生的不同取值 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>，随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 发生的概率 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(X=x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 变成了连续的取值了（学名概率密度函数），于是分布函数（离散的叫分布列）的取值就没那么容易求了（其实一重定积分就可以）。接下来就从定义、性质、应用三个角度出发介绍概率密度函数以及相应的随机变量的分布函数。</p><h4 id="2-3-1-连续型随机变量的密度函数">2.3.1 连续型随机变量的密度函数</h4><p>概率密度函数，简称：密度函数 or 概率密度</p><ul><li><p>定义：设随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的分布函数为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>，如果存在非负可积函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>，使下式成立，则称 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 为连续型随机变量，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的概率密度函数</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">∀</mi><mi>x</mi><mo>∈</mo><mi>R</mi><mo separator="true">,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi>x</mi></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi>d</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">\forall x \in R,F(x) = \int_{-\infty}^{x} p(t)dt</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord">∀</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3846em;vertical-align:-0.9703em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span></span></span></p></li><li><p>性质：</p><ol><li><p>非负性：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">p(x) \ge 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></p></li><li><p>正规性：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\int_{-\infty}^{+\infty} p(x)dx = 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3804em;vertical-align:-0.4142em;"></span><span class="mop"><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9662em;"><span style="top:-2.3442em;margin-left:-0.1945em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.2579em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4142em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></p></li><li><p>可积性：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∀</mi><msub><mi>x</mi><mn>1</mn></msub><mo>≤</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo>≤</mo><mi>X</mi><mo>≤</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>−</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><msub><mi>x</mi><mn>1</mn></msub><msub><mi>x</mi><mn>2</mn></msub></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">\forall x_1 \le x_2,P(x_1 \le X \le x_2) = F(x_2) - F(x_1) = \int_{x_1}^{x_2}p(x)dx</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord">∀</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.3152em;vertical-align:-0.4559em;"></span><span class="mop"><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8593em;"><span style="top:-2.3442em;margin-left:-0.1945em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.2579em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4559em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span></span></p></li><li><p>分布函数可导性：若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 在点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 处连续，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>F</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F'(x) = p(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0019em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></p></li><li><p>已知事件但无意义性：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∀</mi><mi>x</mi><mo>∈</mo><mi>R</mi><mo separator="true">,</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\forall x \in R, P(X=x) = F(x) - F(x) = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord">∀</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></p><ul><li>离散型变量可以通过列举随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的取值来计算概率，但连续型随机变量这么做是无意义的</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">P(A) = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 不能推出 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 是不可能事件，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">P(A)=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 不能推出 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 是必然事件</li><li>对于连续型随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 有：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo>&lt;</mo><mi>X</mi><mo>&lt;</mo><msub><mi>X</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo>&lt;</mo><mi>X</mi><mo>≤</mo><msub><mi>X</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo>≤</mo><mi>X</mi><mo>&lt;</mo><msub><mi>X</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo>≤</mo><mi>X</mi><mo>≤</mo><msub><mi>X</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(x_1 &lt; X &lt; X_2)=P(x_1 &lt; X \le X_2)=P(x_1 \le X &lt; X_2)=P(x_1 \le X \le X_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7224em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></li></ul></li><li><p>实际描述性：密度函数的数值反映了随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 取 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 的临近值的概率的大小，因为</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi mathvariant="normal">Δ</mi><mi>x</mi><mo>≈</mo><msubsup><mo>∫</mo><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mi>d</mi><mi>t</mi><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo>≤</mo><mi>X</mi><mo>≤</mo><mi>x</mi><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(x)\Delta x \approx \int_{x}^{x+\Delta x} p(t)dt = F(x+\Delta x) - F(x) = P(x \le X \le x+\Delta x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord">Δ</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.5032em;vertical-align:-0.9119em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5912em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mbin mtight">+</span><span class="mord mtight">Δ</span><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Δ</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.136em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Δ</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></p></li></ol></li></ul><h4 id="2-3-2-常用连续型随机变量及其密度函数">2.3.2 常用连续型随机变量及其密度函数</h4><table><thead><tr><th style="text-align:center"></th><th style="text-align:center">分布定义式</th><th style="text-align:center">概率密度函数</th><th style="text-align:center">分布函数</th></tr></thead><tbody><tr><td style="text-align:center">均匀分布</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>U</mi><mo stretchy="false">[</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">X \sim U[a,b]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mopen">[</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mclose">]</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mfrac><mn>1</mn><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow></mfrac><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>a</mi><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>b</mi><mo separator="true">,</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>0</mn><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>其他</mtext></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">p(x) = \begin{cases} \frac{1}{b-a}, &amp; a \le x \le b, \\ 0, &amp; \text{其他} \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight">a</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4033em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">0</span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">b</span><span class="mpunct">,</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">其他</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>0</mn><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>x</mi><mo>&lt;</mo><mi>a</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mfrac><mrow><mi>x</mi><mo>−</mo><mi>a</mi></mrow><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow></mfrac><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>a</mi><mo>≤</mo><mi>x</mi><mo>&lt;</mo><mi>b</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>x</mi><mo>≥</mo><mi>b</mi></mrow></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">F(x) = \begin{cases} 0, &amp; x &lt; a \\ \frac{x - a}{b - a}, &amp; a \le x &lt; b \\ 1, &amp; x \ge b \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:4.32em;vertical-align:-1.91em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35em;"><span style="top:-2.2em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.192em;"><span class="pstrut" style="height:3.15em;"></span><span style="height:0.316em;width:0.8889em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.8889em" height="0.316em" style="width:0.8889em" viewBox="0 0 888.89 316" preserveAspectRatio="xMinYMin"><path d="M384 0 H504 V316 H384z M384 0 H504 V316 H384z"></path></svg></span></span><span style="top:-3.15em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.292em;"><span class="pstrut" style="height:3.15em;"></span><span style="height:0.316em;width:0.8889em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.8889em" height="0.316em" style="width:0.8889em" viewBox="0 0 888.89 316" preserveAspectRatio="xMinYMin"><path d="M384 0 H504 V316 H384z M384 0 H504 V316 H384z"></path></svg></span></span><span style="top:-4.6em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.85em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em;"><span style="top:-4.41em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">0</span><span class="mpunct">,</span></span></span><span style="top:-2.97em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8023em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight">a</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4033em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span></span></span><span style="top:-1.53em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">1</span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.91em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em;"><span style="top:-4.41em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">a</span></span></span><span style="top:-2.97em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">b</span></span></span><span style="top:-1.53em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.91em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td></tr><tr><td style="text-align:center">指数分布</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>e</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sim e (\lambda)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">e</span><span class="mopen">(</span><span class="mord mathnormal">λ</span><span class="mclose">)</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>0</mn><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>x</mi><mo>&lt;</mo><mn>0</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>λ</mi><msup><mi>e</mi><mrow><mo>−</mo><mi>λ</mi><mi>x</mi></mrow></msup><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>x</mi><mo>≥</mo><mn>0</mn></mrow></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">p(x) = \begin{cases} 0, &amp; x &lt; 0 \\ \lambda e^{-\lambda x} , &amp; x \ge 0 \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">0</span><span class="mpunct">,</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">λ</span><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>0</mn><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>x</mi><mo>&lt;</mo><mn>0</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><mi>λ</mi><mi>x</mi></mrow></msup><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>x</mi><mo>≥</mo><mn>0</mn></mrow></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">F(x) = \begin{cases} 0, &amp; x &lt; 0 \\ 1- e^{-\lambda x}, &amp; x \ge 0 \end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">0</span><span class="mpunct">,</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight">λ</span><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td></tr><tr><td style="text-align:center">正态分布</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><mi>μ</mi><mo separator="true">,</mo><msup><mi>σ</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sim N(\mu,\sigma^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mopen">(</span><span class="mord mathnormal">μ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><msqrt><mrow><mn>2</mn><mi>π</mi></mrow></msqrt><mi>σ</mi></mrow></mfrac><msup><mi>e</mi><mrow><mo>−</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>μ</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mrow><mn>2</mn><msup><mi>σ</mi><mn>2</mn></msup></mrow></mfrac></mrow></msup><mo separator="true">,</mo><mspace width="1em"></mspace><mo>−</mo><mi mathvariant="normal">∞</mi><mo>&lt;</mo><mi>x</mi><mo>&lt;</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow><annotation encoding="application/x-tex">p(x) = \frac{1}{\sqrt{2 \pi} \sigma } e^{- \frac{(x - \mu)^2}{2 \sigma ^2}} , \quad -\infty &lt; x &lt; + \infty</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.8671em;vertical-align:-0.538em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.551em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord sqrt mtight"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9128em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight" style="padding-left:0.833em;"><span class="mord mtight">2</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">π</span></span></span><span style="top:-2.8728em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail mtight" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1272em;"><span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.538em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.3291em;"><span style="top:-3.4534em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.251em;"><span style="top:-2.5062em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9384em;"><span style="top:-2.9384em;margin-right:0.1em;"><span class="pstrut" style="height:2.6444em;"></span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.5021em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight">x</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight">μ</span><span class="mclose mtight"><span class="mclose mtight">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0484em;"><span style="top:-3.0484em;margin-right:0.1em;"><span class="pstrut" style="height:2.6444em;"></span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4938em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord">∞</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">+</span><span class="mord">∞</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><msqrt><mrow><mn>2</mn><mi>π</mi></mrow></msqrt><mi>σ</mi></mrow></mfrac><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi>x</mi></msubsup><msup><mi>e</mi><mrow><mo>−</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>y</mi><mo>−</mo><mi>μ</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mrow><mn>2</mn><msup><mi>σ</mi><mn>2</mn></msup></mrow></mfrac></mrow></msup><mi>d</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">F(x) = \frac{1}{\sqrt{2 \pi} \sigma } \int_{- \infty}^x e^{- \frac{(y - \mu)^2}{2 \sigma ^2}} dy</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.8671em;vertical-align:-0.538em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.551em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord sqrt mtight"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9128em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight" style="padding-left:0.833em;"><span class="mord mtight">2</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">π</span></span></span><span style="top:-2.8728em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail mtight" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1272em;"><span></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.538em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0006em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8593em;"><span style="top:-2.3442em;margin-left:-0.1945em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.2579em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4142em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.3291em;"><span style="top:-3.4534em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.251em;"><span style="top:-2.5062em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9384em;"><span style="top:-2.9384em;margin-right:0.1em;"><span class="pstrut" style="height:2.6444em;"></span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.5021em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight">μ</span><span class="mclose mtight"><span class="mclose mtight">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0484em;"><span style="top:-3.0484em;margin-right:0.1em;"><span class="pstrut" style="height:2.6444em;"></span><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4938em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span></td></tr></tbody></table><p>补充说明：</p><ul><li><p>指数分布：其中参数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\lambda &gt;0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">λ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></p></li><li><p>正态分布：一般正态函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> <strong>转化为标准正态函数</strong> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Φ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Phi(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Φ</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 公式：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">Φ</mi><mo stretchy="false">(</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mi>μ</mi></mrow><mi>σ</mi></mfrac><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(x) = \Phi(\frac{x - \mu}{\sigma})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.9463em;vertical-align:-0.686em;"></span><span class="mord">Φ</span><span class="mopen">(</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span></span></span></span></span></p><p>于是对于计算一般正态函数的函数值，就可以通过下式将其转化为标准正态函数，最后查表即可：</p></li></ul><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">Φ</mi><mo stretchy="false">(</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mi>μ</mi></mrow><mi>σ</mi></mfrac><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(X \le x) = F(x) = \Phi (\frac{x - \mu}{\sigma})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.9463em;vertical-align:-0.686em;"></span><span class="mord">Φ</span><span class="mopen">(</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span></span></span></span></span></p><h3 id="2-4-随机变量函数的分布">2.4 随机变量函数的分布</h3><div class="note note-light">            <p>本目主要介绍给定一个随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的分布情况，通过一个关系式 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y=g(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 来求解随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span>​ 的分布情况</p>          </div><h4 id="2-4-1-离散型随机变量函数的分布">2.4.1 离散型随机变量函数的分布</h4><p>通过关系式 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y=g(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>​ 将所有的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 的取值全部枚举出来，然后一一统计即可。</p><h4 id="2-4-2-连续型随机变量函数的分布">2.4.2 连续型随机变量函数的分布</h4><p>给定随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的概率密度函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p_X(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>，以及关系式 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y=g(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>，求解随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 的分布函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F_Y(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span>、概率密度函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p_Y(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span></p><ul><li><p><strong>方法一</strong>：先求解随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 的分布函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F_Y(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span>，再通过对其求导得到概率密度函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p_Y(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span>​</p><p>即先 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>P</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>≤</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>P</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>P</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F_Y(y) = P_Y(Y \le y) = P_Y(g(X) \le y) = P_X(X \le f(y)) = F_X(f(y))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">))</span></span></span></span> 得到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 的分布函数</p><p>再对 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F_Y(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> 求导得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><msub><mi>p</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>y</mi></mrow></mfrac><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>y</mi></mrow></mfrac><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>F</mi><mi>X</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msubsup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋅</mo><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>p</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋅</mo><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle p_Y(y) = \frac{d}{dy} F_Y(y) = \frac{d}{dy} F_X(f(y)) = F_X'(f(y)) \cdot f'(y) = p_X(f(y)) \cdot f'(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.2519em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0519em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-2.453em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0519em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0519em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span></p></li><li><p><strong>方法二</strong>：如果关系式 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y=g(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 单调且反函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mi>h</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x=h(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> 连续可导，则可以直接得出随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 的概率密度函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p_Y(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> 为下式。其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span> 为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Y=g(X)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span> 的取值范围（<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 应该怎么取值，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> 就应该怎么取值，从而计算出 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> 的取值范围）</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>p</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>p</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋅</mo><mi mathvariant="normal">∣</mi><msup><mi>h</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mi mathvariant="normal">∣</mi><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>α</mi><mo>&lt;</mo><mi>y</mi><mo>&lt;</mo><mi>β</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>0</mn><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>其他</mtext></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">p_Y(y) = \begin{cases}p_X(h(y)) \cdot |h'(y)|, &amp; \alpha &lt; y &lt; \beta \\0, &amp; \text{其他}\end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">h</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mord">∣</span><span class="mpunct">,</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">0</span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">其他</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p></li></ul><h2 id="第3章-随机向量及其分布">第3章 随机向量及其分布</h2><div class="note note-light">            <p>实际生活中，只采用一个随机变量描述事件往往是不够的。本章引入多维的随机变量概念，构成随机向量，从二维开始，推广到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>​ 维。</p>          </div><h3 id="3-1-二维随机向量的联合分布">3.1 二维随机向量的联合分布</h3><div class="note note-light">            <p>现在我们讨论二维随机向量的联合分布。所谓的联合分布，其实就是一个曲面的概率密度（离散型就是点集），而分布函数就是对其积分得到的三维几何体的体积（散点和）而已。</p>          </div><h4 id="3-1-1-联合分布函数">3.1.1 联合分布函数</h4><p>定义：我们定义满足下式的二元函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> 为二维随机向量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span> 的联合分布函数</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mi>x</mi><mo stretchy="false">)</mo><mo>∩</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>≤</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mi>x</mi><mo separator="true">,</mo><mi>Y</mi><mo>≤</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(x,y) = P((X \le x) \cap (Y \le y)) = P(X \le x, Y \le y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">((</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">))</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span></span></p>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-7ae072b2" role="button" aria-expanded="false" aria-controls="collapse-7ae072b2">        <div class="fold-arrow">▶</div>几何意义：F(x,y) 即左下方无界矩形的面积      </div>      <div class="fold-collapse collapse" id="collapse-7ae072b2">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403271717479.png" alt="联合分布函数的几何意义"></p>        </div>      </div>    </div><p>性质：其实配合几何意义理解就会很容易了</p><ol><li>固定某一维度，另一维度是单调不减的</li><li>对于每个维度都是右连续的</li><li>固定某一维度，另一维度趋近于负无穷对应的函数值为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></li><li>二维前缀和性质，右上角的矩阵面积 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\ge 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></li></ol><h4 id="3-1-2-联合分布列">3.1.2 联合分布列</h4><p>定义：若二维随机向量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span> 的所有可能取值是至多可列的，则称 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span> 为二维离散型随机向量</p><p>表示：有两种表示二维随机向量分布列的方法，如下</p>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-819aef24" role="button" aria-expanded="false" aria-controls="collapse-819aef24">        <div class="fold-arrow">▶</div>二维随机向量分布列的表示方法      </div>      <div class="fold-collapse collapse" id="collapse-819aef24">        <div class="fold-content">          <ol><li><p>公式法</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>p</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><msub><mi>x</mi><mi>i</mi></msub><mo separator="true">,</mo><mi>Y</mi><mo>=</mo><msub><mi>y</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo separator="true">,</mo><mspace width="1em"></mspace><mi>i</mi><mo separator="true">,</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo></mrow><annotation encoding="application/x-tex">p_{ij} = P(X=x_i,Y = y_i), \quad i,j=1,2,\cdots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">i</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span></span></span></span></span></p></li><li><p>表格法：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403271728248.png" alt="二维联合分布列"></p></li></ol>        </div>      </div>    </div><p>性质：</p><ol><li>非负性：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>≥</mo><mn>0</mn><mo separator="true">,</mo><mspace width="1em"></mspace><mi>i</mi><mo separator="true">,</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo></mrow><annotation encoding="application/x-tex">p_{ij} \ge 0, \quad i,j=1,2,\cdots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9221em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">i</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span></span></span></span></li><li>正规性：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><munder><mo>∑</mo><mi>i</mi></munder><munder><mo>∑</mo><mi>j</mi></munder><msub><mi>p</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mn>1</mn></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \sum_{i} \sum_{j} p_{ij} = 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4638em;vertical-align:-1.4138em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4138em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></li></ol><h4 id="3-1-3-联合密度函数">3.1.3 联合密度函数</h4><p>定义：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi>x</mi></msubsup><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi>y</mi></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo stretchy="false">)</mo><mi>d</mi><mi>u</mi><mi>d</mi><mi>v</mi></mrow><annotation encoding="application/x-tex">F(x,y) = \int_{-\infty}^x \int_{-\infty}^y p(u,v)dudv</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3846em;vertical-align:-0.9703em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">u</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span></span></span></p><p>性质：</p><ol><li>非负性：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∀</mi><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>∈</mo><mi>R</mi><mo separator="true">,</mo><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\forall x,y \in R,p(x,y) \ge 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord">∀</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></li><li>正规性：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mi>d</mi><mi>y</mi><mo>=</mo><mn>1</mn></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} p(x,y)dxdy = 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4915em;vertical-align:-0.9703em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></li></ol><p>结论：</p><ol><li><strong>联合分布函数</strong>相比于一元分布函数，其实就是从概率密度函数与 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 轴围成的面积转变为了概率密度曲面与 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mi>O</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">xOy</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>​ 平面围成的<strong>体积</strong></li><li>若概率密度曲面在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mi>O</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">xOy</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> 平面的投影为点集或线集，则对应的概率显然为零</li></ol><p>常见的连续型二维分布：</p><ol><li><p>二维均匀分布：假设该曲面与 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mi>O</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">xOy</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> 面的投影面积为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span>，则分布函数其实就是一个高为定值 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mn>1</mn><mi>S</mi></mfrac></mrow><annotation encoding="application/x-tex">\frac{1}{S}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1901em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">S</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 的柱体，密度函数为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mfrac><mn>1</mn><mi>S</mi></mfrac><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>G</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>0</mn><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>其他</mtext></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">p(x,y) = \begin{cases}\frac{1}{S}, &amp;(x,y) \in G \\0, &amp;\text{其他}\end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8451em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05764em;">S</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">0</span><span class="mpunct">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">G</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">其他</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p></li><li><p>二元正态分布：不要求掌握密度函数，可以感受一下密度函数的图像：</p>     <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-e058054e" role="button" aria-expanded="false" aria-controls="collapse-e058054e">        <div class="fold-arrow">▶</div>二元正态分布 - 密度函数的图像      </div>      <div class="fold-collapse collapse" id="collapse-e058054e">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403290948792.png" alt="二元正态分布 - 密度函数的图像"></p>        </div>      </div>    </div></li></ol><p>计算题：往往给出一个二元密度函数，然后让我们求解（1）密度函数中的参数、（2）分布函数、（3）联合事件某个区域下的概率</p><p>（1）我们利用二元密度函数的正规性，直接积分值为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 即可</p><p>（2）划分区间后进行曲面积分即可，在曲面积分时往往结合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 型和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 型的二重积分进行</p><p>（3）画出概率密度曲面在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mi>O</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">xOy</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> 面的投影，然后积分即可</p><h3 id="3-2-二维随机向量的边缘分布">3.2 二维随机向量的边缘分布</h3><div class="note note-light">            <p>对于二元分布函数，我们也可以研究其中任意一个随机变量的分布情况，而不需要考虑另一个随机变量的取值情况。举一个实例就是，假如当前的随机向量是身高和体重，所谓的只研究其中一个随机变量，即边缘分布函数的情形就是，我们不考虑身高只考虑体重的分布情况；或者我们不考虑体重，只考虑身高的分布情况。接下来，我们将从边缘分布函数入手，逐渐学习离散型的分布列与连续型的分布函数。</p>          </div><h4 id="3-2-1-边缘分布函数">3.2.1 边缘分布函数</h4><p>我们称 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">F_X(x),F_Y(y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>) 分别为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span> 关于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo separator="true">,</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X,Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 的边缘分布函数，定义式为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mi>x</mi><mo separator="true">,</mo><mi>Y</mi><mo>&lt;</mo><mo>+</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>y</mi><mo>→</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mo>+</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>≤</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>&lt;</mo><mo>+</mo><mi mathvariant="normal">∞</mi><mo separator="true">,</mo><mi>Y</mi><mo>≤</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mrow><mi>lim</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>→</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munder><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><mo>+</mo><mi mathvariant="normal">∞</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}F_X(x) = P(X \le x) = P(X \le x,Y &lt; +\infty) = \lim_{y \to +\infty} F(x,y) = F(x,+\infty) \\F_Y(y) = P(Y \le y) = P(X &lt; +\infty, Y \le y) = \lim_{x \to +\infty} F(x,y) = F(+\infty,y)\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.8744em;vertical-align:-1.6872em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.1872em;"><span style="top:-4.3472em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">+</span><span class="mord">∞</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="mrel mtight">→</span><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8361em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">+</span><span class="mord">∞</span><span class="mclose">)</span></span></span><span style="top:-2.3711em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">+</span><span class="mord">∞</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.4em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mrel mtight">→</span><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7583em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord">+</span><span class="mord">∞</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.6872em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><h4 id="3-2-2-边缘分布列">3.2.2 边缘分布列</h4><p>所谓的边缘分布列，就是固定一个随机变量，另外的随机变量取遍，组成的分布列。即：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>p</mi><mrow><mi>i</mi><mo>⋅</mo></mrow></msub><mo>=</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munderover><msub><mi>p</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo separator="true">,</mo><mspace width="1em"></mspace><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><msub><mi>y</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>p</mi><mrow><mo>⋅</mo><mi>j</mi></mrow></msub><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munderover><msub><mi>p</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo separator="true">,</mo><mspace width="1em"></mspace><mi>j</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}P(X=x_i) = p_{i\cdot}=\sum_{j=1}^{+\infty} p_{ij}, \quad i=1,2,\cdots \\P(Y=y_j) = p_{\cdot j}=\sum_{i=1}^{+\infty} p_{ij}, \quad j=1,2,\cdots\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:6.8081em;vertical-align:-3.1541em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.6541em;"><span style="top:-5.6541em;"><span class="pstrut" style="height:3.7583em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mord mtight">⋅</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7583em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4138em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span></span></span><span style="top:-2.1819em;"><span class="pstrut" style="height:3.7583em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">⋅</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7583em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.1541em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>我们称：</p><ul><li><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(X=x_i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 为随机向量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span> 关于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的边缘分布列</p></li><li><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><msub><mi>y</mi><mi>j</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(Y=y_j)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 为随机向量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span> 关于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 的边缘分布列</p></li></ul><h4 id="3-2-3-边缘密度函数">3.2.3 边缘密度函数</h4><p>所谓的的边缘密度函数，可以与边缘分布列进行类比，也就是固定一个随机变量，另外的随机变量取遍。只不过连续型的取遍就是无数个点，而离散型的取遍是可列个点，仅此而已。即：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi>p</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mo>+</mo><mi mathvariant="normal">∞</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi>x</mi></msubsup><mrow><mo fence="true">[</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo stretchy="false">)</mo><mi>d</mi><mi>v</mi><mo fence="true">]</mo></mrow><mi>d</mi><mi>u</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mi>d</mi><mi>y</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}P(X=x) &amp;= p_X(x) \\&amp;= \frac{d}{dx} F_X(x) \\&amp;= \frac{d}{dx} F(x,+\infty) \\&amp;= \frac{d}{dx} \int_{-\infty}^{x} \left [ \int_{-\infty}^{+\infty} p(u,v) dv \right ] du \\&amp;= \int_{-\infty}^{+\infty} p(x,y) dy \\\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:11.7979em;vertical-align:-5.649em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.149em;"><span style="top:-8.8302em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span><span style="top:-6.7987em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"></span></span><span style="top:-4.4413em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"></span></span><span style="top:-1.9341em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"></span></span><span style="top:0.8574em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.649em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.149em;"><span style="top:-8.8302em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span><span style="top:-6.7987em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span><span style="top:-4.4413em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">+</span><span class="mord">∞</span><span class="mclose">)</span></span></span><span style="top:-1.9341em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">u</span></span></span><span style="top:0.8574em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.649em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi>p</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>y</mi></mrow></mfrac><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>y</mi></mrow></mfrac><mi>F</mi><mo stretchy="false">(</mo><mo>+</mo><mi mathvariant="normal">∞</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>y</mi></mrow></mfrac><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mrow><mo fence="true">[</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi>y</mi></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo stretchy="false">)</mo><mi>d</mi><mi>v</mi><mo fence="true">]</mo></mrow><mi>d</mi><mi>u</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>y</mi></mrow></mfrac><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi>y</mi></msubsup><mrow><mo fence="true">[</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo stretchy="false">)</mo><mi>d</mi><mi>u</mi><mo fence="true">]</mo></mrow><mi>d</mi><mi>v</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}P(Y=y) &amp;= p_Y(y) \\&amp;= \frac{d}{dy} F_Y(y) \\&amp;= \frac{d}{dy} F(+\infty,y) \\&amp;= \frac{d}{dy} \int_{-\infty}^{+\infty} \left [ \int_{-\infty}^{y} p(u,v) dv \right ] du \\&amp;= \frac{d}{dy} \int_{-\infty}^{y} \left [ \int_{-\infty}^{+\infty} p(u,v) du \right ] dv \\&amp;= \int_{-\infty}^{+\infty} p(x,y) dx \\\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:14.9783em;vertical-align:-7.2391em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:7.7391em;"><span style="top:-10.4204em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span><span style="top:-8.3889em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"></span></span><span style="top:-5.8371em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"></span></span><span style="top:-3.1354em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"></span></span><span style="top:-0.3439em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"></span></span><span style="top:2.4476em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:7.2391em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:7.7391em;"><span style="top:-10.4204em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span><span style="top:-8.3889em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span><span style="top:-5.8371em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord">+</span><span class="mord">∞</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span><span style="top:-3.1354em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">u</span></span></span><span style="top:-0.3439em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">u</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span><span style="top:2.4476em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:7.2391em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>我们称：</p><ul><li><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(X=x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 为随机向量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span> 关于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的边缘密度函数</p></li><li><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(Y=y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> 为随机向量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span> 关于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 的边缘密度函数</p></li></ul><h3 id="3-3-随机向量的条件分布">3.3 随机向量的条件分布</h3><div class="note note-light">            <p>本目主要介绍的是条件分布。所谓的条件分布，其实就是在约束一个随机变量为定值的情况下，另外一个随机变量的取值情况。与上述联合分布、边缘分布的区别在于：</p><ul><li>联合分布、边缘分布的分布函数是一个体积（散点和），概率密度（分布列）是一个曲面（点集）</li><li>条件分布的分布函数是一个面积（散点和），概率密度（分布列）是一个曲线（点集）</li></ul>          </div><h4 id="3-3-1-离散型随机向量的条件分布列和条件分布函数">3.3.1 离散型随机向量的条件分布列和条件分布函数</h4><p>条件分布列，即散点情况：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>p</mi><mrow><mi>i</mi><mi mathvariant="normal">∣</mi><mi>j</mi></mrow></msub><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><msub><mi>x</mi><mi>i</mi></msub><mtext>&nbsp;</mtext><mi mathvariant="normal">∣</mi><mtext>&nbsp;</mtext><mi>Y</mi><mo>=</mo><msub><mi>y</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><msub><mi>x</mi><mi>i</mi></msub><mo separator="true">,</mo><mi>Y</mi><mo>=</mo><msub><mi>y</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><msub><mi>y</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><msub><mi>p</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>p</mi><mrow><mo>⋅</mo><mi>j</mi></mrow></msub></mfrac><mo separator="true">,</mo><mspace width="1em"></mspace><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>p</mi><mrow><mi>j</mi><mi mathvariant="normal">∣</mi><mi>i</mi></mrow></msub><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><msub><mi>y</mi><mi>j</mi></msub><mtext>&nbsp;</mtext><mi mathvariant="normal">∣</mi><mtext>&nbsp;</mtext><mi>X</mi><mo>=</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><msub><mi>x</mi><mi>i</mi></msub><mo separator="true">,</mo><mi>Y</mi><mo>=</mo><msub><mi>y</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mfrac><msub><mi>p</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>p</mi><mrow><mi>i</mi><mo>⋅</mo></mrow></msub></mfrac><mo separator="true">,</mo><mspace width="1em"></mspace><mi>j</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}p_{i|j} = P(X=x_i\ |\ Y=y_j) = \frac{P(X=x_i,Y=y_i)}{P(Y=y_i)} = \frac{p_{ij}}{p_{\cdot j}}, \quad i=1,2,\cdots \\p_{j|i} = P(Y=y_j\ |\ X=x_i) = \frac{P(X=x_i,Y=y_i)}{P(X=x_i)} = \frac{p_{ij}}{p_{i\cdot }}, \quad j=1,2,\cdots\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.3621em;vertical-align:-2.4311em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9311em;"><span style="top:-4.9311em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mord mtight">∣</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace">&nbsp;</span><span class="mord">∣</span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">⋅</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9721em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span></span></span><span style="top:-2.2319em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mord mtight">∣</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace">&nbsp;</span><span class="mord">∣</span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mord mtight">⋅</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.4311em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>我们称：</p><ul><li><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mrow><mi>i</mi><mi mathvariant="normal">∣</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">p_{i|j}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7858em;vertical-align:-0.3552em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mord mtight">∣</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span></span></span></span> 为在给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><msub><mi>y</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">Y=y_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 的条件下 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的条件分布列</p></li><li><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mrow><mi>j</mi><mi mathvariant="normal">∣</mi><mi>i</mi></mrow></msub></mrow><annotation encoding="application/x-tex">p_{j|i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7858em;vertical-align:-0.3552em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.5198em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mord mtight">∣</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3552em;"><span></span></span></span></span></span></span></span></span></span> 为在给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>=</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">X=x_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的条件下 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 的条件分布列</p></li></ul><p>条件分布函数，即点集情况：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><msub><mi>y</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mi>x</mi><mtext>&nbsp;</mtext><mi mathvariant="normal">∣</mi><mtext>&nbsp;</mtext><mi>Y</mi><mo>=</mo><msub><mi>y</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mo>=</mo><munder><mo>∑</mo><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>≤</mo><mi>x</mi></mrow></munder><mfrac><msub><mi>p</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>p</mi><mrow><mo>⋅</mo><mi>j</mi></mrow></msub></mfrac></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mi mathvariant="normal">∣</mi><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>≤</mo><mi>y</mi><mtext>&nbsp;</mtext><mi mathvariant="normal">∣</mi><mtext>&nbsp;</mtext><mi>X</mi><mo>=</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><munder><mo>∑</mo><mrow><msub><mi>y</mi><mi>j</mi></msub><mo>≤</mo><mi>y</mi></mrow></munder><mfrac><msub><mi>p</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>p</mi><mrow><mi>i</mi><mo>⋅</mo></mrow></msub></mfrac></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}F(x|y_j) = P(X \le x\ | \ Y=y_j) = \sum _{x_i\le x} \frac{p_{ij}}{p_{\cdot j}} \\F(y|x_i) = P(Y \le y\ | \ X=x_i) = \sum _{y_j\le y} \frac{p_{ij}}{p_{i \cdot}}\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.6349em;vertical-align:-2.5675em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.0675em;"><span style="top:-5.0675em;"><span class="pstrut" style="height:3.1076em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mspace">&nbsp;</span><span class="mord">∣</span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.8888em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mrel mtight">≤</span><span class="mord mathnormal mtight">x</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3613em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">⋅</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9721em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-2.2986em;"><span class="pstrut" style="height:3.1076em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace">&nbsp;</span><span class="mord">∣</span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.8888em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:-0.0359em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2819em;"><span></span></span></span></span></span></span><span class="mrel mtight">≤</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4585em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mord mtight">⋅</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.5675em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>我们称：</p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><msub><mi>y</mi><mi>j</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(x|y_j)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 为在给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><msub><mi>y</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">Y=y_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 的条件下 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的条件分布函数</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mi mathvariant="normal">∣</mi><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(y|x_i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 为在给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>=</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">X=x_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的条件下 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 的条件分布函数</li></ul><h4 id="3-3-2-连续型随机向量的条件密度函数和条件分布函数">3.3.2 连续型随机向量的条件密度函数和条件分布函数</h4><p>条件密度函数，即联合分布的概率密度曲面上，约束了某一维度的随机变量为定值，于是条件密度函数的图像就是一个空间曲线：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi>p</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mfrac><mo separator="true">,</mo><mspace width="1em"></mspace><mo>−</mo><mi mathvariant="normal">∞</mi><mo>&lt;</mo><mi>x</mi><mo>&lt;</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>y</mi><mi mathvariant="normal">∣</mi><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi>p</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mo separator="true">,</mo><mspace width="1em"></mspace><mo>−</mo><mi mathvariant="normal">∞</mi><mo>&lt;</mo><mi>y</mi><mo>&lt;</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}p(x|y) = \frac{p(x,y)}{p_Y(y)}, \quad -\infty &lt; x &lt; +\infty \\p(y|x) = \frac{p(x,y)}{p_X(x)}, \quad -\infty &lt; y &lt; +\infty\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.326em;vertical-align:-2.413em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.913em;"><span style="top:-4.913em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord">∞</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">+</span><span class="mord">∞</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord">∞</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">+</span><span class="mord">∞</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.413em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>我们称：</p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(x|y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> 为在给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">Y=y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> 的条件下 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的条件密度函数</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>y</mi><mi mathvariant="normal">∣</mi><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(y|x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 为在给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">X=x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 的条件下 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 的条件密度函数</li></ul><p>条件分布函数，即上述曲线的分段积分结果：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mi>x</mi><mtext>&nbsp;</mtext><mi mathvariant="normal">∣</mi><mtext>&nbsp;</mtext><mi>Y</mi><mo>=</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi>x</mi></msubsup><mfrac><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>u</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi>p</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mfrac><mi>d</mi><mi>u</mi><mo separator="true">,</mo><mspace width="1em"></mspace><mo>−</mo><mi mathvariant="normal">∞</mi><mo>&lt;</mo><mi>x</mi><mo>&lt;</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mi mathvariant="normal">∣</mi><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>≤</mo><mi>y</mi><mtext>&nbsp;</mtext><mi mathvariant="normal">∣</mi><mtext>&nbsp;</mtext><mi>X</mi><mo>=</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi>y</mi></msubsup><mfrac><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi>p</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mi>d</mi><mi>v</mi><mo separator="true">,</mo><mspace width="1em"></mspace><mo>−</mo><mi mathvariant="normal">∞</mi><mo>&lt;</mo><mi>y</mi><mo>&lt;</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}F(x|y) = P(X \le x \ | \ Y=y) = \int_{-\infty}^x \frac{p(u,y)}{p_Y(y)} du,\quad -\infty &lt; x &lt; +\infty \\F(y|x) = P(Y \le y \ | \ X=x) = \int_{-\infty}^y \frac{p(x,v)}{p_X(x)} dv, \quad -\infty &lt; y &lt; +\infty\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.3946em;vertical-align:-2.4473em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9473em;"><span style="top:-4.9473em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mspace">&nbsp;</span><span class="mord">∣</span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">d</span><span class="mord mathnormal">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord">∞</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">+</span><span class="mord">∞</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.427em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace">&nbsp;</span><span class="mord">∣</span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">−</span><span class="mord">∞</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">+</span><span class="mord">∞</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.4473em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>我们称：</p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mi mathvariant="normal">∣</mi><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(x|y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> 为在给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">Y=y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> 的条件下 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的条件分布函数</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mi mathvariant="normal">∣</mi><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(y|x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mord">∣</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> 为在给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">X=x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 的条件下 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 的条件分布函数</li></ul><h3 id="3-4-随机变量的独立性">3.4 随机变量的独立性</h3><div class="note note-light">            <p>本目主要介绍随机变量的独立性。我们知道随机事件之间是有独立性的，即满足 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(AB)=P(A)P(B)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span></span></span></span> 的事件，那么随机变量之间也有独立性吗？答案是有的，以生活中的例子为实例，比如我和某个同学进教室，就是独立的两个随机变量。下面开始介绍。</p>          </div><ul><li><p>定义：我们定义如果两个随机变量的分布函数满足下式，则两个随机变量相互独立：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(x,y)=F_X(x)F_Y(y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span></span></p></li><li><p>性质：对于随机向量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span></p><ol><li><p>随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 相互独立的充分必要条件是：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mtext>离散型:</mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><msub><mi>x</mi><mi>i</mi></msub><mo separator="true">,</mo><mi>Y</mi><mo>=</mo><msub><mi>y</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>=</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><msub><mi>y</mi><mi>j</mi></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mtext>连续型:</mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>p</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mi>p</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\text{离散型:}&amp; P(X=x_i,Y=y_j) = P(X=x_i)P(Y=y_j) \\\text{连续型:}&amp; p(x,y) = p_X(x)p_Y(y)\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">离散型</span><span class="mord">:</span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">连续型</span><span class="mord">:</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li><li><p>若随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 相互独立，且 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h(\cdot)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord">⋅</span><span class="mclose">)</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(\cdot)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord">⋅</span><span class="mclose">)</span></span></span></span> 连续，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>g</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h(X),g(Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span> 也相互独立</p></li></ol></li></ul><h3 id="3-5-随机向量函数的分布">3.5 随机向量函数的分布</h3><div class="note note-light">            <p>在 2.4 目中我们了解到了随机变量函数的分布，现在我们讨论随机向量函数的分布。在生活中，假设我们已经知道了一个人群中所有人的身高和体重的分布情况，现在想要血糖根据身高和体重的分布情况，就需要用到本目的理念。我们从离散型和连续型随机向量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span> 出发，讨论 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(X,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span> 的分布情况。</p>          </div><h4 id="3-5-1-离散型随机向量函数的分布">3.5.1 离散型随机向量函数的分布</h4><p>按照规则枚举即可。</p><h4 id="3-5-2-连续型随机向量函数的分布">3.5.2 连续型随机向量函数的分布</h4><p>与连续型随机变量函数的分布类似，这类题目一般也是：给定随机向量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span> 的密度函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> 和 映射函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span>，现在需要求解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z=g(X,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span> 的分布函数（若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> 二元连续，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span></span></span> 也是连续型随机变量）。方法同理，先求解 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span></span></span> 的分布函数，再对 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span></span></span></span> 求导得到密度函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>Z</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p_Z(z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">Z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span></span>​。接下来我们介绍两种常见随机向量的分布。</p><p><strong>(1) 和的分布：</strong></p><ul><li><p>先求分布函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>F</mi><mi>Z</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F_Z(z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">Z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span></span>：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>F</mi><mi>Z</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>+</mo><mi>Y</mi><mo>≤</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><munder><mo>∬</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi><mo>≤</mo><mi>z</mi></mrow></munder><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mi>d</mi><mi>y</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi>z</mi></msubsup><mrow><mo fence="true">[</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>t</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mo fence="true">]</mo></mrow><mi>d</mi><mi>t</mi></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr><mtr><mtd class="mtr-glue"></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi>z</mi></msubsup><mrow><mo fence="true">[</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>t</mi><mo>−</mo><mi>y</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mi>d</mi><mi>y</mi><mo fence="true">]</mo></mrow><mi>d</mi><mi>t</mi></mrow></mstyle></mtd><mtd class="mtr-glue"></mtd><mtd class="mml-eqn-num"></mtd></mtr></mtable></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}F_Z(z) &amp;= P(X+Y \le z) \\&amp;= \iint\limits_{x+y \le z} p(x,y) dxdy \\&amp;\begin{align}&amp;= \int _{-\infty}^z \left [ \int_{-\infty}^{+\infty} p(x,t-x)dx \right ] dt \\&amp;= \int _{-\infty}^z \left [ \int_{-\infty}^{+\infty} p(t-y,y)dy \right ] dt\end{align}\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:10.7523em;vertical-align:-5.1262em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.6262em;"><span style="top:-9.8277em;"><span class="pstrut" style="height:5.0415em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">Z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:-7.8067em;"><span class="pstrut" style="height:5.0415em;"></span><span class="mord"></span></span><span style="top:-2.7569em;"><span class="pstrut" style="height:5.0415em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.1262em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.6262em;"><span style="top:-9.8277em;"><span class="pstrut" style="height:5.0415em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:-7.8067em;"><span class="pstrut" style="height:5.0415em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop op-limits"><span class="mspace" style="margin-right:0.4445em;"></span><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.361em;"><span style="top:-1.8878em;margin-left:-0.4445em;"><span class="pstrut" style="height:3.36em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mbin mtight">+</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="mrel mtight">≤</span><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span><span style="top:-3.361em;"><span class="pstrut" style="height:3.36em;"></span><span><span class="mop op-symbol large-op" style="margin-right:0.44445em;">∬</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.7083em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-2.7569em;"><span class="pstrut" style="height:5.0415em;"></span><span class="mord"><span class="mord"></span><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.0415em;"><span style="top:-5.0415em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.5415em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.0415em;"><span style="top:-5.0415em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">[</span></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.5415em;"><span></span></span></span></span></span></span><span class="tag"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.0415em;"><span style="top:-5.0415em;"><span class="pstrut" style="height:3.5212em;"></span><span class="eqn-num"></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.5212em;"></span><span class="eqn-num"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.5415em;"><span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.1262em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li><li><p>由分布函数定义：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi>x</mi></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mi>d</mi><mi>u</mi></mrow><annotation encoding="application/x-tex">F_X(x) = \int_{-\infty}^xp(u)du</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3846em;vertical-align:-0.9703em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4143em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">u</span></span></span></span></span></p></li><li><p>所以可得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi><mo>=</mo><mi>X</mi><mo>+</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">Z=X+Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 的密度函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>Z</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p_Z(z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">Z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span></span> 为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>p</mi><mi>Z</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>z</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mspace width="1em"></mspace></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>p</mi><mi>Z</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mi>y</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mi>d</mi><mi>y</mi><mspace width="1em"></mspace></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}p_Z(z) = \int_{-\infty}^{+\infty} p(x,z-x)dx \quad &amp;(1) \\p_Z(z) = \int_{-\infty}^{+\infty} p(z-y,y)dy \quad &amp;(2) \\\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:5.583em;vertical-align:-2.5415em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.0415em;"><span style="top:-5.0415em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">Z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:1em;"></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">Z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:1em;"></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.5415em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.0415em;"><span style="top:-5.0415em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mopen">(</span><span class="mord">2</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.5415em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li><li><p>若 X 和 Y 相互独立，还可得卷积式：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left right" columnspacing="0em 1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>p</mi><mi>Z</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>z</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><msub><mi>p</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⋅</mo><msub><mi>p</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mspace width="1em"></mspace></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>p</mi><mi>Z</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>p</mi><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mi>y</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mi>d</mi><mi>y</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><msub><mi>p</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mi>y</mi><mo stretchy="false">)</mo><mo>⋅</mo><msub><mi>p</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mi>d</mi><mi>y</mi><mspace width="1em"></mspace></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}p_Z(z) &amp;= \int_{-\infty}^{+\infty} p(x,z-x)dx \\&amp;= \int_{-\infty}^{+\infty} p_X(x)\cdot p_Y(z-x) dx \quad &amp;(1) \\p_Z(z) &amp;= \int_{-\infty}^{+\infty} p(z-y,y)dy \\&amp;= \int_{-\infty}^{+\infty} p_X(z-y)\cdot p_Y(y) dy \quad &amp;(2)\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:11.166em;vertical-align:-5.333em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.833em;"><span style="top:-7.833em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">Z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:-5.0415em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">Z</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:0.5415em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.333em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.833em;"><span style="top:-7.833em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span><span style="top:-5.0415em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:1em;"></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:0.5415em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:1em;"></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.333em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.0415em;"><span style="top:-5.0415em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span></span></span><span style="top:0.5415em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">2</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.333em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ul><p><strong>(2) 次序统计量的分布（对于两个相互独立的随机变量 X 和 Y）：</strong></p><ul><li><p>对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mo>=</mo><mi>max</mi><mo>⁡</mo><mrow><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex">M=\max{(X,Y)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">max</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span></span> 的分布函数，有：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>F</mi><mi>M</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>M</mi><mo>≤</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>max</mi><mo>⁡</mo><mrow><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><mo>≤</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mi>z</mi><mo separator="true">,</mo><mi>Y</mi><mo>≤</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≤</mo><mi>z</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>≤</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>⋅</mo><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}F_M(z) &amp;= P(M \le z) \\&amp;= P(\max{(X,Y)} \le z) \\&amp;= P(X \le z, Y \le z) \\&amp;= P(X \le z) \cdot P(Y \le z) \\&amp;= F_X(z) \cdot F_Y(z)\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:7.5em;vertical-align:-3.5em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4em;"><span style="top:-6.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:-4.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-3.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-1.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-0.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.5em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4em;"><span style="top:-6.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:-4.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mop">max</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:-3.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:-1.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:-0.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.5em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li><li><p>对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo>=</mo><mi>min</mi><mo>⁡</mo><mrow><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex">N=\min{(X,Y)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">min</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span></span> 的分布函数，有：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>F</mi><mi>N</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>N</mi><mo>≤</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>min</mi><mo>⁡</mo><mrow><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><mo>≤</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>1</mn><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mi>min</mi><mo>⁡</mo><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>+</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><mo>≥</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>1</mn><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≥</mo><mi>z</mi><mo separator="true">,</mo><mi>Y</mi><mo>≥</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>1</mn><mo>−</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>≥</mo><mi>z</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>≥</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>1</mn><mo>−</mo><mo stretchy="false">[</mo><mn>1</mn><mo>−</mo><msub><mi>F</mi><mi>X</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>⋅</mo><mo stretchy="false">[</mo><mn>1</mn><mo>−</mo><msub><mi>F</mi><mi>Y</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}F_N(z) &amp;= P(N \le z) \\&amp;= P(\min{(X,Y)} \le z) \\&amp;= 1 - P(\min{(X+Y)} \ge z) \\&amp;= 1 - P(X \ge z,Y \ge z) \\&amp;= 1 - P(X \ge z) \cdot P(Y \ge z) \\&amp;= 1 - [1 - F_X(z)] \cdot [1 - F_Y(z)]\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:9em;vertical-align:-4.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.75em;"><span style="top:-6.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:-5.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-0.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:0.59em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.25em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.75em;"><span style="top:-6.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:-5.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mop">min</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mop">min</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:-0.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:0.59em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">[</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">[</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)]</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:4.25em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li><li><p>若拓展到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个相互独立且同分布的随机变量，则有：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>F</mi><mi>M</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo stretchy="false">[</mo><mi>F</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">]</mo><mi>n</mi></msup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>p</mi><mi>M</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>n</mi><mi>p</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>F</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">]</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}F_M(z) &amp;= [F(z)]^n \\p_M(z) &amp;= np(z)[F(z)]^{n-1}\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0241em;vertical-align:-1.2621em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7621em;"><span style="top:-3.9221em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:-2.3979em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2621em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7621em;"><span style="top:-3.9221em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span><span style="top:-2.3979em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2621em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>F</mi><mi>N</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>1</mn><mo>−</mo><mo stretchy="false">[</mo><mn>1</mn><mo>−</mo><mi>F</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">]</mo><mi>n</mi></msup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><msub><mi>p</mi><mi>N</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>n</mi><mi>p</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mn>1</mn><mo>−</mo><mi>F</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><msup><mo stretchy="false">]</mo><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}F_N(z) &amp;= 1 - [1-F(z)]^n \\p_N(z) &amp;= np(z)[1-F(z)]^{n-1}\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0241em;vertical-align:-1.2621em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7621em;"><span style="top:-3.9221em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span><span style="top:-2.3979em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2621em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7621em;"><span style="top:-3.9221em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">[</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span><span style="top:-2.3979em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mopen">[</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mclose"><span class="mclose">]</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2621em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ul><h2 id="第4章-随机变量的数字特征">第4章 随机变量的数字特征</h2><div class="note note-light">            <p>本章我们将学习随机变量的一些数字特征。所谓的数字特征其实就是随机变量分布的一些内在属性，比如均值、方差、协方差等等，有些分布特性甚至可以通过某个数字特征而直接觉得。其中<strong>期望</strong>和<strong>方差</strong>往往用来衡量单个随机变量的特征，而<strong>协方差</strong>与<strong>相关系数</strong>则是用来衡量随机变量之间的数字特征。接下来开始介绍。</p>          </div><h3 id="4-1-数学期望">4.1 数学期望</h3><div class="note note-light">            <p><strong>加权平均</strong>概念的严格数学定义。</p>          </div><h4 id="4-1-1-随机变量的数学期望">4.1.1 随机变量的数学期望</h4><ul><li><p>离散型</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>E</mi><mi>X</mi><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><msub><mi>x</mi><mi>i</mi></msub><msub><mi>p</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">EX = \sum_{i=1}^{\infty} x_i p_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">EX</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p></li><li><p>连续型</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi>E</mi><mi>X</mi><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>x</mi><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}&amp;EX = \int_{-\infty}^{+\infty} xp(x)dx\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.7915em;vertical-align:-1.1458em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6458em;"><span style="top:-3.6458em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.1458em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6458em;"><span style="top:-3.6458em;"><span class="pstrut" style="height:3.5212em;"></span><span class="mord"><span class="mord"></span><span class="mord mathnormal" style="margin-right:0.07847em;">EX</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.1458em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li></ul><h4 id="4-1-2-随机变量函数的数学期望">4.1.2 随机变量函数的数学期望</h4><ul><li><p>离散型</p><ul><li><p>一元</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>E</mi><mi>g</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mi>g</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mi>p</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">Eg(X) = \sum_{i=1}^{\infty}g(x_i)p_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p></li><li><p>二元</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>E</mi><mi>g</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></munderover><mi>g</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>y</mi><mi>i</mi></msub><mo stretchy="false">)</mo><msub><mi>p</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">Eg(X,Y) = \sum_{i=1}^{\infty}\sum_{j=1}^{\infty}g(x_i,y_i)p_{ij}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.0652em;vertical-align:-1.4138em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4138em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span></span></p></li></ul></li><li><p>连续型</p><ul><li><p>一元</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>E</mi><mi>g</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">Eg(X) = \int_{-\infty}^{+\infty}g(x)p(x)dx</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4915em;vertical-align:-0.9703em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span></span></span></span></span></p></li><li><p>二元</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>E</mi><mi>g</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><msubsup><mo>∫</mo><mrow><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mrow><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></msubsup><mi>g</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>y</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi><mi>d</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">Eg(X,Y) = \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}g(x_i,y_i)p(x,y)dxdy</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4915em;vertical-align:-0.9703em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5212em;"><span style="top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.8129em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mord mathnormal">d</span><span class="mord mathnormal">x</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span></span></p></li></ul></li></ul><h4 id="4-1-3-数学期望的性质">4.1.3 数学期望的性质</h4><ol><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mi>C</mi><mo>=</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">EC=C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">EC</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span></li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>C</mi><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mi>C</mi><mi>E</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">E(CX)=CEX</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">CX</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">CEX</span></span></span></span></li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo>+</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mi>X</mi><mo>+</mo><mi>E</mi><mi>Y</mi></mrow><annotation encoding="application/x-tex">E(X+Y)=EX+EY</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">EX</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span></li><li>若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 相互独立，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mi>X</mi><mi>E</mi><mi>Y</mi></mrow><annotation encoding="application/x-tex">E(XY)=EXEY</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">EXE</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span></li></ol><h3 id="4-2-方差">4.2 方差</h3><div class="note note-light">            <p>随机变量的取值与均值之间的<strong>离散程度</strong>。</p>          </div><h4 id="4-2-1-方差的定义">4.2.1 方差的定义</h4><p>我们定义随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的方差 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(X)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span> 为：（全部可由期望的性质推导而来）</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>E</mi><mrow><mo fence="true">[</mo><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>E</mi><mi>X</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo fence="true">]</mo></mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>E</mi><mrow><mo fence="true">(</mo><msup><mi>X</mi><mn>2</mn></msup><mo fence="true">)</mo></mrow><mo>−</mo><mo stretchy="false">(</mo><mi>E</mi><mi>X</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}D(X) &amp;= E\left[(X-EX)^2\right ] \\&amp;= E\left ( X^2 \right ) - (EX)^2\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0482em;vertical-align:-1.2741em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7741em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span><span style="top:-2.3859em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2741em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7741em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">[</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">EX</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">]</span></span></span></span></span><span style="top:-2.3859em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">(</span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">)</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">EX</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2741em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><h4 id="4-2-2-方差的性质">4.2.2 方差的性质</h4><p>下列方差的性质全部可由上述方差的定义式，结合期望的性质推导而来：</p><ol><li><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>a</mi><mi>X</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mi>D</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(aX+b) = a^2D(X)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span></p></li><li><p>若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>X</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>X</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>⋯</mo></mrow><annotation encoding="application/x-tex">X_1,X_2,\cdots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span></span></span></span> 相互独立，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>a</mi><msub><mi>X</mi><mn>1</mn></msub><mo>±</mo><mi>b</mi><msub><mi>X</mi><mn>2</mn></msub><mo>±</mo><mo>⋯</mo><mtext> </mtext><mo stretchy="false">)</mo><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mi>D</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mi>D</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mo>⋯</mo></mrow><annotation encoding="application/x-tex">D(aX_1 \pm bX_2 \pm \cdots) = a^2D(X_1) + b^2D(X_2) + \cdots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord mathnormal">b</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.313em;"></span><span class="minner">⋯</span></span></span></span></p></li><li><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mrow><mo fence="true">[</mo><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>E</mi><mi>X</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo fence="true">]</mo></mrow><mo>≤</mo><mi>E</mi><mrow><mo fence="true">[</mo><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>C</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">E\left[ (X-EX)^2 \right] \le E \left [ (X-C)^2 \right ]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">[</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">EX</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">]</span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">[</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">]</span></span></span></span></span></span>​</p></li><li><p>切比雪夫不等式 <s>（本以为不要求掌握的，但是被小测拷打了，补一下）</s>：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">∀</mi><mi>ϵ</mi><mo>&gt;</mo><mn>0</mn><mo separator="true">,</mo><mi>P</mi><mo stretchy="false">(</mo><mi mathvariant="normal">∣</mi><mi>X</mi><mo>−</mo><mi>E</mi><mi>X</mi><mi mathvariant="normal">∣</mi><mo>&lt;</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mo>≥</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mi>D</mi><mi>X</mi></mrow><msup><mi>ϵ</mi><mn>2</mn></msup></mfrac></mrow><annotation encoding="application/x-tex">\forall \epsilon &gt;0, P(|X - EX| &lt; \epsilon) \ge 1 - \frac{DX}{\epsilon^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.0391em;"></span><span class="mord">∀</span><span class="mord mathnormal">ϵ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">EX</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">ϵ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0463em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">ϵ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p></li></ol><h3 id="4-3-结论与推导（补）">4.3 结论与推导（补）</h3><table><thead><tr><th style="text-align:center">类型</th><th style="text-align:center">分布</th><th style="text-align:center">符号</th><th style="text-align:center">期望 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(X)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span></th><th style="text-align:center">方差 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(X)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span></th></tr></thead><tbody><tr><td style="text-align:center">离散型</td><td style="text-align:center">0-1 分布</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∼</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><mo>−</mo><mi>p</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>p</mi></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">X \sim \begin{pmatrix} 0 &amp; 1 \\ 1-p &amp; p \end{pmatrix}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(1-p)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mclose">)</span></span></span></span></td></tr><tr><td style="text-align:center">—</td><td style="text-align:center">*二项分布</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>B</mi><mo stretchy="false">(</mo><mi>n</mi><mo separator="true">,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sim B(n,p)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mclose">)</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>p</mi></mrow><annotation encoding="application/x-tex">np</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">p</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>p</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">np(1-p)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mclose">)</span></span></span></span></td></tr><tr><td style="text-align:center">—</td><td style="text-align:center">几何分布</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>G</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sim G(p)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">G</span><span class="mopen">(</span><span class="mord mathnormal">p</span><span class="mclose">)</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mi>p</mi></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \frac{1}{p}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2019em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">p</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mn>1</mn><mo>−</mo><mi>p</mi></mrow><msup><mi>p</mi><mn>2</mn></msup></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \frac{1-p}{p^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2019em;vertical-align:-0.8804em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td></tr><tr><td style="text-align:center">—</td><td style="text-align:center">*泊松分布</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>P</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sim P(\lambda)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal">λ</span><span class="mclose">)</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">λ</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">λ</span></span></span></span></td></tr><tr><td style="text-align:center">连续型</td><td style="text-align:center">均匀分布</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>U</mi><mo stretchy="false">[</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">X \sim U[a,b]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mopen">[</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mclose">]</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mn>2</mn></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \frac{a+b}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0574em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>−</mo><mi>a</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mn>12</mn></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \frac{(b-a)^2}{12}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">12</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">a</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td></tr><tr><td style="text-align:center">—</td><td style="text-align:center">指数分布</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>e</mi><mo stretchy="false">(</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sim e(\lambda)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">e</span><span class="mopen">(</span><span class="mord mathnormal">λ</span><span class="mclose">)</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mi>λ</mi></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \frac{1}{\lambda}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">λ</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><msup><mi>λ</mi><mn>2</mn></msup></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \frac{1}{\lambda^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td></tr><tr><td style="text-align:center">—</td><td style="text-align:center">*正态分布</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><mi>μ</mi><mo separator="true">,</mo><msup><mi>σ</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sim N(\mu,\sigma^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mopen">(</span><span class="mord mathnormal">μ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\sigma^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></td></tr></tbody></table><div class="note note-warning">            <p>注：打星号表示在两个随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo separator="true">,</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X,Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 相互独立时，具备可加性。具体的：</p><ol><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mi>μ</mi><mn>1</mn></msub><mo separator="true">,</mo><msubsup><mi>σ</mi><mn>1</mn><mn>2</mn></msubsup><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>Y</mi><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mi>μ</mi><mn>2</mn></msub><mo separator="true">,</mo><msubsup><mi>σ</mi><mn>2</mn><mn>2</mn></msubsup><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi><mo>±</mo><mi>Y</mi><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><msub><mi>μ</mi><mn>1</mn></msub><mo>±</mo><msub><mi>μ</mi><mn>2</mn></msub><mo separator="true">,</mo><msubsup><mi>σ</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mn>2</mn><mn>2</mn></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sim N(\mu_1,\sigma_1^2), Y \sim N(\mu_2,\sigma_2^2) \to X\pm Y\sim N(\mu_1\pm\mu_2,\sigma_1^2+\sigma_2^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4519em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4519em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">±</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0622em;vertical-align:-0.2481em;"></span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4519em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4519em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>B</mi><mo stretchy="false">(</mo><msub><mi>n</mi><mn>1</mn></msub><mo separator="true">,</mo><mi>p</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>Y</mi><mo>∼</mo><mi>B</mi><mo stretchy="false">(</mo><msub><mi>n</mi><mn>2</mn></msub><mo separator="true">,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi><mo>+</mo><mi>Y</mi><mo>∼</mo><mi>B</mi><mo stretchy="false">(</mo><msub><mi>n</mi><mn>1</mn></msub><mo>+</mo><msub><mi>n</mi><mn>2</mn></msub><mo separator="true">,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sim B(n_1,p), Y \sim B(n_2,p) \to X+Y\sim B(n_1+n_2,p)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mclose">)</span></span></span></span></li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>λ</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>Y</mi><mo>∼</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>λ</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi><mo>+</mo><mi>Y</mi><mo>∼</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mi>λ</mi><mn>1</mn></msub><mo>+</mo><msub><mi>λ</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sim P(\lambda_1),Y\sim P(\lambda_2) \to X+Y \sim P(\lambda_1+\lambda_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></li></ol>          </div>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-65e4fb32" role="button" aria-expanded="false" aria-controls="collapse-65e4fb32">        <div class="fold-arrow">▶</div>推导      </div>      <div class="fold-collapse collapse" id="collapse-65e4fb32">        <div class="fold-content">          <p>推导的根本方式还是从定义出发。当然为了省事也可以从性质出发。</p><p>0-1 分布</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404202313030.jpg" alt="0-1 分布"></p><p>二项分布</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404202313503.jpg" alt="二项分布"></p><p>几何分布</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404202313329.jpg" alt="几何分布"></p><p>泊松分布</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404202313316.jpg" alt="泊松分布"></p><p>均匀分布</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404202313239.jpg" alt="均匀分布"></p><p>指数分布</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404202313832.jpg" alt="指数分布"></p>        </div>      </div>    </div><h3 id="4-4-协方差与相关系数">4.4 协方差与相关系数</h3><h4 id="4-4-1-协方差">4.4.1 协方差</h4><p>定义：随机变量 X 与 Y 的协方差 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cov(X,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span> 为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>E</mi><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>E</mi><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>−</mo><mi>E</mi><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mi>Y</mi><mo stretchy="false">)</mo><mo>−</mo><mi>E</mi><mi>X</mi><mi>E</mi><mi>Y</mi></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}Cov(X,Y)&amp;= E[(X-EX)(Y-EY)] \\&amp;= E(XY) - EXEY\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mopen">[(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">EX</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)]</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">EXE</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>特别的：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mi>D</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">Cov(X,X) = DX</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span></span></p><p>性质：</p><ol><li>交换律：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>Y</mi><mo separator="true">,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cov(X,Y)=Cov(Y,X)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span></li><li>提取率：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>a</mi><mi>X</mi><mo separator="true">,</mo><mi>b</mi><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mi>b</mi><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cov(aX,bY)=abCov(X,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">bY</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">ab</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span></li><li>分配率：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo>+</mo><msub><mi>X</mi><mn>2</mn></msub><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>+</mo><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mn>2</mn></msub><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cov(X_1+X_2,Y) = Cov(X_1,Y)+Cov(X_2,Y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span></li><li>独立性：若 X 与 Y 相互独立，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">Cov(X,Y)=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>；反之不一定成立</li><li>放缩性：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mrow><mo fence="true">[</mo><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo fence="true">]</mo></mrow><mn>2</mn></msup><mo>≤</mo><mi>D</mi><mi>X</mi><mo>⋅</mo><mi>D</mi><mi>Y</mi></mrow><annotation encoding="application/x-tex">\left[Cov(X,Y)\right]^2 \le DX \cdot DY</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.204em;vertical-align:-0.25em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mclose delimcenter" style="top:0em;">]</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span></li></ol><h4 id="4-4-2-相关系数">4.4.2 相关系数</h4><p>定义：相关系数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span></span></span></span> 是用来刻画两个随机变量之间<strong>线性</strong>相关关系强弱的一个数字特征，注意是线性关系。<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>ρ</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|\rho|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">ρ</span><span class="mord">∣</span></span></span></span> 越接近 0，则说明两个随机变量越不线性相关；<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>ρ</mi><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|\rho|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">ρ</span><span class="mord">∣</span></span></span></span> 越接近 1，则说明两个随机变量越线性相关，定义式为</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>ρ</mi><mrow><mi>X</mi><mo separator="true">,</mo><mi>Y</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>C</mi><mi>o</mi><mi>v</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><mrow><msqrt><mrow><mi>D</mi><mi>X</mi></mrow></msqrt><msqrt><mrow><mi>D</mi><mi>Y</mi></mrow></msqrt></mrow></mfrac></mrow><annotation encoding="application/x-tex">\rho_{X,Y} = \frac{Cov(X,Y)}{\sqrt{DX}\sqrt{DY}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.22222em;">Y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.357em;vertical-align:-0.93em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.1833em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9267em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-2.8867em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1133em;"><span></span></span></span></span></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9267em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span><span style="top:-2.8867em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1133em;"><span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.93em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><p>特别的：</p><ol><li>若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>&lt;</mo><mi>ρ</mi><mo>&lt;</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">0 &lt; \rho &lt; 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6835em;vertical-align:-0.0391em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，则称 X 与 Y 正相关</li><li>若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>1</mn><mo>&lt;</mo><mi>ρ</mi><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">-1&lt;\rho&lt;0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7335em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">ρ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，则称 X 与 Y 负相关</li></ol><p>性质：</p><ol><li>放缩性（由协方差性质5可得）：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>ρ</mi><mi mathvariant="normal">∣</mi><mo>≤</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">|\rho| \le 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">ρ</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></li><li>独立性（由协方差性质4可得）：若 X 与 Y 相互独立，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">p=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>；反之不一定成立</li><li>线性相关性（不予证明）：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><mi>ρ</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">|\rho|=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathnormal">ρ</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 的充分必要条件是存在常数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo stretchy="false">(</mo><mi>a</mi><mo mathvariant="normal">≠</mo><mn>0</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a(a\ne0),b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span></span></span></span> 使得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>=</mo><mi>a</mi><mi>X</mi><mo>+</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">P(Y=aX+b)=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">a</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></li></ol><h4 id="4-4-3-独立性与线性相关性（补）">4.4.3 独立性与线性相关性（补）</h4><p>一般的：对于两个随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span></p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 相互独立 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 线性无关（可以用线性相关的定义式结合协方差计算公式导出）</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 相互独立 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>↚</mo></mrow><annotation encoding="application/x-tex">\nleftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel amsrm">↚</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 线性无关（因为有可能出现 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 非线性相关）</li></ul><p>特别的：对于满足二维正态分布的随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span>，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>∼</mo><mo stretchy="false">(</mo><msub><mi>μ</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>μ</mi><mn>2</mn></msub><mo separator="true">,</mo><msubsup><mi>σ</mi><mn>1</mn><mn>2</mn></msubsup><mo separator="true">,</mo><msubsup><mi>σ</mi><mn>2</mn><mn>2</mn></msubsup><mo separator="true">,</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,Y) \sim (\mu_1,\mu_2,\sigma_1^2,\sigma_2^2,\rho)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">μ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4519em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4519em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">ρ</span><span class="mclose">)</span></span></span></span></p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 相互独立 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 线性无关</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 相互独立 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>←</mo></mrow><annotation encoding="application/x-tex">\leftarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">←</span></span></span></span> <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span>​ 线性无关</li></ul>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-92251a21" role="button" aria-expanded="false" aria-controls="collapse-92251a21">        <div class="fold-arrow">▶</div>证明 - 二维正态分布的两个随机变量：相互独立 等价于 线性无关      </div>      <div class="fold-collapse collapse" id="collapse-92251a21">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404261112302.jpg" alt="二维正态分布的两个随机变量：相互独立 等价于 线性无关"></p><p>参考：<a href="https://www.zhihu.com/question/29641138">https://www.zhihu.com/question/29641138</a></p>        </div>      </div>    </div><h2 id="第5章-大数定律与中心极限定理">第5章 大数定律与中心极限定理</h2><div class="note note-light">            <p>本章只需要知道一个<strong>独立同分布中心极限定理</strong>即可，至于<strong>棣莫弗-拉普拉斯中心极限定理</strong>其实就是前者的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>X</mi><mi>i</mi></msub><msubsup><mo stretchy="false">}</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></msubsup></mrow><annotation encoding="application/x-tex">\{X_i\}_{i=1}^{\infty}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2587em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">}</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> 服从伯努利 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 重分布罢了。</p>          </div><h3 id="独立同分布中心极限定理">独立同分布中心极限定理</h3><p>定义：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>X</mi><mi>i</mi></msub><msubsup><mo stretchy="false">}</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi mathvariant="normal">∞</mi></msubsup></mrow><annotation encoding="application/x-tex">\{X_i\}_{i=1}^{\infty}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0087em;vertical-align:-0.2587em;"></span><span class="mopen">{</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">}</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-2.4413em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2587em;"><span></span></span></span></span></span></span></span></span></span> 独立同分布且非零方差，其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><msub><mi>X</mi><mi>i</mi></msub><mo>=</mo><mi>μ</mi><mo separator="true">,</mo><mi>D</mi><msub><mi>X</mi><mi>i</mi></msub><mo>=</mo><msup><mi>σ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">EX_i=\mu,DX_i=\sigma^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>，则有：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>X</mi><mi>i</mi></msub></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mo stretchy="false">(</mo><mi>E</mi><msub><mi>X</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo separator="true">,</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mo stretchy="false">(</mo><mi>D</mi><msub><mi>X</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><mi>n</mi><mi>μ</mi><mo separator="true">,</mo><mi>n</mi><msup><mi>σ</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\sum_{i=1}^n X_i &amp;\sim N(\sum_{i=1}^n(EX_i),\sum_{i=1}^n(DX_i)) \\&amp;\sim N(n\mu,n\sigma^2)\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.7532em;vertical-align:-2.1266em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6266em;"><span style="top:-4.6266em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.1848em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1266em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6266em;"><span style="top:-4.6266em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mopen">(</span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">))</span></span></span><span style="top:-2.1848em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mord mathnormal">μ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1266em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>解释：其实就是对于独立同分布的随机事件 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>X</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">X_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，在事件数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 足够大时，就近似为正态分布（术语叫做<strong>依分布</strong>）。这样就可以很方便利用正态分布的性质计算当前事件的概率。至于<strong>棣莫弗-拉普拉斯中心极限定理</strong>就是上述 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi><mo>=</mo><mi>p</mi><mo separator="true">,</mo><msup><mi>σ</mi><mn>2</mn></msup><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mu=p,\sigma^2=p(1-p)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mclose">)</span></span></span></span> 的特殊情况罢了</p><h2 id="第6章-数理统计的基本概念">第6章 数理统计的基本概念</h2><div class="note note-light">            <p>开始统计学之旅。</p>          </div><h3 id="6-1-总体与样本">6.1 总体与样本</h3><p><s>类比 ML：数据集=总体，样本=样本。</s></p><p>我们只研究一种样本：简单随机样本 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>X</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>X</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X_1,X_2,...,X_n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">...</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>。符合下列两种特点：</p><ol><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>X</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>X</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X_1,X_2,...,X_n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">...</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 相互独立</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>X</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>X</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X_1,X_2,...,X_n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">...</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 同分布</li></ol><p>同样的，我们研究总体 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的分布与概率密度，一般概率密度会直接给，需要我们在此基础之上研究所有样本的联合密度：</p><ul><li><p>分布：由于样本相互独立，故：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>⋯</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(x_1,x_2,...,x_n)=F(x_1)F(x_2) \cdots F(x_n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">...</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p></li><li><p>联合密度：同样由于样本相互独立，故：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo>⋯</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(x_1,x_2,...,x_n)=p(x_1)p(x_2) \cdots p(x_n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">...</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p></li></ul><h3 id="6-2-经验分布与频率直方图">6.2 经验分布与频率直方图</h3><p>经验分布函数是利用样本得到的。也是给区间然后统计样本频度进而计算频率，只不过区间长度不是固定的。</p><p>频率直方图就是选定<strong>固定</strong>的区间长度，然后统计频度进而计算频率作图。</p><h3 id="6-3-统计量">6.3 统计量</h3><p><strong>统计量定义</strong>：关于样本不含未知数的表达式。</p><p><strong>常见统计量</strong>：假设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>X</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>X</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X_1,X_2,...,X_n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">...</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 为来自总体 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的简单随机样本</p><p>一、样本均值和样本方差</p><ul><li><p>样本均值：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>X</mi><mi>i</mi></msub></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \overline{X} = \frac{1}{n} \sum_{i=1}^n X_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p></li><li><p>样本方差：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><msubsup><mi>S</mi><mn>0</mn><mn>2</mn></msubsup><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mo stretchy="false">(</mo><msub><mi>X</mi><mi>i</mi></msub><mo>−</mo><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>X</mi><mi>i</mi><mn>2</mn></msubsup><mo>−</mo><msup><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover><mn>2</mn></msup></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle S_0^2 = \frac{1}{n} \sum_{i=1}^n (X_i - \overline{X})^2 = \frac{1}{n}\sum_{i=1}^n X_i^2 - \overline{X}^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1111em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1333em;vertical-align:-0.25em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0873em;"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0873em;"><span style="top:-3.3362em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></p></li><li><p>样本标准差：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><msub><mi>S</mi><mn>0</mn></msub><mo>=</mo><msqrt><msubsup><mi>S</mi><mn>0</mn><mn>2</mn></msubsup></msqrt></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle S_0 = \sqrt{S_0^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.84em;vertical-align:-0.5413em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2987em;"><span class="svg-align" style="top:-3.8em;"><span class="pstrut" style="height:3.8em;"></span><span class="mord" style="padding-left:1em;"><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7959em;"><span style="top:-2.4337em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.0448em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2663em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.2587em;"><span class="pstrut" style="height:3.8em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.88em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.88em" viewBox="0 0 400000 1944" preserveAspectRatio="xMinYMin slice"><path d="M983 90l0 -0c4,-6.7,10,-10,18,-10 H400000v40H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5c53.7,-170.3,84.5,-266.8,92.5,-289.5zM1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.5413em;"><span></span></span></span></span></span></span></span></span></p></li><li><p>修正样本方差：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><msup><mi>S</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mo stretchy="false">(</mo><msub><mi>X</mi><mi>i</mi></msub><mo>−</mo><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle S^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \overline{X})^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8641em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1333em;vertical-align:-0.25em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></p></li><li><p>修正样本标准差：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mi>S</mi><mo>=</mo><msqrt><msup><mi>S</mi><mn>2</mn></msup></msqrt></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle S = \sqrt{S^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.04em;vertical-align:-0.0361em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0039em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-2.9639em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.0361em;"><span></span></span></span></span></span></span></span></span></p>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-0d019804" role="button" aria-expanded="false" aria-controls="collapse-0d019804">        <div class="fold-arrow">▶</div>推导      </div>      <div class="fold-collapse collapse" id="collapse-0d019804">        <div class="fold-content">          <p>设总体 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的数学期望和方差分别为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\sigma^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>X</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>X</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>X</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X_1,X_2,...,X_n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">...</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 是简单随机样本，则：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202405101145088.png" alt="样本均值的数学期望与总体的数学期望相等"></p><p>即：样本均值的数学期望 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo></mrow><annotation encoding="application/x-tex">=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">=</span></span></span></span> 总体的数学期望</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202405101148690.png" alt="样本方差的数学期望与总体的数学期望 不相等"></p><p>即：样本方差的数学期望 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo mathvariant="normal">≠</mo></mrow><annotation encoding="application/x-tex">\ne</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span></span></span></span> 总体的数学期望</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202405101148126.png" alt="修正样本方差推导"></p><p>上图即：修正样本方差推导</p>        </div>      </div>    </div></li><li><p>样本 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 阶原点矩：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><msub><mi>A</mi><mi>k</mi></msub><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>X</mi><mi>i</mi><mi>k</mi></msubsup><mo separator="true">,</mo><mspace width="1em"></mspace><mi>k</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle A_k = \frac{1}{n} \sum_{i=1}^n X_i^k,\quad k=1,2,\cdots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-2.453em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span></span></span></span></p></li><li><p>样本 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 阶中心矩：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><msub><mi>B</mi><mi>k</mi></msub><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mo stretchy="false">(</mo><msub><mi>X</mi><mi>i</mi></msub><mo>−</mo><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover><msup><mo stretchy="false">)</mo><mi>k</mi></msup><mo separator="true">,</mo><mspace width="1em"></mspace><mi>k</mi><mo>=</mo><mn>2</mn><mo separator="true">,</mo><mn>3</mn><mo separator="true">,</mo><mo>⋯</mo></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle B_k = \frac{1}{n} \sum_{i=1}^n (X_i-\overline{X})^k,\quad k=2,3,\cdots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1491em;vertical-align:-0.25em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:1em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.1944em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span></span></span></span></p></li></ul><p>二、次序统计量</p><ul><li>序列最小值</li><li>序列最大值</li><li>极差 = 序列最大值 - 序列最小值</li></ul><h3 id="6-4-正态总体抽样分布定理">6.4 正态总体抽样分布定理</h3><div class="note note-light">            <p>时刻牢记一句话：构造性定义！</p>          </div><h4 id="6-4-1-卡方分布、t-分布、F-分布">6.4.1 卡方分布、t 分布、F 分布</h4><p><strong>分位数</strong></p><ul><li>我们定义实数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mi>α</mi></msub></mrow><annotation encoding="application/x-tex">\lambda_\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 为随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的上侧 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 分位数（点）当且仅当 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>&gt;</mo><msub><mi>λ</mi><mi>α</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>α</mi></mrow><annotation encoding="application/x-tex">P(X &gt; \lambda_\alpha) = \alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span>​</li><li>我们定义实数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>λ</mi><mrow><mn>1</mn><mo>−</mo><mi>β</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\lambda_{1-\beta}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.05278em;">β</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 为随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 的下侧 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span> 分位数（点）当且仅当 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo>&lt;</mo><msub><mi>λ</mi><mrow><mn>1</mn><mo>−</mo><mi>β</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mi>β</mi></mrow><annotation encoding="application/x-tex">P(X &lt; \lambda_{1-\beta})=\beta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.05278em;">β</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span></span></span></span></li></ul><p><strong><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>χ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\chi^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 分布</strong></p>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-dd3d75fc" role="button" aria-expanded="false" aria-controls="collapse-dd3d75fc">        <div class="fold-arrow">▶</div>密度函数图像      </div>      <div class="fold-collapse collapse" id="collapse-dd3d75fc">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202405171013305.webp" alt="密度函数图像"></p>        </div>      </div>    </div><p>定义：</p><ul><li>对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个独立同分布的标准正态随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>X</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>X</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><msub><mi>X</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X_1,X_2,\cdots ,X_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><msubsup><mi>X</mi><mn>1</mn><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>X</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><mo>⋯</mo><mo>+</mo><msubsup><mi>X</mi><mi>n</mi><mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">Y = X_1^2 + X_2^2 + \cdots + X_n^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0622em;vertical-align:-0.2481em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4519em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0622em;vertical-align:-0.2481em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.4519em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2481em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0611em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span></li><li>则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 服从自由度为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>χ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\chi^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 分布，记作：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>∼</mo><msup><mi>χ</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Y \sim \chi^2(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li></ul><p>性质：</p><ul><li><p>可加性：若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Y</mi><mn>1</mn></msub><mo>∼</mo><msup><mi>χ</mi><mn>2</mn></msup><mo stretchy="false">(</mo><msub><mi>n</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo separator="true">,</mo><msub><mi>Y</mi><mn>2</mn></msub><mo>∼</mo><msup><mi>χ</mi><mn>2</mn></msup><mo stretchy="false">(</mo><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Y_1 \sim \chi^2(n_1), Y_2 \sim \chi^2(n_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 且 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Y</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>Y</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">Y_1,Y_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 相互独立，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>Y</mi><mn>1</mn></msub><mo>+</mo><msub><mi>Y</mi><mn>2</mn></msub><mo>∼</mo><msup><mi>χ</mi><mn>2</mn></msup><mo stretchy="false">(</mo><msub><mi>n</mi><mn>1</mn></msub><mo>+</mo><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Y_1+Y_2 \sim \chi^2(n_1+n_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></p></li><li><p>统计性：对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>∼</mo><msup><mi>χ</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Y \sim \chi^2(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>，有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mi>Y</mi><mo>=</mo><mi>n</mi><mo separator="true">,</mo><mi>D</mi><mi>Y</mi><mo>=</mo><mn>2</mn><mi>n</mi></mrow><annotation encoding="application/x-tex">EY = n, DY = 2n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">n</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span><span class="mord mathnormal">n</span></span></span></span></p>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-e5c1222d" role="button" aria-expanded="false" aria-controls="collapse-e5c1222d">        <div class="fold-arrow">▶</div>推导      </div>      <div class="fold-collapse collapse" id="collapse-e5c1222d">        <div class="fold-content">          <p>EY 的推导利用：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><msup><mi>X</mi><mn>2</mn></msup><mo>=</mo><mi>D</mi><mi>X</mi><mo>−</mo><mo stretchy="false">(</mo><mi>E</mi><mi>X</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">EX^2 = DX - (EX)^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">EX</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202405211524059.png" alt="EY 的推导"></p><p>DY 的推导利用：方差计算公式、随机变量函数的数学期望进行计算</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202405211524574.png" alt="DY 的推导"></p>        </div>      </div>    </div></li></ul><p><strong><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span> 分布</strong></p>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-bfc3604c" role="button" aria-expanded="false" aria-controls="collapse-bfc3604c">        <div class="fold-arrow">▶</div>密度函数图像      </div>      <div class="fold-collapse collapse" id="collapse-bfc3604c">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202405171014091.webp" alt="密度函数图像"></p>        </div>      </div>    </div><p>定义：</p><ul><li>若随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>Y</mi><mo>∼</mo><msup><mi>χ</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sim N(0, 1),Y \sim \chi^2 (n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 且 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo separator="true">,</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X,Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span> 相互独立</li><li>则称随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mi>X</mi><msqrt><mrow><mi>Y</mi><mi mathvariant="normal">/</mi><mi>n</mi></mrow></msqrt></mfrac></mstyle></mrow><annotation encoding="application/x-tex">T = \displaystyle \frac{X}{\sqrt{Y/n}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4903em;vertical-align:-1.13em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.175em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.935em;"><span class="svg-align" style="top:-3.2em;"><span class="pstrut" style="height:3.2em;"></span><span class="mord" style="padding-left:1em;"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mord">/</span><span class="mord mathnormal">n</span></span></span><span style="top:-2.895em;"><span class="pstrut" style="height:3.2em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.28em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.28em" viewBox="0 0 400000 1296" preserveAspectRatio="xMinYMin slice"><path d="M263,681c0.7,0,18,39.7,52,119c34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120c340,-704.7,510.7,-1060.3,512,-1067l0 -0c4.7,-7.3,11,-11,19,-11H40000v40H1012.3s-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232c-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1s-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26c-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60zM1001 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305em;"><span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.13em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 为服从自由度为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span> 分布，记作 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi><mo>∼</mo><mi>t</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T \sim t(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">t</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>​</li></ul><p>性质：</p><ul><li>密度函数是偶函数，具备对称性</li></ul><p><strong><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span> 分布</strong></p>    <div class="fold">      <div class="fold-title fold-light collapsed" data-toggle="collapse" href="#collapse-670e1898" role="button" aria-expanded="false" aria-controls="collapse-670e1898">        <div class="fold-arrow">▶</div>密度函数图像      </div>      <div class="fold-collapse collapse" id="collapse-670e1898">        <div class="fold-content">          <p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202405171015933.webp" alt="密度函数图像"></p>        </div>      </div>    </div><p>定义：</p><ul><li>若随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi><mo>∼</mo><msup><mi>χ</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>Y</mi><mo>∼</mo><msup><mi>χ</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \sim \chi^2(m), Y \sim \chi^2(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">m</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 且相互独立</li><li>则称随机变量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi>X</mi><mi mathvariant="normal">/</mi><mi>m</mi></mrow><mrow><mi>Y</mi><mi mathvariant="normal">/</mi><mi>n</mi></mrow></mfrac></mstyle></mrow><annotation encoding="application/x-tex">G=\displaystyle \frac{X/m}{Y/n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">G</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mord">/</span><span class="mord mathnormal">n</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mord">/</span><span class="mord mathnormal">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 服从自由度为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>m</mi><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(m,n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">m</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span> 分布，记作 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>G</mi><mo>∼</mo><mi>F</mi><mo stretchy="false">(</mo><mi>m</mi><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G \sim F(m, n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">G</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">m</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>​</li></ul><p>性质：</p><ul><li>倒数自由度转换：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mi>G</mi></mfrac><mo>∼</mo><mi>F</mi><mo stretchy="false">(</mo><mi>n</mi><mo separator="true">,</mo><mi>m</mi><mo stretchy="false">)</mo></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \frac{1}{G} \sim F(n, m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">G</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span></li><li><a href="https://zhuanlan.zhihu.com/p/382940609">三变性质</a>：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><msub><mi>F</mi><mrow><mn>1</mn><mo>−</mo><mi>α</mi></mrow></msub><mo stretchy="false">(</mo><mi>m</mi><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mrow><mo fence="true">[</mo><msub><mi>F</mi><mi>α</mi></msub><mo stretchy="false">(</mo><mi>n</mi><mo separator="true">,</mo><mi>m</mi><mo stretchy="false">)</mo><mo fence="true">]</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle F_{1-\alpha}(m, n) = \left [F_\alpha (n, m)\right]^{-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">m</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.204em;vertical-align:-0.25em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span><span class="mclose delimcenter" style="top:0em;">]</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.954em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span></li></ul><h4 id="6-4-2-正态总体抽样分布基本定理">6.4.2 正态总体抽样分布基本定理</h4><p>设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>X</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>X</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><msub><mi>X</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X_1,X_2,\cdots ,X_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是来自正态总体 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>μ</mi><mo separator="true">,</mo><msup><mi>σ</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(\mu, \sigma^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mopen">(</span><span class="mord mathnormal">μ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 的简单随机样本，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover><mo separator="true">,</mo><msup><mi>S</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\overline{X},S^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0778em;vertical-align:-0.1944em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 分别是样本均值和修正样本方差。则有：</p><p>定理：</p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><mi>μ</mi><mo separator="true">,</mo><mfrac><msup><mi>σ</mi><mn>2</mn></msup><mi>n</mi></mfrac><mo stretchy="false">)</mo></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \overline{X} \sim N(\mu, \frac{\sigma^2}{n})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mopen">(</span><span class="mord mathnormal">μ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span></span></span></span></li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>S</mi><mn>2</mn></msup></mrow><msup><mi>σ</mi><mn>2</mn></msup></mfrac><mo>∼</mo><msup><mi>χ</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \frac{(n-1)S^2}{\sigma^2} \sim \chi^2(n-1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\overline{X}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>S</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 相互独立</li></ul><p>推论：</p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><msqrt><mi>n</mi></msqrt><mo stretchy="false">(</mo><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover><mo>−</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow><mi>S</mi></mfrac><mo>∼</mo><mi>t</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \frac{\sqrt{n}(\overline{X} - \mu)}{S} \sim t(n-1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2463em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8003em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">n</span></span></span><span style="top:-2.7603em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2397em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">μ</span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">t</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></li></ul><h2 id="第7章-参数估计">第7章 参数估计</h2><div class="note note-light">            <p>有些时候我们知道数据的分布类型，但是不清楚表达式中的某些参数，这就需要我们利用「已有的样本」对分布表达式中的参数进行估计。本章我们将从点估计、估计评价、区间估计三部分出发进行介绍。</p>          </div><h3 id="7-1-点估计">7.1 点估计</h3><div class="note note-light">            <p>所谓点估计策略，就是直接给出参数的一个估计值。本目我们介绍点估计策略中的两个方法：矩估计法、极大似然估计法。</p>          </div><h4 id="7-1-1-矩估计法">7.1.1 矩估计法</h4><p>其实就一句话：我们用样本的原点矩 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">A_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 来代替总体 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><msup><mi>X</mi><mi>k</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(X^k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0991em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个未知参数就需要用到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个原点矩：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><msup><mi>X</mi><mi>k</mi></msup><mo stretchy="false">)</mo><mo>=</mo><msub><mi>A</mi><mi>k</mi></msub><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>X</mi><mi>i</mi><mi>k</mi></msubsup></mrow><annotation encoding="application/x-tex">E(X^k) = A_k =  \frac{1}{n}\sum_{i=1}^nX_i^k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1491em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-2.453em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span></span></p><h4 id="7-1-2-极大似然估计法">7.1.2 极大似然估计法</h4><p>基本原理是：在当前样本数据的局面下，我们希望<strong>找到合适的参数使得当前的样本分布情况发生的概率最大</strong>。由于各样本相互独立，因此我们可以用连乘的概率公式来计算当前局面的概率值：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>θ</mi><mo separator="true">;</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(\theta;x_1,x_2,\cdots,x_n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p><p>上述 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>θ</mi><mo separator="true">;</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><msub><mi>x</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(\theta;x_1,x_2,\cdots,x_n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 即似然函数，目标就是选择适当的参数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span></span> 来最大化似然函数。无论是离散性还是连续型，都可以采用下面的方式来计算极大似然估计：</p><ol><li>写出似然函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(\theta)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span></li><li>将上述似然函数取对数</li><li>求对数似然函数关于所有未知参数的偏导并计算极值点</li><li>解出参数关于样本统计量的表达式</li></ol><p>离散型随机变量的似然函数表达式</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo separator="true">;</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi>P</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mi>i</mi></msub><mo>=</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(\theta) = \prod_{i=1}^n p(x_i;\theta) = \prod_{i=1}^n P(X_i = x_i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∏</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∏</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p><p>连续型随机变量的似然函数表达式</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo separator="true">;</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(\theta) = \prod_{i=1}^n p(x_i;\theta)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∏</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span></span></p><p>可以看出极大似然估计本质上就是一个多元函数求极值的问题。特别地，当我们没法得到参数关于样本统计量的表达式 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(\theta)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">L</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span> 时，可以直接从定义域、原函数恒增或恒减等角度出发求解这个多元函数的极值。</p><h3 id="7-2-估计量的评价标准">7.2 估计量的评价标准</h3><div class="note note-light">            <p>如何衡量不同的点估计方法好坏？我们引入三种点估计量的评价指标：无偏性、有效性、一致性。其中一致性一笔带过，不做详细讨论。补充一点，参数的估计量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span></span> 是关于样本的统计量，因此可以对其进行求期望、方差等操作。</p>          </div><h4 id="7-2-1-无偏性">7.2.1 无偏性</h4><p>顾名思义，就是希望估计出来的参数量尽可能不偏离真实值。我们定义满足下式的估计量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><annotation encoding="application/x-tex">\hat \theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9579em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9579em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span></span></span></span> 为真实参数的无偏估计：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>E</mi><mover accent="true"><mi>θ</mi><mo>^</mo></mover><mo>=</mo><mi>θ</mi></mrow><annotation encoding="application/x-tex">E\hat \theta =\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9579em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9579em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span></span></span></p><h4 id="7-2-2-有效性">7.2.2 有效性</h4><p>有效性是基于比较的定义方法。对于两个无偏估计 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>θ</mi><mo>^</mo></mover><mn>1</mn></msub><mo separator="true">,</mo><msub><mover accent="true"><mi>θ</mi><mo>^</mo></mover><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\hat\theta_1,\hat\theta_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1523em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9579em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9579em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，谁的方差越小谁就越有效。即若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><msub><mover accent="true"><mi>θ</mi><mo>^</mo></mover><mn>1</mn></msub><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>D</mi><mo stretchy="false">(</mo><msub><mover accent="true"><mi>θ</mi><mo>^</mo></mover><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(\hat\theta_1),D(\hat\theta_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2079em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9579em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9579em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 满足下式，则称 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>θ</mi><mo>^</mo></mover><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\hat\theta_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1079em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9579em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 更有效</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><msub><mover accent="true"><mi>θ</mi><mo>^</mo></mover><mn>1</mn></msub><mo stretchy="false">)</mo><mo>&lt;</mo><mi>D</mi><mo stretchy="false">(</mo><msub><mover accent="true"><mi>θ</mi><mo>^</mo></mover><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(\hat\theta_1) &lt; D(\hat\theta_2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2079em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9579em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2079em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mopen">(</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9579em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p><h4 id="7-2-3-一致性">7.2.3 一致性</h4><p>即当样本容量 n 趋近于无穷时，参数的估计值也能趋近于真实值，则称该估计量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>θ</mi><mo>^</mo></mover></mrow><annotation encoding="application/x-tex">\hat\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9579em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9579em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span><span style="top:-3.2634em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span></span></span></span> 为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span></span> 的一致估计量</p><h3 id="7-3-区间估计">7.3 区间估计</h3><div class="note note-light">            <p>由于点估计只能进行比较，无法对单一估计进行性能度量。因此引入「主元法」的概念与「区间估计」策略</p>          </div><h4 id="7-3-1-基本概念">7.3.1 基本概念</h4><p>可靠程度：参数估计区间越长，可靠程度越高</p><p>精确程度：参数估计区间越短，可靠程度越高</p><h4 id="7-3-2-区间估计常用方法之主元法">7.3.2 区间估计常用方法之主元法</h4><p>主元法的核心逻辑就一个：在已知数据总体分布的情况下，构造一个关于样本 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span></span> 和待估参数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span></span> 的函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z(X,\theta)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span>，然后利用置信度和总体分布函数，通过查表得到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z(X,\theta)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span> 的取值范围，最后通过移项变形得到待估参数的区间，也就是估计区间。</p><h4 id="7-3-3-正态总体的区间估计">7.3.3 正态总体的区间估计</h4><p>我们只需要掌握「一个总体服从正态分布」的情况。这种情况下的区间估计分为三种，其中估计均值 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span> 有 2 种，估计方差 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\sigma^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 有 1 种。估计的逻辑我总结为了以下三步：</p><ol><li>构造主元 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z(X,\theta)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span></li><li>利用置信度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>−</mo><mi>α</mi></mrow><annotation encoding="application/x-tex">1-\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 计算主元 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span></span></span> 的取值范围</li><li>对主元 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span></span></span> 的取值范围移项得到参数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span></span> 的取值范围</li></ol><p>为了提升区间估计的可信度，我们希望上述第 2 步计算出来的关于主元的取值范围尽可能准确。我们不加证明的给出以下结论：取主元的取值范围为 <strong>主元服从的分布的上下 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mi>α</mi><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{\alpha}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 分位数之间</strong>。</p><p><strong>(一) 求 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span> 的置信区间，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\sigma^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 已知</strong></p><p>构造主元 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z(X,\theta)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span>：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>Z</mi><mo>=</mo><mfrac><mrow><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover><mo>−</mo><mi>μ</mi></mrow><mrow><mi>σ</mi><mi mathvariant="normal">/</mi><msqrt><mi>n</mi></msqrt></mrow></mfrac><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z = \frac{\overline{X} - \mu}{\sigma / \sqrt{n}} \sim N(0,1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.5006em;vertical-align:-0.9403em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5603em;"><span style="top:-2.3097em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mord">/</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8003em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">n</span></span></span><span style="top:-2.7603em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path 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class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9403em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></span></p><p>利用置信度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>−</mo><mi>α</mi></mrow><annotation encoding="application/x-tex">1-\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 计算主元 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span></span></span> 的取值范围：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>P</mi><mo stretchy="false">(</mo><mi mathvariant="normal">∣</mi><mi>Z</mi><mi mathvariant="normal">∣</mi><mo>≤</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>1</mn><mo>−</mo><mi>α</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>↓</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>Z</mi><mo>∈</mo><mo stretchy="false">[</mo><mo>−</mo><mi>λ</mi><mo separator="true">,</mo><mi>λ</mi><mo stretchy="false">]</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo stretchy="false">[</mo><mo>−</mo><msub><mi>u</mi><mfrac><mi>α</mi><mn>2</mn></mfrac></msub><mo separator="true">,</mo><msub><mi>u</mi><mfrac><mi>α</mi><mn>2</mn></mfrac></msub><mo stretchy="false">]</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}P(|Z| \le \lambda) &amp;= 1-\alpha \\&amp;\downarrow\\Z \in [-\lambda,\lambda] &amp;= [-u_{\frac{\alpha}{2}},u_\frac{\alpha}{2}]\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.5308em;vertical-align:-2.0154em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5154em;"><span style="top:-4.6754em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">λ</span><span class="mclose">)</span></span></span><span style="top:-3.1754em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-1.6754em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">[</span><span class="mord">−</span><span class="mord mathnormal">λ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">λ</span><span class="mclose">]</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.0154em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5154em;"><span style="top:-4.6754em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span><span style="top:-3.1754em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">↓</span></span></span><span style="top:-1.6754em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">[</span><span class="mord">−</span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3341em;"><span style="top:-2.85em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3908em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3341em;"><span style="top:-2.85em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3908em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.0154em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>对主元 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span></span></span> 的取值范围移项得到参数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span></span> 的取值范围：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover><mo>−</mo><mfrac><mi>σ</mi><msqrt><mi>n</mi></msqrt></mfrac><msub><mi>u</mi><mfrac><mi>α</mi><mn>2</mn></mfrac></msub><mo>≤</mo><mi>μ</mi><mo>≤</mo><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover><mo>+</mo><mfrac><mi>σ</mi><msqrt><mi>n</mi></msqrt></mfrac><msub><mi>u</mi><mfrac><mi>α</mi><mn>2</mn></mfrac></msub></mrow><annotation encoding="application/x-tex">\overline{X} - \frac{\sigma}{\sqrt{n}} u_\frac{\alpha}{2} \le \mu \le \overline{X} + \frac{\sigma}{\sqrt{n}} u_\frac{\alpha}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9667em;vertical-align:-0.0833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0376em;vertical-align:-0.93em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.3097em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8003em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">n</span></span></span><span style="top:-2.7603em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2397em;"><span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.93em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3341em;"><span style="top:-2.85em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3908em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9667em;vertical-align:-0.0833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0376em;vertical-align:-0.93em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.3097em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8003em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">n</span></span></span><span style="top:-2.7603em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2397em;"><span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.93em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3341em;"><span style="top:-2.85em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3908em;"><span></span></span></span></span></span></span></span></span></span></span></p><p><strong>(二) 求 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span> 的置信区间，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\sigma^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 未知</strong></p><p>构造主元 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z(X,\theta)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span>：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>Z</mi><mo>=</mo><mfrac><mrow><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover><mo>−</mo><mi>μ</mi></mrow><mrow><mi>S</mi><mi mathvariant="normal">/</mi><msqrt><mi>n</mi></msqrt></mrow></mfrac><mo>∼</mo><mi>t</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z = \frac{\overline{X} - \mu}{S / \sqrt{n}} \sim t(n-1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.5006em;vertical-align:-0.9403em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5603em;"><span style="top:-2.3097em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mord">/</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8003em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">n</span></span></span><span style="top:-2.7603em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2397em;"><span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9403em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">t</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></span></p><p>利用置信度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>−</mo><mi>α</mi></mrow><annotation encoding="application/x-tex">1-\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 计算主元 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span></span></span> 的取值范围：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>P</mi><mo stretchy="false">(</mo><mi mathvariant="normal">∣</mi><mi>Z</mi><mi mathvariant="normal">∣</mi><mo>≤</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>1</mn><mo>−</mo><mi>α</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>↓</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>Z</mi><mo>∈</mo><mo stretchy="false">[</mo><mo>−</mo><mi>λ</mi><mo separator="true">,</mo><mi>λ</mi><mo stretchy="false">]</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo stretchy="false">[</mo><mo>−</mo><msub><mi>t</mi><mfrac><mi>α</mi><mn>2</mn></mfrac></msub><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><msub><mi>t</mi><mfrac><mi>α</mi><mn>2</mn></mfrac></msub><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}P(|Z| \le \lambda) &amp;= 1-\alpha \\&amp;\downarrow\\Z \in [-\lambda,\lambda] &amp;= [-t_{\frac{\alpha}{2}}(n-1),t_\frac{\alpha}{2}(n-1)]\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.5308em;vertical-align:-2.0154em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5154em;"><span style="top:-4.6754em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">λ</span><span class="mclose">)</span></span></span><span style="top:-3.1754em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-1.6754em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">[</span><span class="mord">−</span><span class="mord mathnormal">λ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">λ</span><span class="mclose">]</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.0154em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5154em;"><span style="top:-4.6754em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span><span style="top:-3.1754em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">↓</span></span></span><span style="top:-1.6754em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">[</span><span class="mord">−</span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3341em;"><span style="top:-2.85em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3908em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3341em;"><span style="top:-2.85em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3908em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose">)]</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.0154em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>对主元 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span></span></span> 的取值范围移项得到参数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span></span> 的取值范围：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover><mo>−</mo><mfrac><mi>S</mi><msqrt><mi>n</mi></msqrt></mfrac><msub><mi>t</mi><mfrac><mi>α</mi><mn>2</mn></mfrac></msub><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≤</mo><mi>μ</mi><mo>≤</mo><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover><mo>+</mo><mfrac><mi>S</mi><msqrt><mi>n</mi></msqrt></mfrac><msub><mi>t</mi><mfrac><mi>α</mi><mn>2</mn></mfrac></msub><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\overline{X} - \frac{S}{\sqrt{n}} t_\frac{\alpha}{2}(n-1) \le \mu \le \overline{X} + \frac{S}{\sqrt{n}} t_\frac{\alpha}{2}(n-1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9667em;vertical-align:-0.0833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.2903em;vertical-align:-0.93em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.3097em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8003em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">n</span></span></span><span style="top:-2.7603em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2397em;"><span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.93em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3341em;"><span style="top:-2.85em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3908em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9667em;vertical-align:-0.0833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.2903em;vertical-align:-0.93em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.3097em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8003em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">n</span></span></span><span style="top:-2.7603em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2397em;"><span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.93em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3341em;"><span style="top:-2.85em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3908em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></span></p><p><strong>(三) 求 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\sigma^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 的置信区间，构造的主元与总体均值无关，因此不需要考虑 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span> 的情况：</strong></p><p>构造主元 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><mi>X</mi><mo separator="true">,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z(X,\theta)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span>：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>Z</mi><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>S</mi><mn>2</mn></msup></mrow><msup><mi>σ</mi><mn>2</mn></msup></mfrac><mo>∼</mo><msup><mi>χ</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z = \frac{(n-1)S^2}{\sigma^2}\sim \chi^2(n-1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></span></p><p>利用置信度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>−</mo><mi>α</mi></mrow><annotation encoding="application/x-tex">1-\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 计算主元 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span></span></span> 的取值范围：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mi>λ</mi><mn>1</mn></msub><mo>≤</mo><mi>Z</mi><mo>≤</mo><msub><mi>λ</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mn>1</mn><mo>−</mo><mi>α</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>↓</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>Z</mi><mo>∈</mo><mo stretchy="false">[</mo><msub><mi>λ</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>λ</mi><mn>2</mn></msub><mo stretchy="false">]</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo stretchy="false">[</mo><msubsup><mi>χ</mi><mrow><mn>1</mn><mo>−</mo><mfrac><mi>α</mi><mn>2</mn></mfrac></mrow><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo separator="true">,</mo><msubsup><mi>χ</mi><mfrac><mi>α</mi><mn>2</mn></mfrac><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}P(\lambda_1 \le Z \le \lambda_2) &amp;= 1-\alpha \\&amp;\downarrow\\Z \in [\lambda_1,\lambda_2] &amp;= [\chi^2_{1-\frac{\alpha}{2}}(n-1),\chi^2_\frac{\alpha}{2}(n-1)]\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.6519em;vertical-align:-2.076em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.576em;"><span style="top:-4.736em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-3.236em;"><span class="pstrut" style="height:3em;"></span><span class="mord"></span></span><span style="top:-1.7118em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.076em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.576em;"><span style="top:-4.736em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span><span style="top:-3.236em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">↓</span></span></span><span style="top:-1.7118em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.753em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mbin mtight">−</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span><span style="top:-3.413em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4878em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.753em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span><span style="top:-3.413em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4878em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose">)]</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.076em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>对主元 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span></span></span> 的取值范围移项得到参数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span></span> 的取值范围：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>S</mi><mn>2</mn></msup></mrow><mrow><msubsup><mi>χ</mi><mfrac><mi>α</mi><mn>2</mn></mfrac><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><mo>≤</mo><msup><mi>σ</mi><mn>2</mn></msup><mo>≤</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>S</mi><mn>2</mn></msup></mrow><mrow><msubsup><mi>χ</mi><mrow><mn>1</mn><mo>−</mo><mfrac><mi>α</mi><mn>2</mn></mfrac></mrow><mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{(n-1)S^2}{\chi^2_\frac{\alpha}{2}(n-1)} \le \sigma^2 \le \frac{(n-1)S^2}{\chi^2_{1-\frac{\alpha}{2}}(n-1)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.7172em;vertical-align:-1.2261em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7959em;"><span style="top:-2.7007em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span><span style="top:-3.3448em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.5401em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2261em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0001em;vertical-align:-0.136em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.7172em;vertical-align:-1.2261em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7959em;"><span style="top:-2.7007em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mbin mtight">−</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.2255em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.0037em;">α</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span><span style="top:-3.3448em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.5401em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2261em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><h2 id="第8章-假设检验">第8章 假设检验</h2><div class="note note-light">            <p>第 7 章的参数估计是在总体分布已知且未知分布表达式中某些参数的情况下，基于「抽取的少量样本」进行的参数估计。</p><p>现在的局面同样，我们已知总体分布和不完整的分布表达式参数。现在需要我们利用抽取的少量样本判断样本所在向量空间是否符合某种性质。本章的「假设检验」策略就是为了解决上述情况而诞生的。我们主要讨论单个正态总体的情况并针对均值和方差两个参数进行假设和检验：</p><ul><li>假设均值满足某种趋势，利用已知数据判断假设是否成立</li><li>假设方差满足某种趋势，利用已知数据判断假设是否成立</li></ul>          </div><h3 id="8-1-假设检验的基本概念">8.1 假设检验的基本概念</h3><p>基本思想：首先做出假设并构造一个关于样本观察值和已知参数的检验统计量，接着计算假设发生的情况下小概率事件发生时该检验统计量的取值范围（拒绝域），最终代入已知样本数据判断计算结果是否在拒绝域内。如果在，则说明在当前假设的情况下小概率事件发生了，对应的假设为假；反之说明假设为真。</p><p>为了量化「小概率事件发生」这个指标，我们引入显著性水平 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 这一概念。该参数为一个很小的正数，定义为「小概率事件发生」的概率上界。</p><p>基于数据的实验导致我们无法避免错误，因此我们定义以下两类错误：</p><ul><li>第一类错误：弃真错误。即假设正确，但由于数据采样不合理导致拒绝了真实的假设</li><li>第二类错误：存伪错误。即假设错误，同样因为数据的不合理导致接受了错误的假设</li></ul><h3 id="8-2-单个正态总体均值的假设检验">8.2 单个正态总体均值的假设检验</h3><p>设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>X</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>X</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><msub><mi>X</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X_1,X_2,\cdots ,X_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是来自正态总体 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>μ</mi><mo separator="true">,</mo><msup><mi>σ</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(\mu, \sigma^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mopen">(</span><span class="mord mathnormal">μ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 的简单随机样本。后续进行假设判定计算统计量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span></span></span> 的真实值时，若总体均值 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span> 已知就直接代入，若未知题目也一定会给一个阈值，代这个阈值即可。</p><p>当总体方差 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\sigma^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 已知时，我们构造样本统计量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span></span></span> 为正态分布：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>Z</mi><mo>=</mo><mfrac><mrow><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover><mo>−</mo><mi>μ</mi></mrow><mrow><mi>σ</mi><mi mathvariant="normal">/</mi><msqrt><mi>n</mi></msqrt></mrow></mfrac><mo>∼</mo><mi>N</mi><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z = \frac{\overline{X} - \mu}{\sigma / \sqrt{n}} \sim N(0,1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.5006em;vertical-align:-0.9403em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5603em;"><span style="top:-2.3097em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mord">/</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8003em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">n</span></span></span><span style="top:-2.7603em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2397em;"><span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9403em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></span></p><ul><li>检验是否则求解双侧 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 分位数</li><li>检验单边则求解单侧 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 分位数</li></ul><p>当总体方差 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\sigma^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 未知时，我们构造样本统计量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span></span></span> 为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span></span></span></span> 分布：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>Z</mi><mo>=</mo><mfrac><mrow><mover accent="true"><mi>X</mi><mo stretchy="true">‾</mo></mover><mo>−</mo><mi>μ</mi></mrow><mrow><mi>S</mi><mi mathvariant="normal">/</mi><msqrt><mi>n</mi></msqrt></mrow></mfrac><mo>∼</mo><mi>t</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z = \frac{\overline{X} - \mu}{S / \sqrt{n}} \sim t(n-1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.5006em;vertical-align:-0.9403em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5603em;"><span style="top:-2.3097em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mord">/</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8003em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">n</span></span></span><span style="top:-2.7603em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2397em;"><span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">μ</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9403em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">t</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></span></p><div class="note note-warning">            <p>注：之所以这样构造是因为当总体 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> 未知时，上一个方法构造的主元已经不再是统计量，我们需要找到能够代替未知参数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span></span></span></span> 的变量，这里就采用其无偏估计「修正样本方差 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>S</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>」来代替 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\sigma^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span>。也是说直接拿样本的修正方差来代替总体的方差了。</p>          </div><ul><li>检验是否则求解双侧 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 分位数</li><li>检验单边则求解单侧 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 分位数</li></ul><h3 id="8-3-单个正态总体方差的假设检验">8.3 单个正态总体方差的假设检验</h3><p>设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>X</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>X</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><msub><mi>X</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X_1,X_2,\cdots ,X_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是来自正态总体 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>μ</mi><mo separator="true">,</mo><msup><mi>σ</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(\mu, \sigma^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mopen">(</span><span class="mord mathnormal">μ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 的简单随机样本。后续进行假设判定计算统计量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span></span></span> 的真实值时，若总体方差 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\sigma^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 已知就直接代入，若未知题目也一定会给一个阈值，代这个阈值即可。</p><p>我们直接构造样本统计量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span></span></span></span> 为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>χ</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\chi^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0085em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 分布：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>Z</mi><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>S</mi><mn>2</mn></msup></mrow><msup><mi>σ</mi><mn>2</mn></msup></mfrac><mo>∼</mo><msup><mi>χ</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z = \frac{(n-1)S^2}{\sigma^2} \sim \chi^2(n-1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∼</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">χ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></span></p><ul><li>检验是否则求解双侧 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 分位数</li><li>检验单边则求解单侧 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span></span></span></span> 分位数</li></ul>]]></content>
     
     
     <categories>
@@ -1073,11 +1073,11 @@
   
   
   <entry>
-    <title>DataStructure</title>
-    <link href="/GPA/3rd-term/DataStructure/"/>
-    <url>/GPA/3rd-term/DataStructure/</url>
+    <title>DigitalLogicCircuit</title>
+    <link href="/GPA/3rd-term/DigitalLogicCircuit/"/>
+    <url>/GPA/3rd-term/DigitalLogicCircuit/</url>
     
-    <content type="html"><![CDATA[<blockquote><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">s</mi><mi mathvariant="script">c</mi><mi mathvariant="script">o</mi><mi mathvariant="script">r</mi><mi mathvariant="script">e</mi><mo>:</mo></mrow><annotation encoding="application/x-tex">\mathscr {score:}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord"><span class="mord mathnormal">score</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span></span></span></span></span></p><ul><li>平时 20%（出勤、作业、实验）</li><li>期中 20%</li><li>期末 60%</li></ul></blockquote><h2 id="数据结构">数据结构</h2><p>完整实现代码：<a href="https://github.com/Explorer-Dong/DataStructure">https://github.com/Explorer-Dong/DataStructure</a></p><h2 id="一、绪论">一、绪论</h2><h3 id="1-1-数据分析-结构存储-算法计算">1.1 数据分析+结构存储+算法计算</h3><h4 id="1-1-1-逻辑结构">1.1.1 逻辑结构</h4><p>对于当前的数据之间的关系进行分析，进而思考应该如何存储，有以下几种逻辑结构：集合、线性结构、树形结构、图结构。</p><h4 id="1-1-2-存储结构">1.1.2 存储结构</h4><p>设计一定的方法存储到程序中，需要思考：存什么？怎么存？</p><ul><li>存什么？数值存储、数据与数据之间关系域的存储</li><li>怎么存？顺序存储、链式存储、树形存储、图存储</li></ul><h4 id="1-1-3-算法实现">1.1.3 算法实现</h4><p>设计算法计算实现</p><h3 id="1-2-数据类型">1.2 数据类型</h3><p>约束：值集 + 运算集</p><p>数据类型 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>(Data&nbsp;Type,&nbsp;简称&nbsp;DT)</mtext></mrow><annotation encoding="application/x-tex">\text{(Data Type, 简称 DT)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">(Data&nbsp;Type,&nbsp;</span><span class="mord cjk_fallback">简称</span><span class="mord">&nbsp;DT)</span></span></span></span></span>：一般编程语言已经实现好了</p><p>抽象数据类型 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Abstract&nbsp;Data&nbsp;Type,&nbsp;简称&nbsp;ADT</mtext></mrow><annotation encoding="application/x-tex">\text{Abstract Data Type, 简称 ADT}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">Abstract&nbsp;Data&nbsp;Type,&nbsp;</span><span class="mord cjk_fallback">简称</span><span class="mord">&nbsp;ADT</span></span></span></span></span>：数据结构 + 算法操作</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218513.png" alt="ADT 的不同视图"></p><h3 id="1-3-算法方法">1.3 算法方法</h3><ol><li><p>正确性</p></li><li><p>健壮性（鲁棒性）：对于不合法、异常的输入也有处理能力</p></li><li><p>可读性</p></li><li><p>可扩展性</p></li><li><p>高效率</p><ol><li><p>空间复杂度</p></li><li><p>时间复杂度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi>O</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(n)=O(f(n))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">))</span></span></span></span>，其中有三种表示时间复杂度的公式</p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O()</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mclose">)</span></span></span></span> <code>upper bound</code>：最坏的时间复杂度</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi><mo stretchy="false">(</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega()</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Ω</span><span class="mopen">(</span><span class="mclose">)</span></span></span></span> <code>lower bound</code>：最好的时间复杂度</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Θ</mi><mo stretchy="false">(</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta()</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Θ</span><span class="mopen">(</span><span class="mclose">)</span></span></span></span> <code>average bound</code>：平均时间复杂度</li></ul></li></ol></li></ol><h2 id="二、线性表">二、线性表</h2><h3 id="2-1-线性表的逻辑结构">2.1 线性表的逻辑结构</h3><p>有一个头结点，尾结点，且每一个结点只有一个前驱结点和一个后继结点。</p><h3 id="2-2-线性表的存储结构">2.2 线性表的存储结构</h3><h4 id="2-2-1-顺序存储结构">2.2.1 顺序存储结构</h4><p>存储在一维地址连续的存储单元里</p><p>特点：逻辑位置相邻，物理位置也相邻</p><p>数据结构：一个一个一维数组 + 一个长度变量 n</p><figure class="highlight c++"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br></pre></td><td class="code"><pre><code class="hljs c++"><span class="hljs-keyword">template</span>&lt;<span class="hljs-keyword">class</span> <span class="hljs-title class_">T</span>, <span class="hljs-type">int</span> MaxSize&gt;<br><span class="hljs-keyword">class</span> <span class="hljs-title class_">SeqList</span><br>{<br>    T data[MazSize];<br>    <span class="hljs-type">int</span> length;<br><span class="hljs-keyword">public</span>:<br>    ...<br>}<br></code></pre></td></tr></tbody></table></figure><p>顺序表可以直接存储元素与关系。链表的元素存储也是可以直接实现的，但是关系要通过指针域来实现</p><h4 id="2-2-2-链式存储结构">2.2.2 链式存储结构</h4><ol><li><p>单链表：默认有一个头结点，不存储数据</p></li><li><p>循环链表</p></li><li><p>双向链表</p></li></ol><h3 id="2-3-线性表的操作算法">2.3 线性表的操作算法</h3><h4 id="2-3-1-顺序表的操作算法">2.3.1 顺序表的操作算法</h4><ol><li><p>初始化构造</p></li><li><p>求顺序表长度</p></li><li><p>按位查找</p></li><li><p>按值查找</p></li><li><p>遍历顺序表</p></li><li><p>插入</p></li><li><p>删除</p></li></ol><h4 id="2-3-2-链表的操作算法">2.3.2 链表的操作算法</h4><ol><li><p>单链表初始化构造</p> <figure class="highlight c++"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><code class="hljs c++">head = <span class="hljs-keyword">new</span> Node&lt;T&gt;;<br>head-&gt;next = <span class="hljs-literal">nullptr</span>;<br></code></pre></td></tr></tbody></table></figure><ul><li><p>头插法</p>  <figure class="highlight c++"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><code class="hljs c++">head = <span class="hljs-keyword">new</span> Node&lt;T&gt;;<br>head-&gt;next = <span class="hljs-literal">nullptr</span>;<br></code></pre></td></tr></tbody></table></figure></li><li><p>尾插法</p>  <figure class="highlight c++"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><code class="hljs c++">head = <span class="hljs-keyword">new</span> Node&lt;T&gt;;<br>rear = head;<br></code></pre></td></tr></tbody></table></figure></li></ul></li><li><p>求单链表长度</p></li><li><p>按位查找</p></li><li><p>按值查找</p></li><li><p>遍历单链表</p></li><li><p>插入</p></li><li><p>删除</p></li><li><p>单链表的析构函数</p></li><li><p>其他操作</p></li><li><p>双向链表操作</p><ul><li><p>插入</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218514.png" alt="插入"></p>  <figure class="highlight c++"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br></pre></td><td class="code"><pre><code class="hljs c++"><span class="hljs-comment">// 插入当前结点 s</span><br>s-&gt;prior = p;<br>s-&gt;next = p-&gt;next;<br>p-&gt;next-&gt;prior = s;<br>p-&gt;next = s;<br></code></pre></td></tr></tbody></table></figure></li><li><p>删除</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218515.png" alt="删除"></p>  <figure class="highlight c++"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><code class="hljs c++"><span class="hljs-comment">// 删除当前结点 p</span><br>p-&gt;next-&gt;prior = p-&gt;prior;<br>p-&gt;prior-&gt;next = p-&gt;next;<br></code></pre></td></tr></tbody></table></figure></li></ul></li></ol><h2 id="三、栈和队列">三、栈和队列</h2><h3 id="3-1-栈">3.1 栈</h3><h4 id="3-1-1-栈的基本概念">3.1.1 栈的基本概念</h4><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218516.png" alt="卡特兰数"></p><p>卡特兰数：假设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span> 表示第 k 个数最后一个出栈的总个数，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(k)=f(k-1)f(n-k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi>f</mi><mo stretchy="false">(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfrac><msubsup><mi>C</mi><mrow><mn>2</mn><mi>n</mi></mrow><mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">f(n) = \sum_{k=1}^{n} f(k-1) f(n-k)=\frac{1}{n+1} C_{2n}^{n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9535em;vertical-align:-1.3021em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8479em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3021em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0908em;vertical-align:-0.7693em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight">n</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span></span></p><h4 id="3-1-2-栈的存储结构">3.1.2 栈的存储结构</h4><p>顺序存储</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218517.png" alt="顺序存储"></p><p>链式存储</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218518.png" alt="链式存储"></p><h4 id="3-1-3-栈的操作算法">3.1.3 栈的操作算法</h4><ol><li>顺序栈的操作</li><li>链栈的操作</li></ol><h4 id="3-1-4-栈的应用">3.1.4 栈的应用</h4><ol><li><p>括号匹配</p></li><li><p>算数表达式求值</p><ul><li><p>中缀表达式求值</p><blockquote><p><strong>双栈</strong>思路，算符优先法</p><ul><li><p>遇到数字，直接入数栈</p></li><li><p>遇到符号</p><ul><li>如果是括号，左括号直接入栈，右括号进行<strong>运算</strong>直到遇到左括号</li><li>如果是算符，在入算符栈之前，需要进行<strong>运算</strong>操作直到算符栈顶元素等级小于当前算符等级</li></ul></li></ul></blockquote></li><li><p>中缀表达式转后缀表达式</p><blockquote><p><strong>算符栈</strong>即可</p><p>后缀先遇到就直接计算的运算符 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> 中缀表达式需要先算的运算符，于是转化思路就是：</p><ul><li>遇到数字，直接构造后缀表达式</li><li>遇到算符<ul><li>如果是括号，左括号直接入栈，右括号进行<strong>后缀表达式构造</strong>直到遇到左括号</li><li>如果是算符，在入算符栈之前，需要进行<strong>后缀表达式构造</strong>操作直到算符栈顶元素等级小于当前算符等级</li></ul></li></ul></blockquote></li><li><p>后缀表达式求值</p><blockquote><p><strong>数栈</strong>即可</p><p>遇到数字直接入数栈，遇到算符直接进行运算</p></blockquote></li></ul></li><li><p>栈与递归</p><p>递归工作栈</p></li></ol><h3 id="3-2-队列">3.2 队列</h3><h4 id="3-2-1-队列的基本概念">3.2.1 队列的基本概念</h4><p>先进先出</p><h4 id="3-2-2-队列的存储结构">3.2.2 队列的存储结构</h4><p>顺序存储</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218519.png" alt="顺序存储 - 1"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218520.png" alt="顺序存储 - 2"></p><p>链式存储</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218521.png" alt="链式存储"></p><h4 id="3-2-3-队列的操作算法">3.2.3 队列的操作算法</h4><ol><li><p>循环队列的操作</p><blockquote><p>循环队列的三个注意点</p><ul><li>解决假溢出：采用循环队列，即在入队的时候不是单纯的指针 +1，而是+1后 % MaxSize</li><li>解决队空队满的冲突（真溢出）：<ol><li>浪费一个元素空间：测试rear+1是否==head，</li><li>设置一个辅助标志变量</li><li>设置一个计数器</li></ol></li></ul></blockquote><ol><li>初始化：头尾全部初始化为0</li><li>入队push</li><li>出队pop</li><li>取队头front</li><li>长度size</li><li>队空empty</li></ol></li><li><p>链队列的操作</p></li></ol><h4 id="3-2-4-队列的应用">3.2.4 队列的应用</h4><ol><li><p>报数问题：报到 0 的出队，报到 1 的重新入队，求解出队顺序</p></li><li><p>迷宫最短路问题：开一个记忆数组 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">d[i][j]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">d</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">]</span></span></span></span> 表示从起点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span> 到终点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo separator="true">,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i,j)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">)</span></span></span></span> 点的最短路径的长度。可以将求最短路看做一个<strong>波心扩散</strong>的物理场景，队列中的每一个点都可以作为一个波心，从而实现“两点之间线段最短”的物理场景</p><ul><li>为什么用队列：逐层搜索，每次搜素到的点就是当前点可以搜索到的最短的点，先搜到的点先扩展，于是就是队列的数据结构</li><li>为什么最短：对于每一个点探索到的点都是最短的点，最终的搜索出来的路径就是最短的路径</li></ul></li></ol><h2 id="四、串">四、串</h2><h3 id="4-1-串的基本概念">4.1 串的基本概念</h3><p>由字符组成的串：子串、主串、位置</p><h3 id="4-2-串的存储结构">4.2 串的存储结构</h3><h4 id="4-2-1-串的顺序存储">4.2.1 串的顺序存储</h4><p>使用固定长度的数组来存储，3种存储字符串长度的方法如下：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218522.png" alt="存储字符串长度的方法 - 1"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218523.png" alt="存储字符串长度的方法 - 2"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218524.png" alt="存储字符串长度的方法 - 3"></p><h4 id="4-2-2-串的链式存储">4.2.2 串的链式存储</h4><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>存储密度</mtext><mo>=</mo><mfrac><mtext>串值所占的内存</mtext><mtext>一个结点的总内存</mtext></mfrac></mrow><annotation encoding="application/x-tex">\text{存储密度} = \frac {\text{串值所占的内存}}{\text{一个结点的总内存}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord cjk_fallback">存储密度</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0463em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">一个结点的总内存</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">串值所占的内存</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><p>非压缩形式：一个结点存一个字符</p><figure class="highlight c++"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br></pre></td><td class="code"><pre><code class="hljs c++"><span class="hljs-comment">// 存储密度为：1/9 （64位操作系统）</span><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">String</span> {<br>    <span class="hljs-type">char</span> data;<br>    String* next;<br>};<br></code></pre></td></tr></tbody></table></figure><p>压缩形式（块链）：一个结点存储指定长度的字符</p><figure class="highlight c++"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br></pre></td><td class="code"><pre><code class="hljs c++"><span class="hljs-comment">// 存储密度为：4/12 （64位操作系统）</span><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> MaxSize = <span class="hljs-number">4</span>;<br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">String</span> {<br>    <span class="hljs-type">char</span> data[MaxSize];<br>    String* next;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="4-3-串的操作算法">4.3 串的操作算法</h3><h4 id="4-3-1-串的基本操作算法">4.3.1 串的基本操作算法</h4><p>串连接、串比较、串拷贝</p><h4 id="4-3-2-串的模式匹配">4.3.2 串的模式匹配</h4><p>BF算法（Brute - Force）</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218525.png" alt="BF算法（Brute - Force）"></p><figure class="highlight c++"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br></pre></td><td class="code"><pre><code class="hljs c++"><span class="hljs-comment">// 返回匹配上的所有位置下标（下标从0开始）</span><br><span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">BF</span><span class="hljs-params">(string&amp; s, string&amp; t)</span> </span>{<br>vector&lt;<span class="hljs-type">int</span>&gt; res;<br><span class="hljs-type">int</span> i = <span class="hljs-number">0</span>, j = <span class="hljs-number">0</span>, n = s.<span class="hljs-built_in">size</span>(), m = t.<span class="hljs-built_in">size</span>();<br>    <br><span class="hljs-keyword">while</span> (i &lt; n &amp;&amp; j &lt; m) {<br><span class="hljs-keyword">if</span> (s[i] == t[j]) i++, j++;<br><span class="hljs-keyword">else</span> i = i - j + <span class="hljs-number">1</span>, j = <span class="hljs-number">0</span>;<br>            <br><span class="hljs-keyword">if</span> (j == m) {<br>res.<span class="hljs-built_in">emplace_back</span>(i - j);<br>j = <span class="hljs-number">0</span>;<br>}<br>}<br><br>    <span class="hljs-keyword">return</span> res;<br>}<br></code></pre></td></tr></tbody></table></figure><p>KMP算法</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218526.png" alt="KMP算法"></p><blockquote><p><strong>优化思想</strong>：</p><p>先看暴力思想，我们需要每次将模式串 t 后移一位重新进行比较，其中浪费了已匹配的串数据，优化就是从这块已匹配的串数据入手。而已匹配的串数据就是模式串本身的串数据，因为我们可以直接从模式串本身入手。</p><p><strong>初步猜想</strong>：</p><p>根据模式串的性质，构造一个数表 <code>next</code>，存储模式串应该后移的指针数 <code>k</code></p><p><strong>算法实现</strong>：</p><ol><li>递推求 <code>next</code> 数组</li><li>KMP 中 <code>i</code> 指针不回溯，<code>j</code> 回溯到 <code>next[j]</code></li></ol></blockquote><figure class="highlight c++"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br></pre></td><td class="code"><pre><code class="hljs c++"><span class="hljs-comment">// 求 next 数组下标从1开始</span><br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">2</span>, j = <span class="hljs-number">0</span>; i &lt;= m; i++) {<br>    <span class="hljs-keyword">while</span> (j &amp;&amp; t[i] != t[j + <span class="hljs-number">1</span>])<br>        <span class="hljs-comment">// 未匹配上则不断回溯</span><br>        j = ne[j];<br>    <br>    <span class="hljs-keyword">if</span> (t[i] == t[j + <span class="hljs-number">1</span>])<br>        <span class="hljs-comment">// 匹配上了则j指针后移一位</span><br>        j++;<br>    <br>    ne[i] = j;<br>}<br></code></pre></td></tr></tbody></table></figure><figure class="highlight c++"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br></pre></td><td class="code"><pre><code class="hljs c++"><span class="hljs-comment">// KMP 匹配 下标从1开始</span><br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>, j = <span class="hljs-number">0</span>; i &lt;= n; i++) {<br>    <span class="hljs-keyword">while</span> (j &amp;&amp; news[i] != newt[j + <span class="hljs-number">1</span>])<br>        <span class="hljs-comment">// 未匹配上则不断回溯</span><br>        j = ne[j];<br>    <br>    <span class="hljs-keyword">if</span> (news[i] == newt[j + <span class="hljs-number">1</span>])<br>        <span class="hljs-comment">// 匹配上了则j指针后移一位</span><br>        j++;<br><br>    <span class="hljs-keyword">if</span> (j == m) {<br>        <span class="hljs-comment">// 匹配完全，则统计并且回溯</span><br>        cnt++;<br>        j = ne[j];<br>    }<br>}<br></code></pre></td></tr></tbody></table></figure><h2 id="五、数组和特殊矩阵">五、数组和特殊矩阵</h2><h3 id="5-1-数组">5.1 数组</h3><h4 id="5-1-1-数组的基本概念">5.1.1 数组的基本概念</h4><figure class="highlight c++"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><code class="hljs c++"><span class="hljs-keyword">typedef</span> <span class="hljs-type">int</span> arr[m][n];<br><span class="hljs-comment">// 等价于</span><br><span class="hljs-keyword">typedef</span> <span class="hljs-type">int</span> arr1[n];<br><span class="hljs-keyword">typedef</span> arr1 arr2[m];<br></code></pre></td></tr></tbody></table></figure><h4 id="5-1-2-数组的存储结构">5.1.2 数组的存储结构</h4><ol><li>行优先：按行存储</li><li>列优先：按列存储</li></ol><p>可以按照下标的关系，只需要知道第一个元素的地址，通过矩阵的大小关系即可直接计算出 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi><mo>⋯</mo></mrow></msub></mrow><annotation encoding="application/x-tex">a_{ij\cdots}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span><span class="minner mtight">⋯</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 的地址</p><h3 id="5-2-特殊矩阵的压缩存储">5.2 特殊矩阵的压缩存储</h3><p>对于多个相同的非零元素只分配一个存储空间，对零元素不分配空间</p><h4 id="5-2-1-对称矩阵的压缩存储">5.2.1 对称矩阵的压缩存储</h4><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218527.png" alt="对称矩阵的压缩存储"></p><p>假设现在有一个 n*n 的对称矩阵</p><p>存：行优先存储 <code>data[n * (n + 1) / 2]</code></p><p>取：我们如果要取 <code>data[i][j]</code></p><ul><li><p>对于上三角</p><ul><li><p><code>i &gt;= j</code>：<code>data[i * (i + 1) / 2 + j]</code></p></li><li><p><code>i &lt;  j</code>：<code>data[j * (j + 1) / 2 + i]</code></p></li></ul></li><li><p>对于下三角</p><ul><li><p><code>i &gt;= j</code>：<code>data[i * (i + 1) / 2 + j]</code></p></li><li><p><code>i &lt;  j</code>：<code>data[j * (j + 1) / 2 + i]</code></p></li></ul></li></ul><h4 id="5-2-2-三角矩阵的压缩存储">5.2.2 三角矩阵的压缩存储</h4><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218528.png" alt="三角矩阵的压缩存储"></p><p>假设现在有一个 n*n 的三角矩阵（上三角或下三角为常数c）</p><p>存：行优先存储，常数 c 存储到最后 <code>data[n * (n + 1) / 2 + 1]</code></p><h4 id="5-2-3-对角矩阵的压缩存储">5.2.3 对角矩阵的压缩存储</h4><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218529.png" alt="对角矩阵的压缩存储"></p><p>假设现在有一个 n*n 的对角矩阵（围绕主对角线有数据，其余数据均为0）</p><h4 id="5-2-4-稀疏矩阵的压缩存储">5.2.4 稀疏矩阵的压缩存储</h4><p>假设现在有一个 n*m 的稀疏矩阵（很多零的一个矩阵）</p><ol><li><p>三元组顺序表</p><p>按行存储两个信息，一个是非零元素的数值，还有一个是具体的坐标 <code>(i, j)</code></p></li><li><p>十字链表</p><p>定义两个指针数组，定义两个指针数组，存储行列的头指针即可 <code>vector&lt;CrossNode&lt;T&gt;*&gt; cheads, rheads</code></p></li></ol><h2 id="六、广义表">六、广义表</h2><h3 id="6-1-广义表的概念">6.1 广义表的概念</h3><p>与以往的线性表的区别在于：线性表的元素只能是 DT 或 ADT。而对于广义表，元素还可以是一个广义表，即可递归结构。</p><p>表头、表尾。对于当前序列，第一个元素就是表头，其余元素的集合就是表尾</p><p>特点：层次结构、共享结构、递归结构</p><h3 id="6-2-广义表的存储结构">6.2 广义表的存储结构</h3><h4 id="6-2-1-广义表中结点的结构">6.2.1 广义表中结点的结构</h4><p>采用联合结构体存储结点类型</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218530.png" alt="采用联合结构体存储结点类型"></p><h4 id="6-2-2-广义表的存储结构">6.2.2 广义表的存储结构</h4><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218531.png" alt="广义表的存储结构"></p><h3 id="6-3-广义表的操作算法">6.3 广义表的操作算法</h3><ul><li>直接递归法 - 原子直接操作，子表循环成原子进行操作</li><li>减治法 - 先处理第一个元素（原子：直接操作 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>o</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">or</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">or</span></span></span></span> 子表：递归操作），最后递归操作剩余的元素</li></ul><h4 id="6-3-3-广义表的其他操作算法">6.3.3 广义表的其他操作算法</h4><p>复制广义表、计算广义表的长度、计算广义表的深度、释放广义表的存储空间</p><h2 id="七、树和二叉树">七、树和二叉树</h2><h3 id="7-1-树的概念和性质">7.1 树的概念和性质</h3><h4 id="7-1-1-树的定义">7.1.1 树的定义</h4><h4 id="7-1-2-树的基本术语">7.1.2 树的基本术语</h4><ol><li>结点的度和树的度<ul><li>结点的度：每一个结点孩子结点的数量</li><li>树的度：一棵树中结点度数的最大值</li></ul></li><li>孩子、双亲、兄弟结点</li><li>路径和路径长度</li><li>子孙结点和祖先结点</li><li>结点的层次和树的高度</li><li>有序树和无序树<ul><li>有序树：子集不可以随意交换</li><li>无序树：子集可以随意交换</li></ul></li><li>森林<ul><li>多棵树</li></ul></li></ol><h4 id="7-1-3-树的基本性质">7.1.3 树的基本性质</h4><h3 id="7-2-二叉树的概念和性质">7.2 二叉树的概念和性质</h3><h4 id="7-2-1-二叉树的定义">7.2.1 二叉树的定义</h4><h4 id="7-2-2-二叉树的基本性质">7.2.2 二叉树的基本性质</h4><ol><li><p>根是第一层。第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 层最多有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">2^{i - 1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8247em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span> 个结点</p></li><li><p>树中叶子结点的个数为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>n</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">n_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，度数为2的结点的个数为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>n</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">n_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>。已知树的点数为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>，边数为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span>，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>=</mo><mi>m</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n = m + 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>。而 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>=</mo><msub><mi>n</mi><mn>0</mn></msub><mo>+</mo><msub><mi>n</mi><mn>1</mn></msub><mo>+</mo><msub><mi>n</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">n=n_0+n_1+n_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>=</mo><msub><mi>n</mi><mn>1</mn></msub><mo>+</mo><mn>2</mn><msub><mi>n</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">m=n_1+2n_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7944em;vertical-align:-0.15em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>n</mi><mn>0</mn></msub><mo>+</mo><msub><mi>n</mi><mn>1</mn></msub><mo>+</mo><msub><mi>n</mi><mn>2</mn></msub><mo>=</mo><msub><mi>n</mi><mn>1</mn></msub><mo>+</mo><mn>2</mn><msub><mi>n</mi><mn>2</mn></msub><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n_0+n_1+n_2 = n_1+2n_2 +1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7944em;vertical-align:-0.15em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，则</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>n</mi><mn>0</mn></msub><mo>=</mo><msub><mi>n</mi><mn>2</mn></msub><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n_0=n_2 + 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></p></li><li><p>满二叉树：每一层都是满结点</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218532.png" alt="满二叉树"></p></li><li><p>完全二叉树：对于一个 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 层的二叉树，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>→</mo><mi>k</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">1\to k-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 都是满的，第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 层从左到右连接叶子结点</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218533.png" alt="完全二叉树"></p><p>结点数固定，则完全二叉树的形状唯一</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218534.png" alt="完全二叉树的性质"></p><p>若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 为奇数，且 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo mathvariant="normal">≠</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">i\neq1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，则左兄弟就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">i-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7429em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></p><p>若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 为偶数，则右兄弟就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">i+1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7429em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></p></li></ol><h3 id="7-3-二叉树的存储结构">7.3 二叉树的存储结构</h3><h4 id="7-3-1-二叉树的顺序存储结构">7.3.1 二叉树的顺序存储结构</h4><ul><li>对于一般的二叉树，将其转化为完全二叉树进行存储即可</li><li>插入删除操作都不方便</li></ul><h4 id="7-3-2-二叉树的链式存储结构">7.3.2 二叉树的链式存储结构</h4><h3 id="7-4-二叉树的遍历">7.4 二叉树的遍历</h3><h4 id="7-4-1-二叉树遍历的概念">7.4.1 二叉树遍历的概念</h4><h4 id="7-4-2-二叉树遍历算法">7.4.2 二叉树遍历算法</h4><ol><li>先中后遍历<ol><li>递归遍历</li><li>栈式遍历</li></ol></li><li>层序遍历</li></ol><h4 id="7-4-3-二叉树的构造和析构">7.4.3 二叉树的构造和析构</h4><ol><li><p>由含空指针标记的单个遍历序列构造二叉树</p><p>可以从遍历的逻辑进行逆推。在遍历到空指针的时候输出一个编制符号，然后在构造的时候按照遍历序列进行递归构造即可，如图</p><p><strong>先序序列</strong>进行构造：按照遍历的思路来，对于先序序列而言，第一个元素一定是根元素，因此首先根据“当前局面”的第一个元素创建根结点，接着递归创建左子树和右子树即可。注意传递的序列起始下标是引用类型的变量</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218535.png" alt="示例"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218536.png" alt="示例"></p><p><strong>中序序列</strong>进行构造：</p><p>不可以，因为不能确定根节点以及左子树和右子树的部分</p><p><strong>后序序列</strong>进行构造：与上述先序序列进行构建的逻辑一致，只不过有一个小 trick，即我们从后序序列的最后一个元素开始创建，那么得到的第一个元素就是根结点的值，然后首先递归创建右子树，再递归创建左子树即可。同样需要注意的是传递参数时，序列起始下标是引用类型的变量</p><p>与先序序列构造逻辑相同，只是递归的顺序需要调整一下</p></li><li><p>由两个遍历序列构造二叉树</p><ul><li>先+中：构造逻辑与上述带标记的序列构造逻辑几乎一致，只不过区别在于如何进行递归中参数的传递。传递的参数除了先序和中序的字符串，还有当前局面先序序列的起始下标与当前局面中序序列的起始下标，以及以当前序列进行构造时子树的结点个数。很容易就可以找到当前序列的根结点，接着就是利用很简单的下标关系得到上述的三个参数的过程，最后将新得到的三个参数传递给递归函数进行递归构建左右子树即可，当前的根结点是 <code>pre[ipre]</code></li><li>后+中：逻辑与上述一致，只不过当前的根结点是 <code>post[ipost+n-1]</code></li></ul></li><li><p>由顺序结构构造链式结构</p></li><li><p>拷贝构造</p></li><li><p>析构</p></li></ol><h3 id="7-5-二叉树的其他操作算法">7.5 二叉树的其他操作算法</h3><ol><li>计算二叉树的结点数<ul><li>有返回值的递归</li><li>无返回值的递归</li></ul></li><li>计算二叉树的高度<ul><li>有返回值的递归</li><li>无返回值的递归</li></ul></li><li>根据关键值查找结点</li><li>查找结点的父结点</li></ol><h3 id="7-6-线索二叉树">7.6 线索二叉树</h3><h4 id="7-6-1-线索二叉树的概念">7.6.1 线索二叉树的概念</h4><p>将空指针域用前驱 or 后继结点的地址进行覆盖</p><h4 id="7-6-2-线索二叉树的存储结构">7.6.2 线索二叉树的存储结构</h4><p>依旧是链式存储，只不过增加了结点域中的指针类型，分为链接类型Link与线索类型Thread</p><h4 id="7-6-3-线索二叉树的操作算法">7.6.3 线索二叉树的操作算法</h4><p>以中序线索化的二叉树为例，涉及到以下几种算法：</p><ol><li><p>线索化：设置一个全局变量 pre，为了简化思维，我们可以将一个中序遍历的过程想象成一个线性结构。前驱为 pre，当前为 p</p><ul><li>p 的左子树为空，则 p 的前驱为 pre</li><li>pre 的右子树为空，则 pre 的后继为 p</li></ul></li><li><p>求后继结点和前驱结点</p></li><li><p>遍历</p></li><li><p>求父结点</p><ul><li>首先，若已知当前是左子树，则父结点一定是当前右孩子的中序前驱线索；若已知当前是右子树，则父结点一定是当前左孩子的中序前驱线索</li><li>但是在未知当前结点的位置（未知左右子树）时，同时搜索两边的父结点，然后根据试探出来的父结点，特判父结点的子结点是否是当前结点即可</li></ul></li></ol><h3 id="7-7-树的存储结构与算法">7.7 树的存储结构与算法</h3><h4 id="7-7-1-树的存储结构">7.7.1 树的存储结构</h4><ol><li><p>多叉链表表示法：将每一个结点的子结点都预设置为一个定值（树的最大度数）：浪费空间</p></li><li><p>孩子链表表示法：自顶向下存储边的信息</p> <figure class="highlight c++"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br></pre></td><td class="code"><pre><code class="hljs c++"><span class="hljs-keyword">template</span>&lt;<span class="hljs-keyword">class</span> <span class="hljs-title class_">T</span>&gt;<br>   <span class="hljs-keyword">struct</span> <span class="hljs-title class_">CTBox</span> {<br>       T data;<br>       CTNode* firstchild;<br>   };<br>   <span class="hljs-keyword">struct</span> <span class="hljs-title class_">CTNode</span> {<br>       <span class="hljs-type">int</span> child;<br>       CTNode* next;<br>   };<br></code></pre></td></tr></tbody></table></figure><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218537.png" alt="孩子链表表示法"></p></li><li><p>双亲表示法：自下向上存储边的信息</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218538.png" alt="双亲表示法"></p></li><li><p>孩子兄弟表示法：左结点存储孩子，右结点存储兄弟</p></li></ol><h4 id="7-7-2-树的操作算法">7.7.2 树的操作算法</h4><p>构造、计算树的高度、计算树中所有结点的度</p><h3 id="7-8-哈夫曼树与哈夫曼编码">7.8 哈夫曼树与哈夫曼编码</h3><h4 id="7-8-1-哈夫曼树的定义">7.8.1 哈夫曼树的定义</h4><p>树的路径长度：叶子结点到根结点的路径之和</p><ol><li>树的带权路径长度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>W</mi><mi>P</mi><mi>L</mi></mrow><annotation encoding="application/x-tex">WPL</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">W</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal">L</span></span></span></span> ：叶子结点到根结点的路径之和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>×</mo></mrow><annotation encoding="application/x-tex">\times</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">×</span></span></span></span> 叶子结点的权重，整体之和</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>W</mi><mi>P</mi><mi>L</mi></mrow><annotation encoding="application/x-tex">WPL</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">W</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal">L</span></span></span></span> 最小的树就叫做哈夫曼树：对于一个结点序列n，每次选择其中的两个权值最小的两个结点进行合并，在进行了n-1次以后，得到的二叉树就是哈夫曼树</li><li>哈夫曼编码：<ul><li>编码：利用二叉树进行前缀编码 - 避免解码时的二义性</li><li>解码：根据编码的二叉 trie 树，进行解码</li></ul></li></ol><h4 id="7-8-2-操作算法">7.8.2 操作算法</h4><p>构造 Huffman 树、编码、解码</p><h2 id="八、图">八、图</h2><h3 id="8-1-图的基本概念">8.1 图的基本概念</h3><h4 id="8-1-1-图的定义">8.1.1 图的定义</h4><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>G</mi><mi>r</mi><mi>a</mi><mi>p</mi><mi>h</mi><mo>=</mo><mo stretchy="false">(</mo><mi>V</mi><mo separator="true">,</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Graph = (V,E)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">G</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">a</span><span class="mord mathnormal">p</span><span class="mord mathnormal">h</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mclose">)</span></span></span></span></span></p><p>完全无向图：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi><mi>d</mi><mi>g</mi><mi>e</mi><mo>=</mo><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">edge=n(n-1)/2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">e</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord mathnormal">e</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mord">/2</span></span></span></span></p><p>完全有向图：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi><mi>d</mi><mi>g</mi><mi>e</mi><mo>=</mo><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">edge=n(n-1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">e</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord mathnormal">e</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></p><h4 id="8-1-2-图的基本术语">8.1.2 图的基本术语</h4><ul><li><p>带权图称为网</p></li><li><p>无向图：连通图和连通分量</p><ul><li>连通图：每一个顶点之间都有路径可达</li><li>连通分量：极大连通子图</li></ul></li><li><p>有向图：强连通图和强连通分量</p><ul><li>强连通图：每一个顶点之间都有路径可达</li></ul></li><li><p>强连通分量：极大强连通子图</p></li></ul><h3 id="8-2-图的存储结构">8.2 图的存储结构</h3><p>教材中的点编号统一从 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 开始</p><h4 id="8-2-1-邻接矩阵">8.2.1 邻接矩阵</h4><p>无向图的度：第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 行（列）的非标记数的个数</p><p>有向图的度：入度为第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 行的非标记数的个数；出度为第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 列的非标记数的个数</p><p>类定义：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218539.png" alt="邻接矩阵 - 1"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218540.png" alt="邻接矩阵 - 2"></p><h4 id="8-2-2-邻接表">8.2.2 邻接表</h4><p>存储出边表称为邻接表，存储入编表称为逆邻接表</p><p>类定义：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218541.png" alt="邻接表"></p><h3 id="8-3-图的遍历">8.3 图的遍历</h3><h4 id="8-3-1-图遍历的概念">8.3.1 图遍历的概念</h4><p>每个结点只能访问一次，故需要开启标记数组用来记录是否访问的情况</p><h4 id="8-3-2-深度优先搜索">8.3.2 深度优先搜索</h4><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218542.png" alt="深度优先搜索"></p><p>邻接矩阵：</p><ul><li><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></p></li><li><p>针对<strong>邻接矩阵</strong>的一个<strong>无向</strong>连通图的搜索代码示例</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218543.png" alt="邻接矩阵的一个无向连通图的搜索代码示例"></p></li></ul><p>邻接表：</p><ul><li><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n+e)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">e</span><span class="mclose">)</span></span></span></span></p></li><li><p>针对<strong>邻接表</strong>的一个<strong>无向</strong>连通图的搜索代码示例</p>  <figure class="highlight c++"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br></pre></td><td class="code"><pre><code class="hljs c++"><span class="hljs-function"><span class="hljs-keyword">template</span>&lt;<span class="hljs-keyword">class</span> T&gt;</span><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">ALGraph::DFS</span><span class="hljs-params">(<span class="hljs-type">int</span> v, <span class="hljs-type">bool</span>* visited)</span> </span>{<br>    cout &lt;&lt; vexs[v];<br>    visited[v] = <span class="hljs-literal">true</span>;<br>    <span class="hljs-comment">// 遍历所有的边</span><br>}<br></code></pre></td></tr></tbody></table></figure></li></ul><h4 id="8-3-3-广度优先搜索">8.3.3 广度优先搜索</h4><p>通过队列实现、时间复杂度与上述 DFS 算法类似</p><h4 id="8-3-4-图遍历算法的应用">8.3.4 图遍历算法的应用</h4><ol><li><p>求 <code>(u,v)</code> 的所有简单路径：dfs + 回溯法的简单应用</p></li><li><p>染色法求二部图：bfs 的简单应用。当然 dfs 也是可以的，只需要在染色之后判断是否有相同颜色的邻接点即可</p></li></ol><h3 id="8-4-最小生成树">8.4 最小生成树</h3><h4 id="8-4-1-最小生成树的概念及其性质">8.4.1 最小生成树的概念及其性质</h4><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mi>i</mi><mi>n</mi><mi>i</mi><mi>m</mi><mi>u</mi><mi>m</mi><mtext>&nbsp;</mtext><mi>S</mi><mi>p</mi><mi>a</mi><mi>n</mi><mi>n</mi><mi>i</mi><mi>n</mi><mi>g</mi><mtext>&nbsp;</mtext><mi>T</mi><mi>r</mi><mi>e</mi><mi>e</mi><mo stretchy="false">(</mo><mi>M</mi><mi>S</mi><mi>T</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Minimum\ Spanning\ Tree(MST)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord mathnormal">inim</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mspace">&nbsp;</span><span class="mord mathnormal">Sp</span><span class="mord mathnormal">annin</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mord mathnormal">ree</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">MST</span><span class="mclose">)</span></span></span></span></p><p>证明：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218544.png" alt="最小生成树性质证明 - 图例"></p><p>对于上述的一个割，选择其中权值最小的交叉边。从而对于所有的状态，每次选择最小交叉边即可。</p><h4 id="8-4-2-Prim算法">8.4.2 Prim算法</h4><p>算法标签：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mi>r</mi><mi>e</mi><mi>e</mi><mi>d</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">greedy</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord mathnormal">ree</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span></p><ul><li><p>构造 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 个割的状态</p></li><li><p>起始状态为：顶点集合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span></span></span></span> 含 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 个顶点，顶点集合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>−</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">V-U</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span></span></span></span> 含 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 个顶点</p></li><li><p>状态转移为：</p><ul><li>选完最小交叉边之后，将这条边在集合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>−</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">V-U</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span></span></span></span> 中的顶点加入到最小生成树集合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span></span></span></span> 中</li><li>更新最小交叉边数组 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mi>i</mi><mi>e</mi><mi>d</mi><mi>g</mi><mi>e</mi><mi>s</mi><mo stretchy="false">[</mo><mtext>&nbsp;</mtext><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">miniedges[\ ]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">mini</span><span class="mord mathnormal">e</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord mathnormal">es</span><span class="mopen">[</span><span class="mspace">&nbsp;</span><span class="mclose">]</span></span></span></span></li></ul></li><li><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></p></li></ul><h4 id="8-4-3-Kruskal算法">8.4.3 Kruskal算法</h4><p>算法流标签：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mi>r</mi><mi>e</mi><mi>e</mi><mi>d</mi><mi>y</mi><mo separator="true">,</mo><mi>d</mi><mi>s</mi><mi>u</mi></mrow><annotation encoding="application/x-tex">greedy,dsu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord mathnormal">ree</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">s</span><span class="mord mathnormal">u</span></span></span></span></p><ul><li>初始化 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个顶点作为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个连通分量</li><li>按照边的权值升序进行选择<ul><li>如果选出的边的两个顶点不在同一个集合，则加入最小生成树</li><li>如果选出的边的两个顶点在同一个集合，则不选择（如果选了就会使得生成树形成回路）</li></ul></li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>e</mi><mi>log</mi><mo>⁡</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(e\log e)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">e</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">e</span><span class="mclose">)</span></span></span></span></li></ul><h3 id="8-5-最短路径">8.5 最短路径</h3><h4 id="8-5-1-最短路径的概念">8.5.1 最短路径的概念</h4><p>单源最短路</p><blockquote><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mi>i</mi><mi>j</mi><mi>k</mi><mi>s</mi><mi>t</mi><mi>r</mi><mi>a</mi></mrow><annotation encoding="application/x-tex">Dijkstra</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.03148em;">ijk</span><span class="mord mathnormal">s</span><span class="mord mathnormal">t</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">a</span></span></span></span> 算法无法求解含负边权的单源最短路</p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mi>e</mi><mi>l</mi><mi>l</mi><mi>m</mi><mi>a</mi><mi>n</mi><mo>−</mo><mi>F</mi><mi>o</mi><mi>r</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">Bellman-Ford</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathnormal">e</span><span class="mord mathnormal" style="margin-right:0.01968em;">ll</span><span class="mord mathnormal">man</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mord mathnormal" style="margin-right:0.02778em;">or</span><span class="mord mathnormal">d</span></span></span></span> 算法支持负边权的单源最短路求解</p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mi>p</mi><mi>f</mi><mi>a</mi></mrow><annotation encoding="application/x-tex">Spfa</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Sp</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mord mathnormal">a</span></span></span></span> 算法同样支持负边权的单元最短路，属于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mi>e</mi><mi>l</mi><mi>l</mi><mi>m</mi><mi>a</mi><mi>n</mi><mo>−</mo><mi>F</mi><mi>o</mi><mi>r</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">Bellman-Ford</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathnormal">e</span><span class="mord mathnormal" style="margin-right:0.01968em;">ll</span><span class="mord mathnormal">man</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mord mathnormal" style="margin-right:0.02778em;">or</span><span class="mord mathnormal">d</span></span></span></span> 算法的优化</p></blockquote><p>多源最短路</p><blockquote><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mi>l</mi><mi>o</mi><mi>y</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">Floyd</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">Fl</span><span class="mord mathnormal" style="margin-right:0.03588em;">oy</span><span class="mord mathnormal">d</span></span></span></span> 适用于求解含负边权的多源最短路</p></blockquote><h4 id="8-5-2-单源最短路径-Dijkstra-算法">8.5.2 单源最短路径 - <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mi>i</mi><mi>j</mi><mi>k</mi><mi>s</mi><mi>t</mi><mi>r</mi><mi>a</mi></mrow><annotation encoding="application/x-tex">Dijkstra</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.03148em;">ijk</span><span class="mord mathnormal">s</span><span class="mord mathnormal">t</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">a</span></span></span></span> 算法</h4><p>算法标签：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mi>r</mi><mi>e</mi><mi>e</mi><mi>d</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">greedy</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord mathnormal">ree</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>  <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mi>p</mi></mrow><annotation encoding="application/x-tex">dp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">p</span></span></span></span></p><blockquote><p>其实就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mi>r</mi><mi>i</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">Prim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">im</span></span></span></span> 的另一种应用</p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mi>r</mi><mi>i</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">Prim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">im</span></span></span></span> 是只存储交叉边的最小值</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mi>i</mi><mi>j</mi><mi>k</mi><mi>s</mi><mi>t</mi><mi>r</mi><mi>a</mi></mrow><annotation encoding="application/x-tex">Dijkstra</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.03148em;">ijk</span><span class="mord mathnormal">s</span><span class="mord mathnormal">t</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">a</span></span></span></span> 是存储交叉边的最小值 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>+</mo></mrow><annotation encoding="application/x-tex">+</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">+</span></span></span></span> 这条边在集合 S 中的点已经记录的值</li></ul></blockquote><ol><li><p>朴素版：</p><ul><li><p>邻接矩阵</p></li><li><p>定义 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">d[i]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">d</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span></span></span></span> 表示从起点到当前i号点的最短路径的长度</p></li><li><p>将顶点分为两个集合，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>−</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">V-S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span>，其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span> 表示已经更新了最短路径长度的顶点集合</p></li><li><p>迭代更新过程：依次更新每一个结点，对于当前结点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">v_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，在集合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span> 中的所有结点中，选择其中到当前结点路径最短的顶点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">v_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>，则 <code>d[i]=d[j]+edges[j][i]</code></p></li><li><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></p></li></ul></li><li><p>堆优化：</p><ul><li><p>邻接表</p></li><li><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>e</mi><mi>log</mi><mo>⁡</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(e \log e)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">e</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">e</span><span class="mclose">)</span></span></span></span></p></li></ul></li></ol><h4 id="8-5-3-多源最短路径-Floyd-算法">8.5.3 多源最短路径 - <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mi>l</mi><mi>o</mi><mi>y</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">Floyd</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">Fl</span><span class="mord mathnormal" style="margin-right:0.03588em;">oy</span><span class="mord mathnormal">d</span></span></span></span> 算法</h4><p>算法标签：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mi>p</mi></mrow><annotation encoding="application/x-tex">dp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">p</span></span></span></span></p><blockquote><p>多阶段决策共 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个阶段，<code>dp[i][j]</code> 表示每一个阶段 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span>，从 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> 的选择前 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个顶点后的最短路径的长度</p></blockquote><p>对于当前阶段 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span>，我们利用阶段 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 的状态进行转移更新，其实就是对于新增加的顶点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">v_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是否选择的过程</p><ul><li>选择 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">v_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，则 <code>dp[i][j] = dp[i][k] + dp[k][j]</code></li><li>不选 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">v_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，则 <code>dp[i][j]</code> 就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 状态下的 <code>dp[i][j]</code></li></ul><h3 id="8-6-AOV网与拓扑排序">8.6 AOV网与拓扑排序</h3><h4 id="8-6-1-有向无环图与AOV网的概念">8.6.1 有向无环图与AOV网的概念</h4><ul><li>有向无环图：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mi>A</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">DAG</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal">A</span><span class="mord mathnormal">G</span></span></span></span></li><li>AOV网： <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>y</mi><mtext>&nbsp;</mtext><mi>o</mi><mi>n</mi><mtext>&nbsp;</mtext><mi>v</mi><mi>e</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>x</mi><mtext>&nbsp;</mtext><mi>n</mi><mi>e</mi><mi>t</mi><mi>w</mi><mi>o</mi><mi>r</mi><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(activity\ on\ vertex\ network)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">c</span><span class="mord mathnormal">t</span><span class="mord mathnormal">i</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mord mathnormal">i</span><span class="mord mathnormal">t</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace">&nbsp;</span><span class="mord mathnormal">o</span><span class="mord mathnormal">n</span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span><span class="mord mathnormal">t</span><span class="mord mathnormal">e</span><span class="mord mathnormal">x</span><span class="mspace">&nbsp;</span><span class="mord mathnormal">n</span><span class="mord mathnormal">e</span><span class="mord mathnormal" style="margin-right:0.02691em;">tw</span><span class="mord mathnormal" style="margin-right:0.02778em;">or</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span></li><li>应用场景：在时间先后上有约束关系的工程管理问题</li></ul><h4 id="8-6-2-拓扑排序">8.6.2 拓扑排序</h4><ul><li>定义：顶点线性化</li><li>应用：判环、判断一个图是否可以进行动态规划</li><li>算法设计：对于有向图，从所有的入度为 0 的点开始删点删边，到最后判断有多少点被删除即可</li><li>算法实现：可以采用 dfs 进行缩点删边，也可以采用 bfs 进行缩点删边</li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n+e)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">e</span><span class="mclose">)</span></span></span></span></li></ul><h2 id="九、查找">九、查找</h2><h3 id="9-1-静态查找表">9.1 静态查找表</h3><p>定义：只支持查询和修改，不支持删除与插入</p><h4 id="9-1-1-顺序查找">9.1.1 顺序查找</h4><h4 id="9-1-2-折半查找">9.1.2 折半查找</h4><h4 id="9-1-3-分块查找">9.1.3 分块查找</h4><p>结合顺序查找与分块查找的一种方法</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218545.png" alt="分块查找"></p><ul><li>索引表可以折半或者顺序查找</li><li>块内部只能顺序查找</li></ul><h3 id="9-2-动态查找表">9.2 动态查找表</h3><h4 id="9-2-1-二叉排序树">9.2.1 二叉排序树</h4><p>定义：根结点比左子树所有结点的值都大，比右子树所有结点的值都小。关键字唯一</p><p>操作：查找、插入、删除</p><p>判定：想要判定一棵二叉树是否为二叉排序树，只需要判断中序遍历的结果是不是递增的即可，可以采取中序遍历序列比对的方法，也可以在递归遍历二叉树的过程中通过记录前驱结点的值直接进行比较判断。时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p><h4 id="9-2-2-平衡二叉树">9.2.2 平衡二叉树</h4><p>定义：平衡因子为左子树的高度 - 右子树的高度，平衡二叉树的平衡因子绝对值 &lt;= 1</p><p>构建：当插入结点进行构建时出现了有结点平衡因子的绝对值超过了1，则进行“旋转”调整，旋转共分为4种</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218546.png" alt="旋转 - LL、LR"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218547.png" alt="旋转 - LR"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218548.png" alt="旋转 - RL"></p><p>尝试模拟一遍下列序列的构造过程就可以理解了</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218549.png" alt="例题"></p><h3 id="9-3-Hash-查找">9.3 Hash 查找</h3><p>定义：装填因子 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo>=</mo><mfrac><mi>n</mi><mi>m</mi></mfrac></mrow><annotation encoding="application/x-tex">\alpha=\frac{n}{m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>，其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 表示待填入表中的结点数，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 表示哈希表的空间大小</p><p>哈希函数应该满足以下两点：第一、映射出来的地址不会越界；第二、映射出来的地址是唯一的</p><h4 id="9-3-1-构造表">9.3.1 构造表</h4><p><strong>常用的哈希函数</strong></p><ol><li><p>直接地址法 - 线性函数一对一映射</p><p>优点。计算简单且不可能产生冲突</p><p>缺点。对于空间的要求极高，如果数据过于离散，则会造成很大的空间浪费</p></li><li><p>数字分析法 - 按照数位中的数值分布情况进行哈希</p><p>缺点。需要预先知道数据的数字分布情况</p></li><li><p>平方取中法 - 对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mi>m</mi></msup></mrow><annotation encoding="application/x-tex">10^m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6644em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span></span></span></span></span></span></span></span> 的哈希空间，可以将数字平方后取中间 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 位进行哈希存储</p></li><li><p>折叠法</p><ul><li><p>移位法：将一个数字按照数位拆分为几个部分，然后将几个部分的数值累加出一个数即可，高位抹去不用</p></li><li><p>间隔法：与移位法几乎一致，只不过将其中的部分意义间隔的进行数值反转，最后累计即可，高位抹去不用</p></li></ul></li><li><p>除留余数法 - 按照数值 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>mod&nbsp;</mtext><mi>p</mi></mrow><annotation encoding="application/x-tex">\text{mod}\ p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">mod</span></span><span class="mspace">&nbsp;</span><span class="mord mathnormal">p</span></span></span></span> 后的数值进行哈希，假设哈希表空间大小为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> ，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span> 一般取 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≤</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">\le m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 的质数</p></li></ol><p><strong>处理冲突</strong></p><ol><li>开放定址法 - 探测开放地址，一般有三种<ul><li>连续序列进行线性探测</li><li>左右倍增序列进行探测</li><li>伪随机序列进行探测</li><li>双 hash 探测法</li></ul></li><li>拉链法<ul><li>定义：将产生 hash 冲突的元素放入同一个子集，通过单链表进行存储</li><li>优点：没有堆积现象，从而减少了很多不必要的比价，提升比较效率；适合一开始不知道表长的情况；除结点更加容易。</li></ul></li></ol><h4 id="9-3-2-查找表">9.3.2 查找表</h4><p>按照构造相同的逻辑进行查找即可</p><h2 id="十、排序">十、排序</h2><h3 id="10-1-排序的基本概念">10.1 排序的基本概念</h3><p>关键字：</p><ul><li>主关键字：每一个待排序的该关键字是独一无二的</li><li>次关键字：每一个待排序的该关键字可能是重复的</li></ul><p>稳定性：</p><ul><li>场景：只针对次关键字的情况</li><li>稳定：按照次关键字排序后，原来相同关键字的顺序不变</li><li>不稳定：按照次关键字排序后，原来相同关键字的顺序可能会改变</li></ul><p>内外排序：</p><ul><li>内排序：数据全部存放在内存</li><li>外排序：数据量过大时，待排序的数据在内存与外存之间不断转换</li></ul><h3 id="10-2-冒泡排序">10.2 冒泡排序</h3><p><strong>稳定的</strong>。基于交换的思路进行</p><h3 id="10-3-选择排序">10.3 选择排序</h3><ul><li><p>选择第 1 小的数放在第一个位置，…，选择第 i 小的数放在第 i 个位置</p></li><li><p>共选择 n-1 次</p></li></ul><h3 id="10-4-插入排序">10.4 插入排序</h3><p><strong>稳定的</strong>。</p><ul><li>直接插入排序：依次向前缀已经排好序的序列中进行插入 - <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></li><li>折半插入排序：同上，只是选择插入位置的使用二分 - <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li><li>递归插入排序：排序 <code>[1,i]</code> 等价于先排好 <code>[1,i-1]</code>，然后插入当前 <code>num[i]</code> 即可</li></ul><h3 id="10-5-希尔排序">10.5 希尔排序</h3><p><strong>不稳定</strong>。</p><p>基于插入直接排序的优点：</p><ol><li>当序列基本有序时，效率很高</li><li>当待排序数很少时，效率很高</li></ol><p>于是希尔（Shell）就得出来以下的希尔排序算法：</p><ol><li>将序列划分一定次数，从 d&lt;n 到 1</li><li>每次划分都对组内的元素进行直接插入排序</li><li>最后分为 1 组时，直接排序一趟以后就可以得到 sortrd sequence</li></ol><h3 id="10-6-快速排序">10.6 快速排序</h3><p><strong>不稳定</strong>。</p><p>分治法三步骤：divide、conquer and combine</p><p>每次选择一个 pivot 进行 partition，递归两个 partition</p><figure class="highlight c++"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br></pre></td><td class="code"><pre><code class="hljs c++"><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">Sort</span><span class="hljs-params">(<span class="hljs-type">int</span> l, <span class="hljs-type">int</span> r)</span> </span>{<br>    <span class="hljs-keyword">if</span> (l &gt;= r) <span class="hljs-keyword">return</span>;<br><br>    <span class="hljs-type">int</span> i = l - <span class="hljs-number">1</span>, j = r + <span class="hljs-number">1</span>, x = a[l + r &gt;&gt; <span class="hljs-number">1</span>];<br>    <span class="hljs-keyword">while</span> (i &lt; j) {<br>        <span class="hljs-keyword">while</span> (a[++i] &lt; x);<br>        <span class="hljs-keyword">while</span> (a[--j] &gt; x);<br>        <span class="hljs-keyword">if</span> (i &lt; j) <span class="hljs-built_in">swap</span>(a[i], a[j]);            <br>    }<br>    <br>    <span class="hljs-built_in">Sort</span>(l, j), <span class="hljs-built_in">Sort</span>(j + <span class="hljs-number">1</span>, r);<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="10-7-堆排序">10.7 堆排序</h3><p><strong>不稳定</strong>。</p><p>堆与堆排序的定义：首先我们得知道什么是堆结构。堆是具有下面性质（对于任意的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">1\le i \le n/2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7955em;vertical-align:-0.136em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mord">/2</span></span></span></span> ）的完全二叉树</p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mi>i</mi></msub><mo>≤</mo><msub><mi>k</mi><mrow><mn>2</mn><mi>i</mi></mrow></msub><mo separator="true">,</mo><msub><mi>k</mi><mi>i</mi></msub><mo>≤</mo><msub><mi>k</mi><mrow><mn>2</mn><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">k_i \le k_{2i},k_i \le k_{2i+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9028em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight">i</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span> 叫做 小顶堆</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mi>i</mi></msub><mo>≥</mo><msub><mi>k</mi><mrow><mn>2</mn><mi>i</mi></mrow></msub><mo separator="true">,</mo><msub><mi>k</mi><mi>i</mi></msub><mo>≥</mo><msub><mi>k</mi><mrow><mn>2</mn><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">k_i \ge k_{2i},k_i \ge k_{2i+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9028em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight">i</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span> 叫做 大顶堆</li><li>因此一个堆结构可以采用线性的单元进行存储与维护。而堆排序利用堆顶是最值这一性质，通过不断的<strong>取堆顶，调整堆</strong>的方式获得最终的排好序的序列</li></ul><p>建立初始堆：由于完全二叉树中，每一个叶子结点都已经是堆结构，因此直接从第一个非叶子结点开始建堆即可。对每一个元素与左孩子、 右孩子进行比较</p><ul><li>如果当前结点的值比左右孩子都大，那么无需修改，当前位置就是堆顶</li><li>如果当前结点的值比左孩子或者右孩子中的最大值小，则将最大的孩子作为堆顶，并将当前值不断的“下沉”即可</li></ul><p>交换堆顶与记录位置后重新建堆：交换记录值获取当前堆中最值以后，需要将除了已记录的值的结点以外的所有结点重新调整为堆结构</p><ul><li>调整为堆结构的过程与上述初始建堆的过程完全一致，只是结点数每次 -1</li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p><h3 id="10-8-归并排序">10.8 归并排序</h3><p><strong>稳定的</strong>。</p><p>递归：同样采用分治法，我们按照分治法的三个步骤进行讨论</p><ul><li>divide：将当前序列划分为左右两部分</li><li>conquer：递归处理上述划分出来的两部分</li><li>combine：归并上述递归完的两部分</li></ul><p>非递归：就是模拟上述递归的过程，可以拆分为三步</p><ul><li>归并</li><li>按照指定的长度处理整个序列</li><li>划分局部排序的长度</li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo><mo>←</mo><mi>T</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mi>T</mi><mo stretchy="false">(</mo><mfrac><mi>n</mi><mn>2</mn></mfrac><mo stretchy="false">)</mo><mo>+</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)\leftarrow T(n)=2T(\frac{n}{2}) + O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">←</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.095em;vertical-align:-0.345em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mopen">(</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p>]]></content>
+    <content type="html"><![CDATA[<blockquote><p>联系方式：</p><p><span class="github-emoji"><span>✉</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/2709.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span> <a href="mailto:lijuncst@njnu.edu.cn">lijuncst@njnu.edu.cn</a></p><p><span class="github-emoji"><span>☎</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/260e.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span> 13770610040</p></blockquote><h2 id="数字逻辑电路">数字逻辑电路</h2><h2 id="一、数字逻辑概论">一、数字逻辑概论</h2><h3 id="1-1-数字信号与数字电路">1.1 数字信号与数字电路</h3><h4 id="1-1-1-数字技术的发展及其应用">1.1.1 数字技术的发展及其应用</h4><ol><li><p>电流控制器件：电子管、晶体管（二极管、三极管）、半导体集成电路</p></li><li><p>EDA(Electronic Design Automation)技术（硬件设计软件化）：设计：EWB or Verilog、仿真、下载、验证结果</p></li></ol><h4 id="1-1-2-数字集成电路的分类及特点">1.1.2 数字集成电路的分类及特点</h4><ol><li><p>数字集成电路的分类</p><ol><li><p>从结构特点及其对输入信号的响应规则角度：组合逻辑电路、时序逻辑电路</p></li><li><p>从电路的形式角度：集成电路、分立电路</p></li><li><p>从器件的角度：TTL电路、CMOS电路</p></li><li><p>从集成度（每一个芯片所包含的门个数）的角度：小规模、中规模、大规模、超大规模、甚大规模</p></li></ol></li><li><p>数字集成电路的特点</p><ol><li>稳定性高：抗干扰能力强</li><li>易于设计：对0和1表示的信号进行逻辑运算和处理</li><li>便于集成：体积小、通用性好、成本低</li><li>可编程性：可实现硬件设计软件化</li><li>高速度、低功耗</li><li>便于存储、传输、处理</li></ol></li><li><p>数字电路的<strong>分析</strong>、<strong>设计</strong>与测试</p><ol><li>分析方法<ul><li>目标：确定输入与输出</li><li>工具：逻辑代数</li><li>方法：真值表、功能表、逻辑表达式和波形图</li></ul></li><li>设计方法<ul><li>从功能要求出发，选择合适的逻辑器件进行设计</li><li>设计方式：传统的设计方法 or 基于EDA的软件设计方法</li></ul></li><li>测试技术</li></ol></li></ol><h4 id="1-1-3-模拟信号与数字信号">1.1.3 模拟信号与数字信号</h4><ol><li><p>模拟信号：时间和数值均<strong>连续</strong>变化的信号</p></li><li><p>数字信号：时间和数值均<strong>离散</strong>变化的信号</p></li><li><p>模拟量的数字表示：模数转换，即将连续的模拟信号经过<strong>采样与编码</strong>转化为数字信号</p><ul><li>首先对时间离散</li><li>然后对幅值离散</li><li>最后对得到的数字量进行编码</li></ul></li></ol><h4 id="1-1-4-数字信号的描述方法">1.1.4 数字信号的描述方法</h4><ol><li><p>二值数字逻辑和逻辑电平</p><p>二值数字逻辑：0和1两种状态（定量）</p><p>逻辑电平：高电压和低电压（定性）</p><p>正逻辑关系表（负逻辑相反）：</p><table><thead><tr><th style="text-align:center">电压（V）</th><th style="text-align:center">二值逻辑</th><th style="text-align:center">电平</th></tr></thead><tbody><tr><td style="text-align:center">3.5~5</td><td style="text-align:center">1</td><td style="text-align:center">H（高电压）</td></tr><tr><td style="text-align:center">0~1.5</td><td style="text-align:center">0</td><td style="text-align:center">L（低电压）</td></tr></tbody></table></li><li><p>数字波形</p></li><li><p>实际数字信号波形</p></li><li><p>波形图、时序图或定时图</p></li></ol><h3 id="1-2-数制">1.2 数制</h3><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">(</mo><mi>N</mi><msub><mo stretchy="false">)</mo><mi>r</mi></msub><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></munderover><msub><mi>K</mi><mi>i</mi></msub><mtext>&nbsp;</mtext><msup><mi>r</mi><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">(N)_r=\sum_{i=-\infty}^\infty K_i\ r^i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9874em;vertical-align:-1.336em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">∞</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.336em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace">&nbsp;</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8747em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span></span></span></p><h4 id="1-2-1-十进制">1.2.1 十进制</h4><h4 id="1-2-2-二进制">1.2.2 二进制</h4><p>优点：<strong>易于表达</strong>；二进制数字电路<strong>逻辑简单</strong>，所用元件少；基本运算<strong>规则简单</strong>，运算操作方便</p><p>波形表示：应用比如“计数器”</p><p>数据传输：应用比如“串行传输”</p><h4 id="1-2-3-十-二进制之间的转换">1.2.3 十-二进制之间的转换</h4><p>十进制小数转化为二进制：将小数部位不断×2，取整数，直到没有小数部分为止</p><h4 id="1-2-4-十六进制和八进制">1.2.4 十六进制和八进制</h4><p>二进制转十六进制：从右往左每四位换算成十六进制</p><p>二进制转八进制：从右往左每三位换算成八进制</p><h3 id="1-3-二进制数的算术运算">1.3 二进制数的算术运算</h3><h4 id="1-3-1-无符号二进制数的运算">1.3.1 无符号二进制数的运算</h4><h4 id="1-3-2-有符号二进制数的运算">1.3.2 有符号二进制数的运算</h4><p>定义：其实就是多了一个符号位，且不可以省略。其中0表示正数，1表示负数：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mo stretchy="false">(</mo><mo>+</mo><mn>11</mn><msub><mo stretchy="false">)</mo><mi>D</mi></msub><mo>=</mo><mo stretchy="false">(</mo><mn>01011</mn><msub><mo stretchy="false">)</mo><mi>B</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>11</mn><msub><mo stretchy="false">)</mo><mi>D</mi></msub><mo>=</mo><mo stretchy="false">(</mo><mn>11011</mn><msub><mo stretchy="false">)</mo><mi>B</mi></msub></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}(+11)_D=(01011)_B \\(-11)_D=(11011)_B \\\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">+</span><span class="mord">11</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">D</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord">01011</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">−</span><span class="mord">11</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">D</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord">11011</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>补码、反码和原码：</p><ul><li><p>对于正数，补码反码原码全部一样</p></li><li><p>对于负数，反码为：符号位不动，原码按位取反；补码为：反码最低位+1即可</p></li></ul><p>加法：与十进制竖式计算类似</p><p>减法：与十进制竖式计算类似</p><p>溢出：是因为数值位不够了，解决方法是进行<strong>位扩展</strong></p><p>溢出的判别：</p><ul><li><p>两个正数的求和，得到的补码的最高位如果为 1，则溢出</p></li><li><p>两个负数的求和，得到的补码的最高位如果为 0，则溢出</p></li></ul><h3 id="1-4-二进制代码">1.4 二进制代码</h3><h4 id="1-4-1-二-十进制码">1.4.1 二-十进制码</h4><p>其实就是在表示 0-15 的十六个二进制里面，按照不同的规则选取 10 个二进制数来进行转换</p><ol><li>有权码 - 最接近逻辑的：8421BCD码</li><li>无权码</li></ol><h4 id="1-4-2-格雷码">1.4.2 格雷码</h4><h4 id="1-4-3-ASCII码">1.4.3 ASCII码</h4><h3 id="1-5-二值逻辑变量与基本逻辑运算">1.5 二值逻辑变量与基本逻辑运算</h3><h4 id="1-5-1-常见逻辑符号示例">1.5.1 常见逻辑符号示例</h4><table><thead><tr><th style="text-align:center">运算类型</th><th style="text-align:center">逻辑符号</th><th style="text-align:center">逻辑表达式</th></tr></thead><tbody><tr><td style="text-align:center">与</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245091.png" alt="image-20231008095031818" style="zoom:33%;"></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><mi>A</mi><mi>B</mi></mrow><annotation encoding="application/x-tex">Y=AB</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></td></tr><tr><td style="text-align:center">或</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245093.png" alt="image-20231008095047759" style="zoom:33%;"></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><mi>A</mi><mo>+</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">Y=A+B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></td></tr><tr><td style="text-align:center">非</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245094.png" alt="image-20231008094808523" style="zoom:33%;"></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><mover accent="true"><mi>A</mi><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">Y=\overline{A}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span></span></td></tr><tr><td style="text-align:center">与非</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245095.png" alt="image-20231008095111128" style="zoom:33%;"></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">Y=\overline{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span></span></td></tr><tr><td style="text-align:center">或非</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245096.png" alt="image-20231008095809755" style="zoom:33%;"></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><mover accent="true"><mrow><mi>A</mi><mo>+</mo><mi>B</mi></mrow><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">Y=\overline{A+B}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9667em;vertical-align:-0.0833em;"></span><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.0833em;"><span></span></span></span></span></span></span></span></span></td></tr><tr><td style="text-align:center">异或</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245097.png" alt="image-20231008095902796" style="zoom:33%;"></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><mover accent="true"><mi>A</mi><mo stretchy="true">‾</mo></mover><mi>B</mi><mo>+</mo><mi>A</mi><mover accent="true"><mi>B</mi><mo stretchy="true">‾</mo></mover><mo>=</mo><mi>A</mi><mo>⊕</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">Y=\overline A B+A \overline B=A \oplus B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9667em;vertical-align:-0.0833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8833em;"></span><span class="mord mathnormal">A</span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊕</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></td></tr><tr><td style="text-align:center">同或</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245098.png" alt="image-20231008095922663" style="zoom:33%;"></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><mi>A</mi><mi>B</mi><mo>+</mo><mover accent="true"><mi>A</mi><mo>ˉ</mo></mover><mover accent="true"><mi>B</mi><mo>ˉ</mo></mover><mo>=</mo><mi>A</mi><mo>⊙</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">Y = AB + \bar A \bar B = A\odot B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8201em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8201em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">A</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1111em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8201em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊙</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></td></tr><tr><td style="text-align:center">与或非</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245099.png" alt="image-20231008095947154" style="zoom:33%;"></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><mover accent="true"><mrow><mi>A</mi><mi>B</mi><mo>+</mo><mi>C</mi><mi>D</mi></mrow><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">Y=\overline{AB+CD}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9667em;vertical-align:-0.0833em;"></span><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.0833em;"><span></span></span></span></span></span></span></span></span></td></tr></tbody></table><h4 id="1-5-2-使用逻辑函数表示实际问题">1.5.2 使用逻辑函数表示实际问题</h4><table><thead><tr><th style="text-align:center">实际问题</th><th style="text-align:center">图片示例</th><th style="text-align:center">变量表示</th><th style="text-align:center">列真值表</th><th style="text-align:center">逻辑函数</th></tr></thead><tbody><tr><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245100.png" alt="image-20231008102102546" style="zoom:25%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245101.png" alt="image-20231008102037276" style="zoom:25%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245102.png" alt="image-20231008101253710" style="zoom:50%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245104.png" alt="image-20231008101303842" style="zoom: 33%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245105.png" alt="image-20231008101318582" style="zoom:33%;"></td></tr></tbody></table><h3 id="1-6-逻辑函数及其表示方法">1.6 逻辑函数及其表示方法</h3><p>描述输入逻辑变量和输出逻辑变量之间的因果关系，称为逻辑函数</p><h4 id="1-6-1-逻辑函数的几种表示方法">1.6.1 逻辑函数的几种表示方法</h4><table><thead><tr><th style="text-align:center">方法</th><th style="text-align:center">示例</th></tr></thead><tbody><tr><td style="text-align:center">真值表</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245106.png" alt="image-20231008110656451" style="zoom: 33%;"></td></tr><tr><td style="text-align:center">逻辑函数表达式</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245107.png" alt="image-20231008110713908" style="zoom:25%;"></td></tr><tr><td style="text-align:center">逻辑图</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245108.png" alt="image-20240116131615935" style="zoom: 25%;"></td></tr><tr><td style="text-align:center">波形图</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245109.png" alt="image-20231008111013474" style="zoom:33%;"></td></tr></tbody></table><h4 id="1-6-2-逻辑函数表示方法之间的转换">1.6.2 逻辑函数表示方法之间的转换</h4><p><strong>真值表到逻辑图的转换</strong></p><ul><li><p>查看真值表</p>  <img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245110.png" alt="image-20231008112605100" style="zoom:33%;"></li><li><p>根据真值表写出逻辑表达式</p>  <img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245111.png" alt="image-20231008112628066" style="zoom: 50%;"></li><li><p>化简（上式不用化简）</p></li><li><p>绘制逻辑图</p>  <img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245112.png" alt="image-20231008112708161" style="zoom:50%;"></li></ul><p><strong>逻辑图到真值表的转换</strong></p><ul><li><p>根据逻辑图逐级写出表达式</p>  <img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245113.png" alt="image-20231008112800422" style="zoom:50%;"></li><li><p>化简</p></li><li><p>代入所有输入变量求真值表</p>  <img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245114.png" alt="image-20231008112839220" style="zoom:50%;"></li></ul><h2 id="二、逻辑代数与硬件描述语言基础">二、逻辑代数与硬件描述语言基础</h2><h3 id="2-1-逻辑代数的基本定律和规则">2.1 逻辑代数的基本定律和规则</h3><h4 id="2-1-1-逻辑代数的基本定律和恒等式">2.1.1 逻辑代数的基本定律和恒等式</h4><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245115.png" alt="image-20231008114849627"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245116.png" alt="image-20231008114943429"></p><h4 id="2-1-2-逻辑代数的基本规则">2.1.2 逻辑代数的基本规则</h4><ol><li><p><strong>代入规则</strong> - 类似于换元</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245117.png" alt="image-20231008115435355"></p></li><li><p><strong>反演规则</strong>（获得反函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>Y</mi><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\overline Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span></span> ）</p><blockquote><p>觉得烦可以直接进行取反运算，简单明了不会错</p></blockquote><ul><li>对于任意一个逻辑表达式L，与门 &amp; 或门取反，变量取反，0 &amp; 1取反</li><li>保持原来的运算优先顺序（即如果在原函数表达式中，AB之间先运算，再和其他变量进行运算，那么非函数的表达式中，仍然是AB之间先运算）</li><li>对于反变量以外的非号应保留不变</li></ul><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245118.png" alt="image-20231008115328932"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245119.png" alt="image-20231008115337125"></p></li><li><p><strong>对偶规则</strong>（获得对偶式 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>L</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">L'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> ）</p><ul><li>对于任何逻辑函数式：与门、或门取反，0、1 取反</li></ul></li></ol><h3 id="2-2-逻辑函数表达式的形式">2.2 逻辑函数表达式的形式</h3><h4 id="2-2-1-逻辑函数表达式的基本形式">2.2.1 逻辑函数表达式的基本形式</h4><ol><li><p>与或表达式：若干个与项相或</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245120.png" alt="image-20231013094928938"></p></li><li><p>或与表达式：若干个或项相与</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245121.png" alt="image-20231013094938441"></p></li></ol><h4 id="2-2-2-最小项与最小项表达式">2.2.2 最小项与最小项表达式</h4><ol><li><p>最小项的定义和性质：n 个变量的最小项一共有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">2^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6644em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> 个</p></li><li><p>最小项表达式：所有的最小项相或</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245122.png" alt="image-20231013100105922"></p></li></ol><h4 id="2-2-3-最大项与最大项表达式">2.2.3 最大项与最大项表达式</h4><ol><li><p>最大项的定义和性质：n 个变量的最大项一共有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">2^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6644em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> 个</p></li><li><p>最大项表达式：所有的最大项相与</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245123.png" alt="image-20240116134303065"></p></li></ol><h4 id="2-2-4-最大项和最小项的关系">2.2.4 最大项和最小项的关系</h4><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>m</mi><mi>i</mi></msub><mo>=</mo><mover accent="true"><msub><mi>M</mi><mi>i</mi></msub><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">m_i=\overline {M_i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0333em;vertical-align:-0.15em;"></span><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p><h3 id="2-3-逻辑函数的代数化简法">2.3 逻辑函数的代数化简法</h3><p>为什么要学化简？因为化简之后可以减少门的使用，从而增强电路可靠性、降低成本</p><h4 id="2-3-1-逻辑函数的最简形式">2.3.1 逻辑函数的最简形式</h4><p>最简与或表达式：包含的与项数最少，且每个与项中变量数最少的与或表达式</p><h4 id="2-3-2-逻辑函数的代数化简法">2.3.2 逻辑函数的代数化简法</h4><ol><li><p>逻辑函数的<strong>化简</strong></p><table><thead><tr><th style="text-align:center">方法</th><th style="text-align:center">逻辑函数</th><th style="text-align:center">证明</th></tr></thead><tbody><tr><td style="text-align:center">并项法</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>+</mo><mover accent="true"><mi>A</mi><mo stretchy="true">‾</mo></mover><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">A+\overline A = 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></td><td style="text-align:center">显然</td></tr><tr><td style="text-align:center">吸收法</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>+</mo><mi>A</mi><mi>B</mi><mo>=</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">A+AB = A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span></td><td style="text-align:center">提取公因子</td></tr><tr><td style="text-align:center">消去法</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>+</mo><mover accent="true"><mi>A</mi><mo stretchy="true">‾</mo></mover><mi>B</mi><mo>=</mo><mi>A</mi><mo>+</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A+\overline A B = A+B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></td><td style="text-align:center">摩根定律使用两次</td></tr><tr><td style="text-align:center">配项法</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>=</mo><mi>A</mi><mo stretchy="false">(</mo><mi>B</mi><mo>+</mo><mover accent="true"><mi>B</mi><mo stretchy="true">‾</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A = A(B + \overline B)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">A</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1333em;vertical-align:-0.25em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></td><td style="text-align:center">显然</td></tr></tbody></table></li><li><p>逻辑函数形式的<strong>变换</strong></p><blockquote><p>使用场景：通常在一片集成电路芯片中只有一种门电路，为了减少门电路的种类，需要对逻辑函数表达式进行变换</p><p>变换方法：常常使用两次取反的套路进行变换</p></blockquote> <img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245124.png" alt="image-20231013112834593" style="zoom:50%;"></li></ol><h3 id="2-4-逻辑函数的卡诺图化简法">2.4 逻辑函数的卡诺图化简法</h3><h4 id="2-4-1-用卡诺图表示逻辑函数">2.4.1 用卡诺图表示逻辑函数</h4><p>首先写出逻辑函数的表达式并且转化为最小项表达式，最后将最小项填入相应的矩阵中即可</p><h4 id="2-4-2-用卡诺图化简逻辑函数">2.4.2 用卡诺图化简逻辑函数</h4><p>尽可能使得圈出来的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>k</mi></msup></mrow><annotation encoding="application/x-tex">2^k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span></span></span> 圈中包含的数尽可能的多，即让 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 尽可能的大。注意：圈中的数全部都得是最小项的数</p><h3 id="2-5-硬件描述语言-Verilog-HDL-基础">2.5 硬件描述语言 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>l</mi><mi>o</mi><mi>g</mi><mtext>&nbsp;</mtext><mi>H</mi><mi>D</mi><mi>L</mi></mrow><annotation encoding="application/x-tex">Verilog\ HDL</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span><span class="mord mathnormal">i</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right:0.02778em;">HD</span><span class="mord mathnormal">L</span></span></span></span> 基础</h3><p>为了从软件代码的角度描述电路，从下面三个方面介绍如何用相关的方式进行描述</p><h4 id="2-5-1-门级描述">2.5.1 门级描述</h4><p>门级元件中，第一个位置是输出变量，之后的都是输入变量，可解释为：多输入门</p><table><thead><tr><th style="text-align:center">门级元件</th><th style="text-align:center">元件符号</th></tr></thead><tbody><tr><td style="text-align:center">与</td><td style="text-align:center">and</td></tr><tr><td style="text-align:center">或</td><td style="text-align:center">or</td></tr><tr><td style="text-align:center">非</td><td style="text-align:center">not</td></tr><tr><td style="text-align:center">与非</td><td style="text-align:center">nand</td></tr><tr><td style="text-align:center">或非</td><td style="text-align:center">nor</td></tr><tr><td style="text-align:center">异或</td><td style="text-align:center">xor</td></tr><tr><td style="text-align:center">同或</td><td style="text-align:center">xnor</td></tr></tbody></table><h4 id="2-5-2-数据流描述">2.5.2 数据流描述</h4><p>简单的概括就是使用相关的位运算进行表述，因为电路逻辑本就是二元逻辑，因此位运算就刚好匹配。在使用数据流进行电路描述时，采用的语句都是连续赋值语句，由 assign 关键词开始，多条 assign 语句是<strong>并行</strong>运行的</p><p>需要注意的是，在连续赋值语句中，被赋值的变量一定是 wire 的<strong>线网</strong>类型的变量，示例如下</p><figure class="highlight verilog"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><code class="hljs verilog"><span class="hljs-comment">// 其中 Y 为 wire 类型的变量</span><br><span class="hljs-keyword">assign</span> Y = (~S &amp; D0) | (S &amp; D1)<br></code></pre></td></tr></tbody></table></figure><h4 id="2-5-3-行为描述">2.5.3 行为描述</h4><p>简单的概括就是使用底层语言进行编程，类似于最开始的 C 语言。使用行为描述语句进行描述时，使用 always 关键字开始变量赋值逻辑，多条 always 语句是<strong>串行</strong>运行的</p><p>需要注意的是，在行为描述语句中，被赋值的变量一定是 reg 等<strong>寄存器</strong>类型的变量，这与上述数据流描述的方式不同，示例如下</p><figure class="highlight verilog"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><code class="hljs verilog"><span class="hljs-comment">// 其中 Y 为 reg 类型的变量</span><br><span class="hljs-keyword">always</span>@(*)<span class="hljs-comment">// * 为敏感变量，对于组合电路而言，所有的输入都是敏感变量</span><br>    <span class="hljs-keyword">if</span> (S) Y = D1;<br><span class="hljs-keyword">else</span>   Y = D0;<br></code></pre></td></tr></tbody></table></figure><h2 id="三、逻辑门电路">三、逻辑门电路</h2><h3 id="3-1-逻辑门电路简介">3.1 逻辑门电路简介</h3><p>MOS 管含有 NMOS 管和 PMOS 管，NMOS 管与 PMOS 管的组合称为互补 MOS，或称为 CMOS 电路</p><h3 id="3-2-基本CMOS逻辑门电路">3.2 基本CMOS逻辑门电路</h3><p>附上启蒙的博客：<a href="https://zhuanlan.zhihu.com/p/258852620">MOS管及简单CMOS逻辑门电路原理图解析！</a></p><table><thead><tr><th style="text-align:center">器件</th><th style="text-align:center">电路</th></tr></thead><tbody><tr><td style="text-align:center">开关</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245125.png" alt="image-20231201090156879" style="zoom:50%;"></td></tr><tr><td style="text-align:center">反相器（非门）</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245126.png" alt="image-20231201090250860" style="zoom:33%;"></td></tr><tr><td style="text-align:center">与非门</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245127.png" alt="image-20231201090233084" style="zoom: 33%;"></td></tr><tr><td style="text-align:center">或非门</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245128.png" alt="image-20231201090319423" style="zoom:33%;"></td></tr><tr><td style="text-align:center">传输门（开关）</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245129.png" alt="image-20231222104533397" style="zoom:33%;"></td></tr><tr><td style="text-align:center">与门</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245130.png" alt="image-20240117003028351" style="zoom:33%;"></td></tr><tr><td style="text-align:center">或门</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245131.png" alt="image-20240117003058029" style="zoom:33%;"></td></tr></tbody></table><p><strong>应用示例</strong></p><p>解读的逻辑其实很简单，在理解之前，应该首先观看上面给出的连接中的 MOS 电路的简化版，从而理解电路的正确结构！即，每一个 MOS 管都理解为一个开关，何时闭合与断开完全取决于相应的 MOS 管的种类与电平，如果是 NMOS 管，即箭头指向左边的，为高电平导通，PMOS 管则相反，只需要知道此电路基本逻辑，那么接下来的分析结果就是水到渠成的事</p><p>需要知道一个理念就是，两个电路如果是并联的存在，那么逻辑表达式就是或，简称为<strong>并联相或</strong>；对应的，两个电路如果是串联的存在，那么逻辑表达式就是与，简称为<strong>串联相与</strong>。最后需要补充一点的就是关于取反的辨识，我们知道一个反相器 MOS 管的逻辑是非常简单的，就是一个 NMOS 管和一个 PMOS 管的组合，那么只需要在分析多个线路是串联还是并联的关系之后，最后经过一个反相器就是一个<strong>取反</strong>逻辑</p><table><thead><tr><th style="text-align:center">电路</th><th style="text-align:center">逻辑表达式</th><th style="text-align:center">功能描述</th></tr></thead><tbody><tr><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245132.png" alt="image-20231201090643503" style="zoom:50%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245133.png" alt="image-20231201090727291" style="zoom:50%;"></td><td style="text-align:center">异或门</td></tr><tr><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245134.png" alt="image-20240117103214449"></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi><mo>=</mo><mover accent="true"><mrow><mo stretchy="false">(</mo><mi>B</mi><mi>C</mi><mo>+</mo><mi>D</mi><mo stretchy="false">)</mo><mi>A</mi></mrow><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">L=\overline{(BC+D)A}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.25em;"></span><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">BC</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mclose">)</span><span class="mord mathnormal">A</span></span></span><span style="top:-3.87em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.25em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:center"></td></tr><tr><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245135.png" alt="image-20240117103238963"></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi><mo>=</mo><mover accent="true"><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>+</mo><mi>B</mi><mo stretchy="false">)</mo><mi>X</mi></mrow><mo stretchy="true">‾</mo></mover><mo>=</mo><mover accent="true"><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>+</mo><mi>B</mi><mo stretchy="false">)</mo><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">‾</mo></mover></mrow><mo stretchy="true">‾</mo></mover><mo>=</mo><mi>A</mi><mo>⊙</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">L=\overline{(A+B)X}=\overline{(A+B)\overline{AB}}=A\odot B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.25em;"></span><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.87em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.25em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.3333em;vertical-align:-0.25em;"></span><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span style="top:-4.0033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.25em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊙</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></td><td style="text-align:center">同或门</td></tr><tr><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245136.png" alt="image-20231201091736402" style="zoom:50%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245137.png" alt="image-20231201091750341" style="zoom:50%;"></td><td style="text-align:center">异或门</td></tr><tr><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245138.png" alt="image-20231201091831389" style="zoom:50%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245139.png" alt="image-20231201091858929" style="zoom:50%;"></td><td style="text-align:center">2选1数据选择器</td></tr></tbody></table><h2 id="四、组合逻辑电路">四、组合逻辑电路</h2><h3 id="4-1-组合逻辑电路的分析">4.1 组合逻辑电路的分析</h3><h4 id="4-1-1-定义">4.1.1 定义</h4><p>只取决于实时输入，从而给出相应的输出，与之前的运行结果无关。<strong>没有反馈</strong>，<strong>没有记忆单元</strong></p><h4 id="4-1-2-分析步骤与方法">4.1.2 分析步骤与方法</h4><ol><li>由逻辑图得到<strong>逻辑表达式</strong></li><li><strong>化简和变换</strong></li><li>列<strong>真值表</strong></li><li>根据真值表（或者波形图）<strong>分析电路功能</strong></li></ol><h3 id="4-2-组合逻辑电路的设计">4.2 组合逻辑电路的设计</h3><h4 id="4-2-1-设计过程">4.2.1 设计过程</h4><ol><li>明确逻辑含义：确定输入输出并定义逻辑状态的含义</li><li>列出真值表：根据逻辑描述写出真值表</li><li>写出逻辑表达式：由真值表写出逻辑表达式，真值取原、假值取反</li><li>化简逻辑表达式：代数化简法 or 卡诺图化简法</li><li>画出逻辑图：使用相应的门级元件进行组合连接</li></ol><h4 id="4-2-2-优化实现">4.2.2 优化实现</h4><table><thead><tr><th style="text-align:center">电路类型</th><th style="text-align:center">优化策略</th><th style="text-align:center">电路图</th><th style="text-align:center">优化结果</th></tr></thead><tbody><tr><td style="text-align:center">单输出电路</td><td style="text-align:center">统一元件类型</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245140.png" alt="image-20231117112320588" style="zoom: 33%;"></td><td style="text-align:center">见左图文字</td></tr><tr><td style="text-align:center">多输出电路</td><td style="text-align:center">共享相同逻辑项</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245141.png" alt="image-20231117112426358" style="zoom:33%;"></td><td style="text-align:center">见左图文字</td></tr><tr><td style="text-align:center">多级逻辑电路（限定入数）</td><td style="text-align:center">提取公因项</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245142.png" alt="image-20231117112559081" style="zoom:33%;"></td><td style="text-align:center">见左图文字</td></tr><tr><td style="text-align:center">多级逻辑电路（限定入数）</td><td style="text-align:center">提取公因项</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245143.png" alt="image-20231117112618856" style="zoom: 33%;"></td><td style="text-align:center">见左图文字</td></tr></tbody></table><h3 id="4-3-组合逻辑电路中的竞争与冒险">4.3 组合逻辑电路中的竞争与冒险</h3><h4 id="4-3-1-产生的原因">4.3.1 产生的原因</h4><p>门级元件的延时效应</p><h4 id="4-3-2-消去的方法">4.3.2 消去的方法</h4><ol><li><p>消除互补变量</p></li><li><p>增加乘积项，避免互补项相加</p></li><li><p>输出端并联电容器</p> <img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245144.png" alt="image-20231117115217185" style="zoom: 50%;"></li></ol><h3 id="4-4-若干典型的组合逻辑电路">4.4 若干典型的组合逻辑电路</h3><h4 id="4-4-1-编码器">4.4.1 编码器</h4><p>普通编码器：只允许有一个输入，从而进行编码，一旦出现多输入就会发生错误</p><p>优先编码器：无论多少输入，都会按照一开始设定的优先级进行最高等级的那一个信号位的编码</p><table><thead><tr><th style="text-align:center">名称</th><th style="text-align:center">型号</th><th style="text-align:center">逻辑符号</th><th style="text-align:center">功能分析</th><th style="text-align:center">逻辑图</th></tr></thead><tbody><tr><td style="text-align:center">4-2优先编码器</td><td style="text-align:center">74LS00</td><td style="text-align:center">2片7400（4个2输入与非门）实现</td><td style="text-align:center">需要将4-2优先编码器的两个逻辑函数转化为与非式，从而进行电路逻辑的搭建。化简后发现需要7个2输入与非门，故需要2片7400才能实现4-2线优先编码器</td><td style="text-align:center"></td></tr><tr><td style="text-align:center">8-3优先编码器</td><td style="text-align:center">CD4532</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245145.png" alt="image-20231208104642447" style="zoom: 50%;"></td><td style="text-align:center">除了8个输入端与3个输出端，还有EI、EO与GS端。其中<strong>GS</strong>是用来标明当前电路是否处于工作状态的，即如果没有输入端为有效信号，GS就是低电平，反之则是高电平。而<strong>EI与EO</strong>是为了电路扩展而诞生的，当EI为高电平且没有任何输入的情况下，EO也是1，此时的4532就相当于一根导线，从而可以进行片子的扩展</td><td style="text-align:center">由于有现成的集成电路板，故就是逻辑符号</td></tr><tr><td style="text-align:center">16-4优先编码器</td><td style="text-align:center">CD4532</td><td style="text-align:center">2片4532实现</td><td style="text-align:center">首先确保EI始终为高电平。<strong>输出后三位</strong>就是两个4532片子的3输出分别或的结果，<strong>最高位的输出</strong>是高位片的GS端的结果</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245146.png" alt="image-20240117120736794" style="zoom:200%;"></td></tr><tr><td style="text-align:center">32-5优先编码器</td><td style="text-align:center">74LS00+CD4532</td><td style="text-align:center">1片7400+4片4532实现</td><td style="text-align:center">首先确保EI始终为高电平。<strong>输出后三位</strong>就是四个4532片子的3输出分别或的结果，<strong>最高位的两个输出</strong>取决于4个片子GS端4-2优先编码的结果。</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245147.png" alt="image-20240117121434992" style="zoom:200%;"></td></tr></tbody></table><h4 id="4-4-2-译码器-数据分配器">4.4.2 译码器/数据分配器</h4><table><thead><tr><th style="text-align:center">名称</th><th style="text-align:center">型号</th><th style="text-align:center">逻辑符号</th><th style="text-align:center">功能分析</th><th style="text-align:center">逻辑图</th></tr></thead><tbody><tr><td style="text-align:center">2-4译码器</td><td style="text-align:center">74X139</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245148.png" alt="image-20231208101820876" style="zoom: 33%;"></td><td style="text-align:center">使能端有效时。按照对应的输出给出相应输出的低电平信号</td><td style="text-align:center">由于有现成的集成电路板，故就是逻辑符号</td></tr><tr><td style="text-align:center">3-8译码器</td><td style="text-align:center">74X138</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245149.png" alt="image-20231208101658175" style="zoom:25%;"></td><td style="text-align:center">使能端有效时。按照对应的输出给出相应输出的低电平信号</td><td style="text-align:center">由于有现成的集成电路板，故就是逻辑符号</td></tr><tr><td style="text-align:center">4-16译码器</td><td style="text-align:center">74X138或74X139</td><td style="text-align:center">2片74X138或5片74X139</td><td style="text-align:center">使能端有效时。<strong>输入的前三位</strong>分别接入两片3-8译码器的输入端，<strong>输入的最后一位</strong>接入两片3-8译码器的高电平使能端即可；如果用2-4译码器来实现，<strong>输入的前两位</strong>分别接入四片2-4译码器的输入端，<strong>输入的后两位</strong>通过一个2-4译码器的四个输入分别接入4片2-4译码器的低电平使能端即可</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245150.png" alt="image-20240117125311036"></td></tr><tr><td style="text-align:center">5-32译码器</td><td style="text-align:center">74X139+74X138</td><td style="text-align:center">1片74X139+4片74X138</td><td style="text-align:center">使能端有效时。<strong>输入的前三位</strong>分别接入四片3-8译码器的输入端，<strong>输入的后两位</strong>通过2-4译码的4个结果分别接入四片3-8译码器的低电平使能端，从而决定是哪一个3-8译码器在工作</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245151.png" alt="image-20231208104231966" style="zoom: 200%;"></td></tr></tbody></table><p><strong>使用译码器实现逻辑函数</strong></p><p>我们知道译码器的每一个输出代表一个最小项，那么对于一个 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 变量的逻辑函数，可以通过以下步骤用 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>−</mo><msup><mn>2</mn><mi>x</mi></msup></mrow><annotation encoding="application/x-tex">x-2^x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6644em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span></span> 译码器实现任意 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 变量的逻辑函数</p><ol><li>将逻辑函数转化为最小项表达式（大量使用摩根定律）</li><li>转化为译码器的输出（写成 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∑</mo><msub><mi>m</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\sum m_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的形式）</li><li>在译码器的输出端加一个多输入与非门即可（对结果进行与非）</li></ol><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245152.png" alt="image-20231208110328594" style="zoom:50%;"><p><strong>数据分配器</strong></p><p>功能：相当于多输出的单刀多掷开关，是将公共数据线上的数据按需要送到不同的通道上去的逻辑电路。</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245153.png" alt="image-20231208114700981"></p><center>图一：示意图</center><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245154.png" alt="image-20231208114749275"></p><center>图二：功能仿真图</center><h4 id="4-4-3-数据选择器">4.4.3 数据选择器</h4><table><thead><tr><th style="text-align:center">名称</th><th style="text-align:center">型号</th><th style="text-align:center">逻辑符号</th><th style="text-align:center">功能分析</th><th style="text-align:center">逻辑图</th></tr></thead><tbody><tr><td style="text-align:center">2选1</td><td style="text-align:center">—</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245155.png" alt="image-20240117135219030" style="zoom:33%;"></td><td style="text-align:center">通过控制端 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span> 来选择 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>D</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">D_0,D_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245156.png" alt="image-20240117135236405" style="zoom:33%;"></td></tr><tr><td style="text-align:center">4选1</td><td style="text-align:center">—</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245157.png" alt="image-20240117135258894" style="zoom:33%;"></td><td style="text-align:center">通过控制端 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>S</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>S</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">S_0,S_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 来选择 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>D</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>D</mi><mn>2</mn></msub><mo separator="true">,</mo><msub><mi>D</mi><mn>3</mn></msub></mrow><annotation encoding="application/x-tex">D_0,D_1,D_2,D_3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245158.png" alt="image-20240117135326247" style="zoom:33%;"></td></tr><tr><td style="text-align:center">8选1</td><td style="text-align:center">74HC151</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245159.png" alt="image-20240117135401170" style="zoom: 25%;"></td><td style="text-align:center">通过控制端 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>S</mi><mn>0</mn></msub><mo>−</mo><msub><mi>S</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">S_0-S_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 来选择 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mn>0</mn></msub><mo>−</mo><msub><mi>D</mi><mn>7</mn></msub></mrow><annotation encoding="application/x-tex">D_0-D_7</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">7</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></td><td style="text-align:center">由于有现成的集成电路板，故就是逻辑符号</td></tr><tr><td style="text-align:center">16选1</td><td style="text-align:center">2片74HC151</td><td style="text-align:center">—</td><td style="text-align:center">通过控制端 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>S</mi><mn>0</mn></msub><mo>−</mo><msub><mi>S</mi><mn>3</mn></msub></mrow><annotation encoding="application/x-tex">S_0-S_3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 来选择 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mn>0</mn></msub><mo>−</mo><msub><mi>D</mi><mn>15</mn></msub></mrow><annotation encoding="application/x-tex">D_0-D_{15}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">15</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，<strong>输入的前三位</strong>连接三个控制端，<strong>输入的最后一位</strong>连接两片74151的使能端，其实就是译码器的魔改版，让输出为相应的译码结果的高电平而已</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245160.png" alt="image-20240117135555070"></td></tr></tbody></table><p><strong>使用数据选择器实现逻辑函数</strong></p><ul><li>变量个数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>&lt;</mo></mrow><annotation encoding="application/x-tex">&lt;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">&lt;</span></span></span></span> 数据选择端个数：变量直接对应数据选择端，多余的选择端置0，最后相应的信号输入端进行赋1或赋0的操作即可</li><li>变量个数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo></mrow><annotation encoding="application/x-tex">=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">=</span></span></span></span> 数据选择端个数：本质上就是将逻辑函数转化为最小项表达式，然后与标准与或式进行比对，已出现的最小项与1，未出现的最小项与0，从而配凑产生了数据选择器最开始的式子。落到逻辑图上就是，数据选择端接入函数变量，信号输入端接入相应的高低电平，出现的最小项就输入1，未出现的就输入0即可</li><li>变量个数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>&gt;</mo></mrow><annotation encoding="application/x-tex">&gt;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">&gt;</span></span></span></span> 数据选择端个数：<ul><li>刚好多 1 个：变量 or 变量的非接入信号输入端</li><li>不止多 1 个：同样采用将变量作为数据信号输入端，此外可能需要借助相关的门电路辅助进行</li></ul></li></ul><h4 id="4-4-4-数值比较器">4.4.4 数值比较器</h4><p>略</p><h4 id="4-4-5-算术运算电路">4.4.5 算术运算电路</h4><p>半加器：即不考虑低位进位的一位二进制加法器。其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span> 为输出位，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> 为进位，没有考虑低位的进位</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>S</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">=</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>A</mi><mo>⊕</mo><mi>B</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>C</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">=</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>A</mi><mi>B</mi></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases}S &amp;=&amp; A \oplus B \\C &amp;=&amp; AB\end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mrel">=</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mrel">=</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊕</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><p>全加器：即考虑低位进位的一位二进制加法器。其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span> 为输出位，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>C</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">C_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 为低位的进位，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>C</mi><mi>o</mi></msub></mrow><annotation encoding="application/x-tex">C_o</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">o</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 为进位</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>S</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">=</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>A</mi><mo>⊕</mo><mi>B</mi><mo>⊕</mo><msub><mi>C</mi><mi>i</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>C</mi><mi>o</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">=</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>A</mi><mi>B</mi><mo>+</mo><mo stretchy="false">(</mo><mi>A</mi><mo>⊕</mo><mi>B</mi><mo stretchy="false">)</mo><msub><mi>C</mi><mi>i</mi></msub><mo>=</mo><mi>A</mi><mi>B</mi><mo>+</mo><mo stretchy="false">(</mo><mi>A</mi><mo>+</mo><mi>B</mi><mo stretchy="false">)</mo><msub><mi>C</mi><mi>i</mi></msub></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases}S &amp;=&amp; A \oplus B \oplus C_i \\C_o &amp;=&amp; AB + (A \oplus B)C_i = AB+(A+B)C_i\end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">o</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mrel">=</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mrel">=</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊕</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊕</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊕</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><h3 id="4-5-用-verilog-描述组合逻辑电路">4.5 用 verilog 描述组合逻辑电路</h3><h2 id="五、锁存器和触发器">五、锁存器和触发器</h2><p>本章介绍时序逻辑电路的存储单元，分别为锁存器和触发器。其中锁存器对电平敏感，触发器对边沿敏感</p><h3 id="5-1-基本双稳态电路">5.1 基本双稳态电路</h3><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245162.png" alt="image-20231222110345818" style="zoom:80%;"><h3 id="5-2-SR-锁存器">5.2 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">SR</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">SR</span></span></span></span> 锁存器</h3><table><thead><tr><th style="text-align:center">门级元件组成</th><th style="text-align:center">电路图</th><th style="text-align:center">功能分析</th></tr></thead><tbody><tr><td style="text-align:center">或非门实现</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245163.png" alt="image-20231222100822028" style="zoom:50%;"></td><td style="text-align:center">高电平有效。全0不变，谁1谁有效，都1不确定状态</td></tr><tr><td style="text-align:center">与非门实现</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245164.png" alt="image-20231222100904959" style="zoom:50%;"></td><td style="text-align:center">低电平有效。全1不变，谁0谁有效，都0不确定状态</td></tr></tbody></table><table><thead><tr><th style="text-align:center">应用</th><th style="text-align:center">电路图</th><th style="text-align:center">功能分析</th></tr></thead><tbody><tr><td style="text-align:center">开关电路</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245165.png" alt="image-20231222101855246" style="zoom:50%;"><img src="https://s2.loli.net/2023/12/22/tzFvJp8ulGnRsj5.png" alt="image-20231222102251343" style="zoom:35%;"></td><td style="text-align:center">无论开关如何震动，输出始终正常</td></tr><tr><td style="text-align:center">门控 SR 锁存器</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245166.png" alt="image-20231222103758704" style="zoom:33%;"></td><td style="text-align:center">就是加了一个使能端 E，如果 E 为 1，则就是一个基本的 SR 锁存器，如果 E 为 0，则保持</td></tr></tbody></table><h3 id="5-3-D-锁存器">5.3 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 锁存器</h3><table><thead><tr><th style="text-align:center">电路名称</th><th style="text-align:center">逻辑电路图</th><th style="text-align:center">功能分析</th></tr></thead><tbody><tr><td style="text-align:center">传输门控制的D锁存器</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245167.png" alt="image-20231222105139215" style="zoom: 50%;"></td><td style="text-align:center">E=0, Q=不变；E=1, Q=D</td></tr><tr><td style="text-align:center">逻辑门控制的D锁存器</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245168.png" alt="image-20240117191849592" style="zoom:67%;"></td><td style="text-align:center">E=0, Q=不变；E=1, Q=D</td></tr></tbody></table><h3 id="5-4-触发器">5.4 触发器</h3><h4 id="5-4-1-主从-D-触发器的电路结构和工作原理">5.4.1 主从 D 触发器的电路结构和工作原理</h4><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245169.png" alt="image-20231222112319354" style="zoom:80%;"><h4 id="5-4-2-典型的主从-D-触发器集成电路">5.4.2 典型的主从 D 触发器集成电路</h4><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245170.png" alt="image-20231222113334822" style="zoom:50%;"><center>    <img src="https://s2.loli.net/2024/01/17/qzpDMR7BXtQETbr.png" alt="image-20231222113438371" style="zoom:67%;">    <img src="https://s2.loli.net/2024/01/17/GY3msiIEMtXr1kU.png" alt="image-20231222113452256" style="zoom:50%;"></center>### 5.5 触发器的逻辑功能<p>本目需要掌握有关触发器的<strong>特性表</strong>、<strong>特性方程</strong>、<strong>状态图</strong>三者的单独书写以及相互转化的逻辑过程，还需要掌握不同的触发器之间的相互<strong>替换实现</strong></p><table><thead><tr><th style="text-align:center">类型</th><th style="text-align:center">逻辑符号</th><th style="text-align:center">特性表</th><th style="text-align:center">特性方程</th><th style="text-align:center">状态图</th></tr></thead><tbody><tr><td style="text-align:center">D 触发器</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245171.png" alt="image-20240117190140089" style="zoom: 33%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245172.png" alt="image-20240117190521349" style="zoom:25%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245173.png" alt="image-20240117190538232" style="zoom:25%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245174.png" alt="image-20240117190553444" style="zoom:33%;"></td></tr><tr><td style="text-align:center">JK 触发器</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245175.png" alt="image-20240117190400613" style="zoom:33%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245176.png" alt="image-20240117190647444" style="zoom:25%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245177.png" alt="image-20240117190703599" style="zoom:25%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245178.png" alt="image-20240117190716843" style="zoom: 67%;"></td></tr><tr><td style="text-align:center">T 触发器</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245179.png" alt="image-20240117190415482" style="zoom:33%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245180.png" alt="image-20240117190800382" style="zoom:33%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245181.png" alt="image-20240117190818921" style="zoom:25%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245182.png" alt="image-20240117190838344" style="zoom:67%;"></td></tr><tr><td style="text-align:center">T’ 触发器</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245183.png" alt="image-20240117190915612" style="zoom:33%;"></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi><mo>≡</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">T\equiv 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245184.png" alt="image-20240117190926990" style="zoom:33%;"></td><td style="text-align:center"></td></tr><tr><td style="text-align:center">SR 触发器</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245185.png" alt="image-20240117190440059" style="zoom:33%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245186.png" alt="image-20240117191017989" style="zoom:25%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245187.png" alt="image-20240117191040611" style="zoom:33%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245188.png" alt="image-20240117191057024" style="zoom:33%;"></td></tr></tbody></table><h2 id="六、时序逻辑电路-star">六、时序逻辑电路<span class="github-emoji"><span>⭐</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/2b50.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span></h2><h3 id="6-1-同步时序逻辑电路的分析">6.1 同步时序逻辑电路的分析</h3><p>本部分只需要掌握同步时序逻辑电路的分析即可，具体直接从例题出发。三道同步时序逻辑电路分析的例题见教材 P282 ~ P286，分别为</p><ul><li>例一：可控二进制计数器</li><li>例二：可控双向二进制计数器</li><li>例三：脉冲分配器</li></ul><p>下面介绍同步时序逻辑电路分析的五个步骤。在分析之前我们要知道我们的最终目标是什么，可以知道，我们分析电路的最终目标是想要量化的确定电路的物理实现的功能，至于如何设计，此处不予讨论。现在给定了一个同步时序逻辑电路的<strong>逻辑电路图</strong>，接下来我们应该：</p><ol><li><p>了解电路组成：同步 or 异步？穆尔型输出（与输入无关） or 米利型输出（与输入有关） or 都有？由什么触发器组成的？触发器类型是上升沿出发 or 下降沿触发？</p></li><li><p>列出三个方程：</p><ul><li><p>输出方程：电路的最终输出</p></li><li><p>激励方程：触发器的输入</p></li><li><p>状态方程：触发器的输出（将触发器的输入也就是激励方程代入触发器的特性方程即可）</p></li></ul></li><li><p>写出转换表（分析功能用）</p></li><li><p>写出状态图（分析功能用）</p></li><li><p>写出时序图（分析功能用）默认状态的初值设置为 0</p></li></ol><h3 id="6-2-典型的时序逻辑电路-计数器">6.2 典型的时序逻辑电路 - 计数器</h3><p>主要讲 N 位二进制计数器中，利用集成电路板 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>74</mn><mi>L</mi><mi>V</mi><mi>C</mi><mn>161</mn></mrow><annotation encoding="application/x-tex">74LVC161</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">74</span><span class="mord mathnormal">L</span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord">161</span></span></span></span> 实现的 <strong>4 位同步二进制递增计数器</strong>。进而引出利用该 4 位计数器<strong>实现模 N 计数器</strong>的分析与设计思路。同时补充 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>74</mn><mi>L</mi><mi>V</mi><mi>C</mi><mn>162</mn></mrow><annotation encoding="application/x-tex">74LVC162</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">74</span><span class="mord mathnormal">L</span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord">162</span></span></span></span> 实现的 <strong>4 位同步十进制递增计时器</strong>，进而引出相关的模 N 设计思路。下面分析 74LVC161 4 位同步二进制递增计数器集成板</p><center><img src="https://s2.loli.net/2024/01/11/Ivm2g5NrxeCL7Du.png" alt="image-20240111233651905"></center><center>图1 74LVC161集成板</center><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245189.png" alt="image-20240111234456513" style="zoom: 50%;"><center>图2 74LVC161逻辑功能表</center><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>C</mi><mi>R</mi></mrow><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\overline{CR}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">CR</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span></span>: <strong>异步清零</strong>。即无视时钟脉冲信号，直接清零</p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>P</mi><mi>E</mi></mrow><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\overline{PE}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">PE</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span></span>: <strong>同步预置</strong>。即当有效始终脉冲沿到来时，实现 4 个预置位的输出，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mn>3</mn></msub><mo separator="true">,</mo><msub><mi>D</mi><mn>2</mn></msub><mo separator="true">,</mo><msub><mi>D</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>D</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">D_3,D_2,D_1,D_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi><mi>E</mi><mi>P</mi><mo separator="true">,</mo><mi>C</mi><mi>E</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">CEP,CET</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">CEP</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">CET</span></span></span></span>: 使能端。同时为高电平电路才能正常工作</p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi><mi>C</mi></mrow><annotation encoding="application/x-tex">TC</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">TC</span></span></span></span>: 进位输出</p><p><strong>本目小结</strong></p><p>多个集成板进行计数</p><p>考虑如何通信：低位进位作为高位使能？</p><p>考虑如何清零：同步（异步）清零？同步（异步）置数？</p><p>学会利用 74LVC161 的“反馈清零法”实现模 N 计数器</p><p>学会利用 74LVC161 的”反馈置数法“实现模 N 计数器</p><p>学会实现 74LVC162 十进制递增功能功能（<strong>同步清零</strong>、<strong>同步置数</strong>）</p><p>学会利用 74LVC162 的同步清零的特性实现模 9 的九进制计数器功能</p><p>学会利用 74LVC162 实现模 24 的二十四进制计数器功能</p><blockquote><p>00-09与10-19的计数：通过低位片的进位端，作为高位片的使能端即可</p><p>20-23与23-00的计数：通过将低位片的两个低位与高位片通过 4023 三输入与非门连接起来，当全为1时，就是计数到23的状态，此时对高低片进行同步清零即可</p></blockquote><h3 id="6-3-用-verilog-描述同步时序逻辑电路">6.3 用 verilog 描述同步时序逻辑电路</h3>]]></content>
     
     
     <categories>
@@ -1094,11 +1094,11 @@
   
   
   <entry>
-    <title>DataStructureClassDesign</title>
-    <link href="/GPA/3rd-term/DataStructureClassDesign/"/>
-    <url>/GPA/3rd-term/DataStructureClassDesign/</url>
+    <title>DataStructure</title>
+    <link href="/GPA/3rd-term/DataStructure/"/>
+    <url>/GPA/3rd-term/DataStructure/</url>
     
-    <content type="html"><![CDATA[<h2 id="数据结构课程设计">数据结构课程设计</h2><h2 id="效果演示">效果演示</h2><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403021802720.gif" alt="sort 窗口程序"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403021802061.gif" alt="hash 窗口程序"></p><blockquote><p>[!note]</p><p>以下为课设报告内容</p></blockquote><h2 id="一、必做题">一、必做题</h2><p>题目：编程实现希尔、快速、堆排序、归并排序算法。要求随机产生 10000 个数据存入磁盘文件，然后读入数据文件，分别采用不同的排序方法进行排序，并将结果存入文件中。</p><h3 id="1-1-数据结构设计与算法思想">1.1 数据结构设计与算法思想</h3><p>本程序涉及到四种排序算法，其中：</p><ul><li>希尔排序：通过倍增方法逐渐扩大选择插入数据量的规模，而达到减小数据比较的次数，时间复杂度优化到近似 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>1.65</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^{1.65})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1.65</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></li><li>快速排序：通过分治的算法策略，不断确定每一个数的最终位置，从而达到近似 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 的时间复杂度</li><li>堆排序：通过堆的树形结构，减小两数的比较次数，最终通过不断维护堆结构来获得有序序列，时间复杂度为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li><li>归并排序：通过分治排序回溯时左右两支有序的特点，进行归并，使算法整体的时间复杂度为固定的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li></ul><h3 id="1-2-程序结构">1.2 程序结构</h3><p>为了更好的了解排序算法内部的机制，我统计了每一种排序算法内部的比较次数,并结合了 Qt 的可视化框架进行编写。将四种排序作为算法内核嵌入 GUI 界面，程序结构如下：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403011712981.png" alt="sort 程序结构图"></p><h4 id="UI">UI</h4><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-comment">// 窗口 继承窗口库组件</span><br>&lt;widget <span class="hljs-keyword">class</span>=<span class="hljs-string">"QWidget"</span> name=<span class="hljs-string">"SortWidget"</span>&gt;<br>    <span class="hljs-comment">// 标题 标签组件</span><br>    &lt;widget <span class="hljs-keyword">class</span>=<span class="hljs-string">"QLabel"</span> name=<span class="hljs-string">"titleLabel"</span>&gt;&lt;/widget&gt;<br>    <span class="hljs-comment">// 输入 网格布局组件</span><br>    &lt;widget <span class="hljs-keyword">class</span>=<span class="hljs-string">"QGridLayout"</span> name=<span class="hljs-string">"inputGridLayout"</span>&gt;&lt;/widget&gt;<br>    <span class="hljs-comment">// 输出 网格布局组件</span><br>    &lt;widget <span class="hljs-keyword">class</span>=<span class="hljs-string">"QGridLayout"</span> name=<span class="hljs-string">"outputGridLayout"</span>&gt;&lt;/widget&gt;  <br>    <span class="hljs-comment">// 交互 水平布局组件</span><br>    &lt;widget <span class="hljs-keyword">class</span>=<span class="hljs-string">"QHBoxLayout"</span> name=<span class="hljs-string">"buttonHLayout"</span>&gt;&lt;/widget&gt;     <br>&lt;/widget&gt;<br></code></pre></td></tr></tbody></table></figure><h4 id="Qt-Application">Qt Application</h4><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">ifndef</span> DATASTRUCTURECLASSDESIGN_SORTWIDGET_H</span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> DATASTRUCTURECLASSDESIGN_SORTWIDGET_H</span><br><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;QWidget&gt;</span></span><br><br><br>QT_BEGIN_NAMESPACE<br><span class="hljs-keyword">namespace</span> Ui {<br>    <span class="hljs-keyword">class</span> <span class="hljs-title class_">SortWidget</span>;<br>}<br>QT_END_NAMESPACE<br><br><span class="hljs-keyword">class</span> <span class="hljs-title class_">SortWidget</span> : <span class="hljs-keyword">public</span> QWidget {<br>Q_OBJECT<br><br><span class="hljs-keyword">private</span>:<br>    Ui::SortWidget* ui;         <span class="hljs-comment">// 窗口对象指针</span><br>    QString path;               <span class="hljs-comment">// 文件存储路径</span><br><br><span class="hljs-keyword">private</span> slots:<br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">pushFolderButton</span><span class="hljs-params">()</span></span>;    <span class="hljs-comment">// 槽函数 - 触发事件：获取存储路径的窗口对话</span><br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">pushCommitButton</span><span class="hljs-params">()</span></span>;    <span class="hljs-comment">// 槽函数 - 触发事件：根据输入数据量执行算法</span><br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">pushCancelButton</span><span class="hljs-params">()</span></span>;    <span class="hljs-comment">// 槽函数 - 触发事件：清空窗口所有标签的数据</span><br><br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-function"><span class="hljs-keyword">explicit</span> <span class="hljs-title">SortWidget</span><span class="hljs-params">(QWidget* parent = <span class="hljs-literal">nullptr</span>)</span></span>; <span class="hljs-comment">// 窗口构造函数</span><br><br>    ~<span class="hljs-built_in">SortWidget</span>() <span class="hljs-keyword">override</span>;                         <span class="hljs-comment">// 窗口析构函数</span><br>};<br></code></pre></td></tr></tbody></table></figure><h4 id="Algorithm">Algorithm</h4><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;vector&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;fstream&gt;</span></span><br><br><span class="hljs-keyword">class</span> <span class="hljs-title class_">sortAlgorithm</span> {<br><span class="hljs-keyword">private</span>:<br>    <span class="hljs-type">int</span> Size, Range;<br>    std::string Path;<br>    std::vector&lt;<span class="hljs-type">int</span>&gt; arr;<br><br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">Generate</span><span class="hljs-params">(<span class="hljs-type">int</span> num, <span class="hljs-type">int</span> range)</span></span>;                 <span class="hljs-comment">// 数据生成</span><br>    <br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">ShellSort</span><span class="hljs-params">(std::vector&lt;<span class="hljs-type">int</span>&gt; a)</span></span>;                 <span class="hljs-comment">// 希尔排序</span><br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">QuickSort</span><span class="hljs-params">(std::vector&lt;<span class="hljs-type">int</span>&gt; a)</span></span>;                 <span class="hljs-comment">// 快速排序</span><br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">HeapSort</span><span class="hljs-params">(std::vector&lt;<span class="hljs-type">int</span>&gt; a)</span></span>;                  <span class="hljs-comment">// 堆排序</span><br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">MergeSort</span><span class="hljs-params">(std::vector&lt;<span class="hljs-type">int</span>&gt; a)</span></span>;                 <span class="hljs-comment">// 归并排序</span><br>    <br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">WriteToFile</span><span class="hljs-params">(std::string path, std::vector&lt;<span class="hljs-type">int</span>&gt;&amp; a)</span></span>;   <span class="hljs-comment">// 写入文件</span><br><br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-built_in">sortAlgorithm</span>(<span class="hljs-type">int</span> _Size, <span class="hljs-type">int</span> _Range, std::string _Path);   <span class="hljs-comment">// 构造函数</span><br><br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">ShellSort</span><span class="hljs-params">()</span></span>;                                   <span class="hljs-comment">// 用户调用希尔排序</span><br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">QuickSort</span><span class="hljs-params">()</span></span>;                                   <span class="hljs-comment">// 用户调用快速排序</span><br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">HeapSort</span><span class="hljs-params">()</span></span>;                                    <span class="hljs-comment">// 用户调用堆排序</span><br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">MergeSort</span><span class="hljs-params">()</span></span>;                                   <span class="hljs-comment">// 用户调用归并排序</span><br>};<br></code></pre></td></tr></tbody></table></figure><h3 id="1-3-实验数据与测试结果分析">1.3 实验数据与测试结果分析</h3><p>原始数据序列生成算法使用时间种子的除留余数法，在数据规模为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10000</mn></mrow><annotation encoding="application/x-tex">10000</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10000</span></span></span></span> 的情况下，四种算法的比较次数如下：</p><table><thead><tr><th style="text-align:center"></th><th style="text-align:center">shell</th><th style="text-align:center">heap</th><th style="text-align:center">quick</th><th style="text-align:center">merge</th></tr></thead><tbody><tr><td style="text-align:center">1</td><td style="text-align:center">129419</td><td style="text-align:center">231824</td><td style="text-align:center">39061</td><td style="text-align:center">120488</td></tr><tr><td style="text-align:center">2</td><td style="text-align:center">128051</td><td style="text-align:center">232204</td><td style="text-align:center">38723</td><td style="text-align:center">120487</td></tr><tr><td style="text-align:center">3</td><td style="text-align:center">134873</td><td style="text-align:center">231945</td><td style="text-align:center">39052</td><td style="text-align:center">120251</td></tr><tr><td style="text-align:center">4</td><td style="text-align:center">135719</td><td style="text-align:center">232096</td><td style="text-align:center">39158</td><td style="text-align:center">120510</td></tr><tr><td style="text-align:center">5</td><td style="text-align:center">128821</td><td style="text-align:center">232051</td><td style="text-align:center">38747</td><td style="text-align:center">120434</td></tr><tr><td style="text-align:center">average</td><td style="text-align:center">131376</td><td style="text-align:center">232024</td><td style="text-align:center">38948</td><td style="text-align:center">120434</td></tr></tbody></table><p>可以发现：在数据规模在  <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi>e</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">1e4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span><span class="mord mathnormal">e</span><span class="mord">4</span></span></span></span> 的情况下，堆排序的比较次数最多，而快速排序的比较次数最少</p><h3 id="1-4-程序清单">1.4 程序清单</h3><ul><li><code>/Src/Algorithm/sortAlgorithm.cpp</code>：内核排序算法</li><li><code>/Src/Forms/SortWidget.ui</code>：ui 界面</li><li><code>/Src/Headers/SortWidget.h</code>：排序组件逻辑头文件</li><li><code>/Src/Post/SortWidget.cpp</code>：排序组件逻辑源文件</li><li><code>/Src/Test/TestSort.cpp</code>：程序测试文件</li></ul><p>其中，除了 ui 界面使用 Qt Designer 设计师工具进行编写，其余文件均为与文件名同名的 C++ 类，便于维护与扩展</p><h2 id="二、选做题">二、选做题</h2><p>题三：扫描一个 C 源程序，利用两种方法统计该源程序中的关键字出现的频度，并比较各自查找的比较次数。</p><ol><li><p>用 Hash 表存储源程序中出现的关键字，利用 Hash 查找技术统计该程序中的关键字出现的频度。用线性探测法解决 Hash 冲突。设 Hash 函数为 Hash(key) = [(key 的第一个字母序号) × 100 + (key 的最后一个字母序号)] % 41</p></li><li><p>用顺序表存储源程序中出现的关键字，利用二分查找技术统计该程序中的关键字出现的频度。</p></li></ol><h3 id="2-1-数据结构设计与算法思想">2.1 数据结构设计与算法思想</h3><ul><li>第一问：<ul><li><p>首先用关键字序列构造哈希表，哈希函数使用题中所给，容易算出哈希表的最大容量为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>25</mn><mo>×</mo><mn>100</mn><mo>+</mo><mn>40</mn></mrow><annotation encoding="application/x-tex">25 \times 100+40</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">25</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">100</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">40</span></span></span></span> ，我们设置为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3000</mn></mrow><annotation encoding="application/x-tex">3000</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3000</span></span></span></span> ，哈希表数据结构为：<code>int: &lt;string, int&gt;</code> 的键值对形式，可以使用动态数组容器 <code>std::vector&lt;std::pair&gt;</code> 进行存储。其中键设为 <code>int</code> 可以使得对于一个 <code>int</code> 类型的哈希值进行 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 的存储与查找，获得 <code>&lt;word, cnt&gt;</code>，即 <code>&lt;std::string, int&gt;</code></p></li><li><p>然后对于文件中的每一个单词采用哈希搜索。利用哈希函数计算出每一个单词的哈希值，然后通过线性探测法进行搜索比对。关键在于如何解析出一个完整的单词，对于流读入的字符串（不包含空格、换行符、制表符等空白符），我们删除其中的符号后将剩余部分进行合并，之后进行异常处理，排除空串后进行哈希查找。查找次数包含成功和失败的比较次数</p></li></ul></li><li>第二问：将关键字存储于顺序表中，排序后，利用二分查找技术进行搜索统计。查找次数包含成功和失败的比较次数</li></ul><h3 id="2-2-程序结构">2.2 程序结构</h3><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403011712321.png" alt="hash 程序结构图"></p><h4 id="UI-2">UI</h4><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-comment">// 窗口 继承窗口库组件</span><br>&lt;widget <span class="hljs-keyword">class</span>=<span class="hljs-string">"QWidget"</span> name=<span class="hljs-string">"SortWidget"</span>&gt;                         <br>    <span class="hljs-comment">// 标题 标签组件</span><br>    &lt;widget <span class="hljs-keyword">class</span>=<span class="hljs-string">"QLabel"</span> name=<span class="hljs-string">"titleLabel"</span>&gt;&lt;/widget&gt;             <br>    <span class="hljs-comment">// 输入 网格布局组件</span><br>    &lt;widget <span class="hljs-keyword">class</span>=<span class="hljs-string">"QGridLayout"</span> name=<span class="hljs-string">"inputGridLayout"</span>&gt;&lt;/widget&gt;   <br>    <span class="hljs-comment">// 输出 网格布局组件</span><br>    &lt;widget <span class="hljs-keyword">class</span>=<span class="hljs-string">"QGridLayout"</span> name=<span class="hljs-string">"outputGridLayout"</span>&gt;&lt;/widget&gt;<br>    <span class="hljs-comment">// 交互 水平布局组件</span><br>    &lt;widget <span class="hljs-keyword">class</span>=<span class="hljs-string">"QHBoxLayout"</span> name=<span class="hljs-string">"buttonHLayout"</span>&gt;&lt;/widget&gt;     <br>&lt;/widget&gt;<br></code></pre></td></tr></tbody></table></figure><h4 id="Qt-Application-2">Qt Application</h4><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">ifndef</span> DATASTRUCTURECLASSDESIGN_HASHWIDGET_H</span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> DATASTRUCTURECLASSDESIGN_HASHWIDGET_H</span><br><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;QWidget&gt;</span></span><br><br><br>QT_BEGIN_NAMESPACE<br><span class="hljs-keyword">namespace</span> Ui {<br>    <span class="hljs-keyword">class</span> <span class="hljs-title class_">HashWidget</span>;<br>}<br>QT_END_NAMESPACE<br><br><span class="hljs-keyword">class</span> <span class="hljs-title class_">HashWidget</span> : <span class="hljs-keyword">public</span> QWidget {<br>Q_OBJECT<br><br><span class="hljs-keyword">private</span>:<br>    Ui::HashWidget* ui;                       <span class="hljs-comment">// 窗口对象指针</span><br>    QString inputFilePath, outputFolderPath;  <span class="hljs-comment">// 输入输出路径</span><br><br><span class="hljs-keyword">private</span> slots:<br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">pushInputFileButton</span><span class="hljs-params">()</span></span>;     <span class="hljs-comment">// 槽函数 - 触发事件：获取输入文件路径的窗口对话</span><br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">pushOutputFolderButton</span><span class="hljs-params">()</span></span>;  <span class="hljs-comment">// 槽函数 - 触发事件：获取存储文件路径的窗口对话</span><br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">pushCommitButton</span><span class="hljs-params">()</span></span>;        <span class="hljs-comment">// 槽函数 - 触发事件：根据输入的文件开始执行算法</span><br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">pushCancelButton</span><span class="hljs-params">()</span></span>;        <span class="hljs-comment">// 槽函数 - 触发事件：清除当前窗口所有标签的内容</span><br><br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-function"><span class="hljs-keyword">explicit</span> <span class="hljs-title">HashWidget</span><span class="hljs-params">(QWidget* parent = <span class="hljs-literal">nullptr</span>)</span></span>; <span class="hljs-comment">// 窗口构造函数</span><br><br>    ~<span class="hljs-built_in">HashWidget</span>() <span class="hljs-keyword">override</span>;                         <span class="hljs-comment">// 窗口析构函数</span><br><br>};<br><br><span class="hljs-meta">#<span class="hljs-keyword">endif</span> <span class="hljs-comment">//DATASTRUCTURECLASSDESIGN_HASHWIDGET_H</span></span><br></code></pre></td></tr></tbody></table></figure><h4 id="Algorithm-2">Algorithm</h4><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;unordered_map&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;fstream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;vector&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;algorithm&gt;</span></span><br><br><br><span class="hljs-keyword">class</span> <span class="hljs-title class_">hashAlgorithm</span> {<br><span class="hljs-keyword">private</span>:<br>    std::vector&lt;std::string&gt; keywords = {<br>            <span class="hljs-string">"auto"</span>, <span class="hljs-string">"break"</span>, <span class="hljs-string">"case"</span>, <span class="hljs-string">"char"</span>, <span class="hljs-string">"const"</span>,<br>            <span class="hljs-string">"continue"</span>, <span class="hljs-string">"default"</span>, <span class="hljs-string">"do"</span>, <span class="hljs-string">"double"</span>, <span class="hljs-string">"else"</span>,<br>            <span class="hljs-string">"enum"</span>, <span class="hljs-string">"extern"</span>, <span class="hljs-string">"float"</span>, <span class="hljs-string">"for"</span>, <span class="hljs-string">"goto"</span>,<br>            <span class="hljs-string">"if"</span>, <span class="hljs-string">"int"</span>, <span class="hljs-string">"long"</span>, <span class="hljs-string">"register"</span>, <span class="hljs-string">"return"</span>,<br>            <span class="hljs-string">"short"</span>, <span class="hljs-string">"signed"</span>, <span class="hljs-string">"sizeof"</span>, <span class="hljs-string">"static"</span>, <span class="hljs-string">"struct"</span>,<br>            <span class="hljs-string">"switch"</span>, <span class="hljs-string">"typedef"</span>, <span class="hljs-string">"union"</span>, <span class="hljs-string">"unsigned"</span>, <span class="hljs-string">"void"</span>,<br>            <span class="hljs-string">"volatile"</span>, <span class="hljs-string">"while"</span><br>    };<br><br>    std::string inputPath, outputPath;<br>    <span class="hljs-type">int</span> hashSize;<br><br>    <span class="hljs-comment">// std 检验</span><br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">writeToFile</span><span class="hljs-params">(std::unordered_map&lt;std::string, <span class="hljs-type">int</span>&gt;&amp; keywordCount, std::string path)</span></span>;<br><br>    <span class="hljs-comment">// 自定义哈希表检验</span><br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">writeToFile</span><span class="hljs-params">(std::vector&lt;std::pair&lt;std::string, <span class="hljs-type">int</span>&gt;&gt;&amp; keywordCount, std::string path)</span></span>;<br><br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-comment">// 构造函数</span><br>    <span class="hljs-built_in">hashAlgorithm</span>(std::string _inputPath, std::string _outputPath);   <br><br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">stdHash</span><span class="hljs-params">()</span></span>;                 <span class="hljs-comment">// 标准哈希 - 用于测试检验</span><br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">selfHash</span><span class="hljs-params">()</span></span>;                 <span class="hljs-comment">// 自定义哈希</span><br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">binaryHash</span><span class="hljs-params">()</span></span>;               <span class="hljs-comment">// 二分哈希</span><br>};<br></code></pre></td></tr></tbody></table></figure><h3 id="2-3-实验数据与测试结果分析">2.3 实验数据与测试结果分析</h3><p>代码文件从 github 上下载而来，其中 100 行与 500 行的代码文件为完整程序，2500 行代码文件为手动构造程序，三种文件数据规模的查找比较次数如下：</p><table><thead><tr><th style="text-align:center"></th><th style="text-align:center">self hash</th><th style="text-align:center">binary hash</th></tr></thead><tbody><tr><td style="text-align:center">100 lines</td><td style="text-align:center">120</td><td style="text-align:center">775</td></tr><tr><td style="text-align:center">500 lines</td><td style="text-align:center">1546</td><td style="text-align:center">10300</td></tr><tr><td style="text-align:center">2500 lines</td><td style="text-align:center">5561</td><td style="text-align:center">39097</td></tr></tbody></table><p>可以发现：使用线性探测法进行哈希的比较次数远小于二分搜索的比较次数</p><h3 id="2-4-程序清单">2.4 程序清单</h3><ul><li><code>/Src/Algorithm/hashAlgorithm.cpp</code>：内核哈希算法</li><li><code>/Src/Forms/HashWidget.ui</code>：ui 界面</li><li><code>/Src/Headers/HashWidget.h</code>：哈希组件逻辑头文件</li><li><code>/Src/Post/HashWidget.cpp</code>：哈希组件逻辑源文件</li><li><code>/Src/Test/TestHash.cpp</code>：程序测试文件</li></ul><p>其中，除了 ui 界面使用 Qt Designer 设计师工具进行编写，其余文件均为与文件名同名的 C++ 类，便于维护与扩展</p><h2 id="三、收获与体会">三、收获与体会</h2><p><strong>算法思维</strong>。四种排序算法总共也就 100 行，加上堆排序采用树形迭代，快排与归并排序采用递归，整体编码难度并不算高。即使加上了文件 I/O 操作也不过 150 行。而第二道题也是 hash 查找与 binary 查找加上文件 I/O 操作，整体算法难度也十分有限。但是对于我来说也加强了 hash 构建与查找的逻辑。</p><p><strong>前后端开发</strong>。为了锻炼设计模式思维与软件工程思维，我加入了 Qt 集成的 GUI 界面，不仅体会到了算法对于软件的核心作用所在，也体会到了窗口程序的数据交互与数据接口的开发过程。比如 Qt 中信号与槽的概念，其实就是一种数据交互的 API 格式。同时在体验开发过程的同时，也感受到了用户体验的前端板块，通过使用 Qt Designer 的设计师开发界面，可以通过拖动组件的形式进行可视化设计界面，提高了开发效率，如果可以多人组队，也实现了前后端分离的同步开发模式。</p><p><strong>异常处理</strong>。与传统 OJ 不同的是，一个软件还需要考虑更多的异常处理，没有标准的 std 数据，加上用户输入、操作的无穷性，注定了软件的异常处理不是一个简单的事，比如在解析用户输入的字符串以及解析用户按钮操作的过程中，需要添加很多的字符串处理结构以及事件触发处理结构，这对于我强化软件开发的健壮性很有帮助。</p><p><strong>版本管理</strong>。通过 Git 版本管理与 Github 的云端同步功能，也更能体会到开发留痕与 bug 检测的优势。</p><p>当然美中不足的也有很多，比如单人全栈开发并不利于锻炼团队协作的能力，同时对于程序的架构设计也没有经过深度考虑，仅仅有界面，数据正常交互就戛然而止。希望在未来的算法与开发路上可以走的更远、更坚定。</p><h2 id="仓库地址">仓库地址</h2><p>github: <a href="https://github.com/Explorer-Dong/DataStructureClassDesign">https://github.com/Explorer-Dong/DataStructureClassDesign</a></p><p>gitee: <a href="https://gitee.com/explorer-dong/DataStructureClassDesign">https://gitee.com/explorer-dong/DataStructureClassDesign</a></p>]]></content>
+    <content type="html"><![CDATA[<blockquote><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="script">s</mi><mi mathvariant="script">c</mi><mi mathvariant="script">o</mi><mi mathvariant="script">r</mi><mi mathvariant="script">e</mi><mo>:</mo></mrow><annotation encoding="application/x-tex">\mathscr {score:}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord"><span class="mord mathnormal">score</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span></span></span></span></span></p><ul><li>平时 20%（出勤、作业、实验）</li><li>期中 20%</li><li>期末 60%</li></ul></blockquote><h2 id="数据结构">数据结构</h2><p>完整实现代码：<a href="https://github.com/Explorer-Dong/DataStructure">https://github.com/Explorer-Dong/DataStructure</a></p><h2 id="一、绪论">一、绪论</h2><h3 id="1-1-数据分析-结构存储-算法计算">1.1 数据分析+结构存储+算法计算</h3><h4 id="1-1-1-逻辑结构">1.1.1 逻辑结构</h4><p>对于当前的数据之间的关系进行分析，进而思考应该如何存储，有以下几种逻辑结构：集合、线性结构、树形结构、图结构。</p><h4 id="1-1-2-存储结构">1.1.2 存储结构</h4><p>设计一定的方法存储到程序中，需要思考：存什么？怎么存？</p><ul><li>存什么？数值存储、数据与数据之间关系域的存储</li><li>怎么存？顺序存储、链式存储、树形存储、图存储</li></ul><h4 id="1-1-3-算法实现">1.1.3 算法实现</h4><p>设计算法计算实现</p><h3 id="1-2-数据类型">1.2 数据类型</h3><p>约束：值集 + 运算集</p><p>数据类型 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>(Data&nbsp;Type,&nbsp;简称&nbsp;DT)</mtext></mrow><annotation encoding="application/x-tex">\text{(Data Type, 简称 DT)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">(Data&nbsp;Type,&nbsp;</span><span class="mord cjk_fallback">简称</span><span class="mord">&nbsp;DT)</span></span></span></span></span>：一般编程语言已经实现好了</p><p>抽象数据类型 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Abstract&nbsp;Data&nbsp;Type,&nbsp;简称&nbsp;ADT</mtext></mrow><annotation encoding="application/x-tex">\text{Abstract Data Type, 简称 ADT}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">Abstract&nbsp;Data&nbsp;Type,&nbsp;</span><span class="mord cjk_fallback">简称</span><span class="mord">&nbsp;ADT</span></span></span></span></span>：数据结构 + 算法操作</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218513.png" alt="ADT 的不同视图"></p><h3 id="1-3-算法方法">1.3 算法方法</h3><ol><li><p>正确性</p></li><li><p>健壮性（鲁棒性）：对于不合法、异常的输入也有处理能力</p></li><li><p>可读性</p></li><li><p>可扩展性</p></li><li><p>高效率</p><ol><li><p>空间复杂度</p></li><li><p>时间复杂度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi>O</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(n)=O(f(n))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">))</span></span></span></span>，其中有三种表示时间复杂度的公式</p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O()</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mclose">)</span></span></span></span> <code>upper bound</code>：最坏的时间复杂度</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Ω</mi><mo stretchy="false">(</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega()</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Ω</span><span class="mopen">(</span><span class="mclose">)</span></span></span></span> <code>lower bound</code>：最好的时间复杂度</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Θ</mi><mo stretchy="false">(</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta()</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Θ</span><span class="mopen">(</span><span class="mclose">)</span></span></span></span> <code>average bound</code>：平均时间复杂度</li></ul></li></ol></li></ol><h2 id="二、线性表">二、线性表</h2><h3 id="2-1-线性表的逻辑结构">2.1 线性表的逻辑结构</h3><p>有一个头结点，尾结点，且每一个结点只有一个前驱结点和一个后继结点。</p><h3 id="2-2-线性表的存储结构">2.2 线性表的存储结构</h3><h4 id="2-2-1-顺序存储结构">2.2.1 顺序存储结构</h4><p>存储在一维地址连续的存储单元里</p><p>特点：逻辑位置相邻，物理位置也相邻</p><p>数据结构：一个一个一维数组 + 一个长度变量 n</p><figure class="highlight c++"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br></pre></td><td class="code"><pre><code class="hljs c++"><span class="hljs-keyword">template</span>&lt;<span class="hljs-keyword">class</span> <span class="hljs-title class_">T</span>, <span class="hljs-type">int</span> MaxSize&gt;<br><span class="hljs-keyword">class</span> <span class="hljs-title class_">SeqList</span><br>{<br>    T data[MazSize];<br>    <span class="hljs-type">int</span> length;<br><span class="hljs-keyword">public</span>:<br>    ...<br>}<br></code></pre></td></tr></tbody></table></figure><p>顺序表可以直接存储元素与关系。链表的元素存储也是可以直接实现的，但是关系要通过指针域来实现</p><h4 id="2-2-2-链式存储结构">2.2.2 链式存储结构</h4><ol><li><p>单链表：默认有一个头结点，不存储数据</p></li><li><p>循环链表</p></li><li><p>双向链表</p></li></ol><h3 id="2-3-线性表的操作算法">2.3 线性表的操作算法</h3><h4 id="2-3-1-顺序表的操作算法">2.3.1 顺序表的操作算法</h4><ol><li><p>初始化构造</p></li><li><p>求顺序表长度</p></li><li><p>按位查找</p></li><li><p>按值查找</p></li><li><p>遍历顺序表</p></li><li><p>插入</p></li><li><p>删除</p></li></ol><h4 id="2-3-2-链表的操作算法">2.3.2 链表的操作算法</h4><ol><li><p>单链表初始化构造</p> <figure class="highlight c++"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><code class="hljs c++">head = <span class="hljs-keyword">new</span> Node&lt;T&gt;;<br>head-&gt;next = <span class="hljs-literal">nullptr</span>;<br></code></pre></td></tr></tbody></table></figure><ul><li><p>头插法</p>  <figure class="highlight c++"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><code class="hljs c++">head = <span class="hljs-keyword">new</span> Node&lt;T&gt;;<br>head-&gt;next = <span class="hljs-literal">nullptr</span>;<br></code></pre></td></tr></tbody></table></figure></li><li><p>尾插法</p>  <figure class="highlight c++"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><code class="hljs c++">head = <span class="hljs-keyword">new</span> Node&lt;T&gt;;<br>rear = head;<br></code></pre></td></tr></tbody></table></figure></li></ul></li><li><p>求单链表长度</p></li><li><p>按位查找</p></li><li><p>按值查找</p></li><li><p>遍历单链表</p></li><li><p>插入</p></li><li><p>删除</p></li><li><p>单链表的析构函数</p></li><li><p>其他操作</p></li><li><p>双向链表操作</p><ul><li><p>插入</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218514.png" alt="插入"></p>  <figure class="highlight c++"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br></pre></td><td class="code"><pre><code class="hljs c++"><span class="hljs-comment">// 插入当前结点 s</span><br>s-&gt;prior = p;<br>s-&gt;next = p-&gt;next;<br>p-&gt;next-&gt;prior = s;<br>p-&gt;next = s;<br></code></pre></td></tr></tbody></table></figure></li><li><p>删除</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218515.png" alt="删除"></p>  <figure class="highlight c++"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><code class="hljs c++"><span class="hljs-comment">// 删除当前结点 p</span><br>p-&gt;next-&gt;prior = p-&gt;prior;<br>p-&gt;prior-&gt;next = p-&gt;next;<br></code></pre></td></tr></tbody></table></figure></li></ul></li></ol><h2 id="三、栈和队列">三、栈和队列</h2><h3 id="3-1-栈">3.1 栈</h3><h4 id="3-1-1-栈的基本概念">3.1.1 栈的基本概念</h4><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218516.png" alt="卡特兰数"></p><p>卡特兰数：假设 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span> 表示第 k 个数最后一个出栈的总个数，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(k)=f(k-1)f(n-k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span></p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi>f</mi><mo stretchy="false">(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfrac><msubsup><mi>C</mi><mrow><mn>2</mn><mi>n</mi></mrow><mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">f(n) = \sum_{k=1}^{n} f(k-1) f(n-k)=\frac{1}{n+1} C_{2n}^{n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9535em;vertical-align:-1.3021em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8479em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3021em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0908em;vertical-align:-0.7693em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.7693em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight">n</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span></span></p><h4 id="3-1-2-栈的存储结构">3.1.2 栈的存储结构</h4><p>顺序存储</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218517.png" alt="顺序存储"></p><p>链式存储</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218518.png" alt="链式存储"></p><h4 id="3-1-3-栈的操作算法">3.1.3 栈的操作算法</h4><ol><li>顺序栈的操作</li><li>链栈的操作</li></ol><h4 id="3-1-4-栈的应用">3.1.4 栈的应用</h4><ol><li><p>括号匹配</p></li><li><p>算数表达式求值</p><ul><li><p>中缀表达式求值</p><blockquote><p><strong>双栈</strong>思路，算符优先法</p><ul><li><p>遇到数字，直接入数栈</p></li><li><p>遇到符号</p><ul><li>如果是括号，左括号直接入栈，右括号进行<strong>运算</strong>直到遇到左括号</li><li>如果是算符，在入算符栈之前，需要进行<strong>运算</strong>操作直到算符栈顶元素等级小于当前算符等级</li></ul></li></ul></blockquote></li><li><p>中缀表达式转后缀表达式</p><blockquote><p><strong>算符栈</strong>即可</p><p>后缀先遇到就直接计算的运算符 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> 中缀表达式需要先算的运算符，于是转化思路就是：</p><ul><li>遇到数字，直接构造后缀表达式</li><li>遇到算符<ul><li>如果是括号，左括号直接入栈，右括号进行<strong>后缀表达式构造</strong>直到遇到左括号</li><li>如果是算符，在入算符栈之前，需要进行<strong>后缀表达式构造</strong>操作直到算符栈顶元素等级小于当前算符等级</li></ul></li></ul></blockquote></li><li><p>后缀表达式求值</p><blockquote><p><strong>数栈</strong>即可</p><p>遇到数字直接入数栈，遇到算符直接进行运算</p></blockquote></li></ul></li><li><p>栈与递归</p><p>递归工作栈</p></li></ol><h3 id="3-2-队列">3.2 队列</h3><h4 id="3-2-1-队列的基本概念">3.2.1 队列的基本概念</h4><p>先进先出</p><h4 id="3-2-2-队列的存储结构">3.2.2 队列的存储结构</h4><p>顺序存储</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218519.png" alt="顺序存储 - 1"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218520.png" alt="顺序存储 - 2"></p><p>链式存储</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218521.png" alt="链式存储"></p><h4 id="3-2-3-队列的操作算法">3.2.3 队列的操作算法</h4><ol><li><p>循环队列的操作</p><blockquote><p>循环队列的三个注意点</p><ul><li>解决假溢出：采用循环队列，即在入队的时候不是单纯的指针 +1，而是+1后 % MaxSize</li><li>解决队空队满的冲突（真溢出）：<ol><li>浪费一个元素空间：测试rear+1是否==head，</li><li>设置一个辅助标志变量</li><li>设置一个计数器</li></ol></li></ul></blockquote><ol><li>初始化：头尾全部初始化为0</li><li>入队push</li><li>出队pop</li><li>取队头front</li><li>长度size</li><li>队空empty</li></ol></li><li><p>链队列的操作</p></li></ol><h4 id="3-2-4-队列的应用">3.2.4 队列的应用</h4><ol><li><p>报数问题：报到 0 的出队，报到 1 的重新入队，求解出队顺序</p></li><li><p>迷宫最短路问题：开一个记忆数组 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">d[i][j]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">d</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">]</span></span></span></span> 表示从起点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span> 到终点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo separator="true">,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i,j)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">)</span></span></span></span> 点的最短路径的长度。可以将求最短路看做一个<strong>波心扩散</strong>的物理场景，队列中的每一个点都可以作为一个波心，从而实现“两点之间线段最短”的物理场景</p><ul><li>为什么用队列：逐层搜索，每次搜素到的点就是当前点可以搜索到的最短的点，先搜到的点先扩展，于是就是队列的数据结构</li><li>为什么最短：对于每一个点探索到的点都是最短的点，最终的搜索出来的路径就是最短的路径</li></ul></li></ol><h2 id="四、串">四、串</h2><h3 id="4-1-串的基本概念">4.1 串的基本概念</h3><p>由字符组成的串：子串、主串、位置</p><h3 id="4-2-串的存储结构">4.2 串的存储结构</h3><h4 id="4-2-1-串的顺序存储">4.2.1 串的顺序存储</h4><p>使用固定长度的数组来存储，3种存储字符串长度的方法如下：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218522.png" alt="存储字符串长度的方法 - 1"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218523.png" alt="存储字符串长度的方法 - 2"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218524.png" alt="存储字符串长度的方法 - 3"></p><h4 id="4-2-2-串的链式存储">4.2.2 串的链式存储</h4><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>存储密度</mtext><mo>=</mo><mfrac><mtext>串值所占的内存</mtext><mtext>一个结点的总内存</mtext></mfrac></mrow><annotation encoding="application/x-tex">\text{存储密度} = \frac {\text{串值所占的内存}}{\text{一个结点的总内存}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord cjk_fallback">存储密度</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0463em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">一个结点的总内存</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">串值所占的内存</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><p>非压缩形式：一个结点存一个字符</p><figure class="highlight c++"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br></pre></td><td class="code"><pre><code class="hljs c++"><span class="hljs-comment">// 存储密度为：1/9 （64位操作系统）</span><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">String</span> {<br>    <span class="hljs-type">char</span> data;<br>    String* next;<br>};<br></code></pre></td></tr></tbody></table></figure><p>压缩形式（块链）：一个结点存储指定长度的字符</p><figure class="highlight c++"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br></pre></td><td class="code"><pre><code class="hljs c++"><span class="hljs-comment">// 存储密度为：4/12 （64位操作系统）</span><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> MaxSize = <span class="hljs-number">4</span>;<br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">String</span> {<br>    <span class="hljs-type">char</span> data[MaxSize];<br>    String* next;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="4-3-串的操作算法">4.3 串的操作算法</h3><h4 id="4-3-1-串的基本操作算法">4.3.1 串的基本操作算法</h4><p>串连接、串比较、串拷贝</p><h4 id="4-3-2-串的模式匹配">4.3.2 串的模式匹配</h4><p>BF算法（Brute - Force）</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218525.png" alt="BF算法（Brute - Force）"></p><figure class="highlight c++"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br></pre></td><td class="code"><pre><code class="hljs c++"><span class="hljs-comment">// 返回匹配上的所有位置下标（下标从0开始）</span><br><span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">BF</span><span class="hljs-params">(string&amp; s, string&amp; t)</span> </span>{<br>vector&lt;<span class="hljs-type">int</span>&gt; res;<br><span class="hljs-type">int</span> i = <span class="hljs-number">0</span>, j = <span class="hljs-number">0</span>, n = s.<span class="hljs-built_in">size</span>(), m = t.<span class="hljs-built_in">size</span>();<br>    <br><span class="hljs-keyword">while</span> (i &lt; n &amp;&amp; j &lt; m) {<br><span class="hljs-keyword">if</span> (s[i] == t[j]) i++, j++;<br><span class="hljs-keyword">else</span> i = i - j + <span class="hljs-number">1</span>, j = <span class="hljs-number">0</span>;<br>            <br><span class="hljs-keyword">if</span> (j == m) {<br>res.<span class="hljs-built_in">emplace_back</span>(i - j);<br>j = <span class="hljs-number">0</span>;<br>}<br>}<br><br>    <span class="hljs-keyword">return</span> res;<br>}<br></code></pre></td></tr></tbody></table></figure><p>KMP算法</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218526.png" alt="KMP算法"></p><blockquote><p><strong>优化思想</strong>：</p><p>先看暴力思想，我们需要每次将模式串 t 后移一位重新进行比较，其中浪费了已匹配的串数据，优化就是从这块已匹配的串数据入手。而已匹配的串数据就是模式串本身的串数据，因为我们可以直接从模式串本身入手。</p><p><strong>初步猜想</strong>：</p><p>根据模式串的性质，构造一个数表 <code>next</code>，存储模式串应该后移的指针数 <code>k</code></p><p><strong>算法实现</strong>：</p><ol><li>递推求 <code>next</code> 数组</li><li>KMP 中 <code>i</code> 指针不回溯，<code>j</code> 回溯到 <code>next[j]</code></li></ol></blockquote><figure class="highlight c++"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br></pre></td><td class="code"><pre><code class="hljs c++"><span class="hljs-comment">// 求 next 数组下标从1开始</span><br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">2</span>, j = <span class="hljs-number">0</span>; i &lt;= m; i++) {<br>    <span class="hljs-keyword">while</span> (j &amp;&amp; t[i] != t[j + <span class="hljs-number">1</span>])<br>        <span class="hljs-comment">// 未匹配上则不断回溯</span><br>        j = ne[j];<br>    <br>    <span class="hljs-keyword">if</span> (t[i] == t[j + <span class="hljs-number">1</span>])<br>        <span class="hljs-comment">// 匹配上了则j指针后移一位</span><br>        j++;<br>    <br>    ne[i] = j;<br>}<br></code></pre></td></tr></tbody></table></figure><figure class="highlight c++"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br></pre></td><td class="code"><pre><code class="hljs c++"><span class="hljs-comment">// KMP 匹配 下标从1开始</span><br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>, j = <span class="hljs-number">0</span>; i &lt;= n; i++) {<br>    <span class="hljs-keyword">while</span> (j &amp;&amp; news[i] != newt[j + <span class="hljs-number">1</span>])<br>        <span class="hljs-comment">// 未匹配上则不断回溯</span><br>        j = ne[j];<br>    <br>    <span class="hljs-keyword">if</span> (news[i] == newt[j + <span class="hljs-number">1</span>])<br>        <span class="hljs-comment">// 匹配上了则j指针后移一位</span><br>        j++;<br><br>    <span class="hljs-keyword">if</span> (j == m) {<br>        <span class="hljs-comment">// 匹配完全，则统计并且回溯</span><br>        cnt++;<br>        j = ne[j];<br>    }<br>}<br></code></pre></td></tr></tbody></table></figure><h2 id="五、数组和特殊矩阵">五、数组和特殊矩阵</h2><h3 id="5-1-数组">5.1 数组</h3><h4 id="5-1-1-数组的基本概念">5.1.1 数组的基本概念</h4><figure class="highlight c++"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><code class="hljs c++"><span class="hljs-keyword">typedef</span> <span class="hljs-type">int</span> arr[m][n];<br><span class="hljs-comment">// 等价于</span><br><span class="hljs-keyword">typedef</span> <span class="hljs-type">int</span> arr1[n];<br><span class="hljs-keyword">typedef</span> arr1 arr2[m];<br></code></pre></td></tr></tbody></table></figure><h4 id="5-1-2-数组的存储结构">5.1.2 数组的存储结构</h4><ol><li>行优先：按行存储</li><li>列优先：按列存储</li></ol><p>可以按照下标的关系，只需要知道第一个元素的地址，通过矩阵的大小关系即可直接计算出 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi><mo>⋯</mo></mrow></msub></mrow><annotation encoding="application/x-tex">a_{ij\cdots}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">ij</span><span class="minner mtight">⋯</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 的地址</p><h3 id="5-2-特殊矩阵的压缩存储">5.2 特殊矩阵的压缩存储</h3><p>对于多个相同的非零元素只分配一个存储空间，对零元素不分配空间</p><h4 id="5-2-1-对称矩阵的压缩存储">5.2.1 对称矩阵的压缩存储</h4><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218527.png" alt="对称矩阵的压缩存储"></p><p>假设现在有一个 n*n 的对称矩阵</p><p>存：行优先存储 <code>data[n * (n + 1) / 2]</code></p><p>取：我们如果要取 <code>data[i][j]</code></p><ul><li><p>对于上三角</p><ul><li><p><code>i &gt;= j</code>：<code>data[i * (i + 1) / 2 + j]</code></p></li><li><p><code>i &lt;  j</code>：<code>data[j * (j + 1) / 2 + i]</code></p></li></ul></li><li><p>对于下三角</p><ul><li><p><code>i &gt;= j</code>：<code>data[i * (i + 1) / 2 + j]</code></p></li><li><p><code>i &lt;  j</code>：<code>data[j * (j + 1) / 2 + i]</code></p></li></ul></li></ul><h4 id="5-2-2-三角矩阵的压缩存储">5.2.2 三角矩阵的压缩存储</h4><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218528.png" alt="三角矩阵的压缩存储"></p><p>假设现在有一个 n*n 的三角矩阵（上三角或下三角为常数c）</p><p>存：行优先存储，常数 c 存储到最后 <code>data[n * (n + 1) / 2 + 1]</code></p><h4 id="5-2-3-对角矩阵的压缩存储">5.2.3 对角矩阵的压缩存储</h4><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218529.png" alt="对角矩阵的压缩存储"></p><p>假设现在有一个 n*n 的对角矩阵（围绕主对角线有数据，其余数据均为0）</p><h4 id="5-2-4-稀疏矩阵的压缩存储">5.2.4 稀疏矩阵的压缩存储</h4><p>假设现在有一个 n*m 的稀疏矩阵（很多零的一个矩阵）</p><ol><li><p>三元组顺序表</p><p>按行存储两个信息，一个是非零元素的数值，还有一个是具体的坐标 <code>(i, j)</code></p></li><li><p>十字链表</p><p>定义两个指针数组，定义两个指针数组，存储行列的头指针即可 <code>vector&lt;CrossNode&lt;T&gt;*&gt; cheads, rheads</code></p></li></ol><h2 id="六、广义表">六、广义表</h2><h3 id="6-1-广义表的概念">6.1 广义表的概念</h3><p>与以往的线性表的区别在于：线性表的元素只能是 DT 或 ADT。而对于广义表，元素还可以是一个广义表，即可递归结构。</p><p>表头、表尾。对于当前序列，第一个元素就是表头，其余元素的集合就是表尾</p><p>特点：层次结构、共享结构、递归结构</p><h3 id="6-2-广义表的存储结构">6.2 广义表的存储结构</h3><h4 id="6-2-1-广义表中结点的结构">6.2.1 广义表中结点的结构</h4><p>采用联合结构体存储结点类型</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218530.png" alt="采用联合结构体存储结点类型"></p><h4 id="6-2-2-广义表的存储结构">6.2.2 广义表的存储结构</h4><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218531.png" alt="广义表的存储结构"></p><h3 id="6-3-广义表的操作算法">6.3 广义表的操作算法</h3><ul><li>直接递归法 - 原子直接操作，子表循环成原子进行操作</li><li>减治法 - 先处理第一个元素（原子：直接操作 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>o</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">or</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">or</span></span></span></span> 子表：递归操作），最后递归操作剩余的元素</li></ul><h4 id="6-3-3-广义表的其他操作算法">6.3.3 广义表的其他操作算法</h4><p>复制广义表、计算广义表的长度、计算广义表的深度、释放广义表的存储空间</p><h2 id="七、树和二叉树">七、树和二叉树</h2><h3 id="7-1-树的概念和性质">7.1 树的概念和性质</h3><h4 id="7-1-1-树的定义">7.1.1 树的定义</h4><h4 id="7-1-2-树的基本术语">7.1.2 树的基本术语</h4><ol><li>结点的度和树的度<ul><li>结点的度：每一个结点孩子结点的数量</li><li>树的度：一棵树中结点度数的最大值</li></ul></li><li>孩子、双亲、兄弟结点</li><li>路径和路径长度</li><li>子孙结点和祖先结点</li><li>结点的层次和树的高度</li><li>有序树和无序树<ul><li>有序树：子集不可以随意交换</li><li>无序树：子集可以随意交换</li></ul></li><li>森林<ul><li>多棵树</li></ul></li></ol><h4 id="7-1-3-树的基本性质">7.1.3 树的基本性质</h4><h3 id="7-2-二叉树的概念和性质">7.2 二叉树的概念和性质</h3><h4 id="7-2-1-二叉树的定义">7.2.1 二叉树的定义</h4><h4 id="7-2-2-二叉树的基本性质">7.2.2 二叉树的基本性质</h4><ol><li><p>根是第一层。第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 层最多有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">2^{i - 1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8247em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span> 个结点</p></li><li><p>树中叶子结点的个数为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>n</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">n_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，度数为2的结点的个数为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>n</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">n_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>。已知树的点数为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>，边数为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span>，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>=</mo><mi>m</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n = m + 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>。而 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>=</mo><msub><mi>n</mi><mn>0</mn></msub><mo>+</mo><msub><mi>n</mi><mn>1</mn></msub><mo>+</mo><msub><mi>n</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">n=n_0+n_1+n_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>=</mo><msub><mi>n</mi><mn>1</mn></msub><mo>+</mo><mn>2</mn><msub><mi>n</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">m=n_1+2n_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7944em;vertical-align:-0.15em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>n</mi><mn>0</mn></msub><mo>+</mo><msub><mi>n</mi><mn>1</mn></msub><mo>+</mo><msub><mi>n</mi><mn>2</mn></msub><mo>=</mo><msub><mi>n</mi><mn>1</mn></msub><mo>+</mo><mn>2</mn><msub><mi>n</mi><mn>2</mn></msub><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n_0+n_1+n_2 = n_1+2n_2 +1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7944em;vertical-align:-0.15em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，则</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>n</mi><mn>0</mn></msub><mo>=</mo><msub><mi>n</mi><mn>2</mn></msub><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n_0=n_2 + 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></p></li><li><p>满二叉树：每一层都是满结点</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218532.png" alt="满二叉树"></p></li><li><p>完全二叉树：对于一个 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 层的二叉树，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>→</mo><mi>k</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">1\to k-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 都是满的，第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 层从左到右连接叶子结点</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218533.png" alt="完全二叉树"></p><p>结点数固定，则完全二叉树的形状唯一</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218534.png" alt="完全二叉树的性质"></p><p>若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 为奇数，且 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo mathvariant="normal">≠</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">i\neq1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel"><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，则左兄弟就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">i-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7429em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></p><p>若 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 为偶数，则右兄弟就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">i+1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7429em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></p></li></ol><h3 id="7-3-二叉树的存储结构">7.3 二叉树的存储结构</h3><h4 id="7-3-1-二叉树的顺序存储结构">7.3.1 二叉树的顺序存储结构</h4><ul><li>对于一般的二叉树，将其转化为完全二叉树进行存储即可</li><li>插入删除操作都不方便</li></ul><h4 id="7-3-2-二叉树的链式存储结构">7.3.2 二叉树的链式存储结构</h4><h3 id="7-4-二叉树的遍历">7.4 二叉树的遍历</h3><h4 id="7-4-1-二叉树遍历的概念">7.4.1 二叉树遍历的概念</h4><h4 id="7-4-2-二叉树遍历算法">7.4.2 二叉树遍历算法</h4><ol><li>先中后遍历<ol><li>递归遍历</li><li>栈式遍历</li></ol></li><li>层序遍历</li></ol><h4 id="7-4-3-二叉树的构造和析构">7.4.3 二叉树的构造和析构</h4><ol><li><p>由含空指针标记的单个遍历序列构造二叉树</p><p>可以从遍历的逻辑进行逆推。在遍历到空指针的时候输出一个编制符号，然后在构造的时候按照遍历序列进行递归构造即可，如图</p><p><strong>先序序列</strong>进行构造：按照遍历的思路来，对于先序序列而言，第一个元素一定是根元素，因此首先根据“当前局面”的第一个元素创建根结点，接着递归创建左子树和右子树即可。注意传递的序列起始下标是引用类型的变量</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218535.png" alt="示例"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218536.png" alt="示例"></p><p><strong>中序序列</strong>进行构造：</p><p>不可以，因为不能确定根节点以及左子树和右子树的部分</p><p><strong>后序序列</strong>进行构造：与上述先序序列进行构建的逻辑一致，只不过有一个小 trick，即我们从后序序列的最后一个元素开始创建，那么得到的第一个元素就是根结点的值，然后首先递归创建右子树，再递归创建左子树即可。同样需要注意的是传递参数时，序列起始下标是引用类型的变量</p><p>与先序序列构造逻辑相同，只是递归的顺序需要调整一下</p></li><li><p>由两个遍历序列构造二叉树</p><ul><li>先+中：构造逻辑与上述带标记的序列构造逻辑几乎一致，只不过区别在于如何进行递归中参数的传递。传递的参数除了先序和中序的字符串，还有当前局面先序序列的起始下标与当前局面中序序列的起始下标，以及以当前序列进行构造时子树的结点个数。很容易就可以找到当前序列的根结点，接着就是利用很简单的下标关系得到上述的三个参数的过程，最后将新得到的三个参数传递给递归函数进行递归构建左右子树即可，当前的根结点是 <code>pre[ipre]</code></li><li>后+中：逻辑与上述一致，只不过当前的根结点是 <code>post[ipost+n-1]</code></li></ul></li><li><p>由顺序结构构造链式结构</p></li><li><p>拷贝构造</p></li><li><p>析构</p></li></ol><h3 id="7-5-二叉树的其他操作算法">7.5 二叉树的其他操作算法</h3><ol><li>计算二叉树的结点数<ul><li>有返回值的递归</li><li>无返回值的递归</li></ul></li><li>计算二叉树的高度<ul><li>有返回值的递归</li><li>无返回值的递归</li></ul></li><li>根据关键值查找结点</li><li>查找结点的父结点</li></ol><h3 id="7-6-线索二叉树">7.6 线索二叉树</h3><h4 id="7-6-1-线索二叉树的概念">7.6.1 线索二叉树的概念</h4><p>将空指针域用前驱 or 后继结点的地址进行覆盖</p><h4 id="7-6-2-线索二叉树的存储结构">7.6.2 线索二叉树的存储结构</h4><p>依旧是链式存储，只不过增加了结点域中的指针类型，分为链接类型Link与线索类型Thread</p><h4 id="7-6-3-线索二叉树的操作算法">7.6.3 线索二叉树的操作算法</h4><p>以中序线索化的二叉树为例，涉及到以下几种算法：</p><ol><li><p>线索化：设置一个全局变量 pre，为了简化思维，我们可以将一个中序遍历的过程想象成一个线性结构。前驱为 pre，当前为 p</p><ul><li>p 的左子树为空，则 p 的前驱为 pre</li><li>pre 的右子树为空，则 pre 的后继为 p</li></ul></li><li><p>求后继结点和前驱结点</p></li><li><p>遍历</p></li><li><p>求父结点</p><ul><li>首先，若已知当前是左子树，则父结点一定是当前右孩子的中序前驱线索；若已知当前是右子树，则父结点一定是当前左孩子的中序前驱线索</li><li>但是在未知当前结点的位置（未知左右子树）时，同时搜索两边的父结点，然后根据试探出来的父结点，特判父结点的子结点是否是当前结点即可</li></ul></li></ol><h3 id="7-7-树的存储结构与算法">7.7 树的存储结构与算法</h3><h4 id="7-7-1-树的存储结构">7.7.1 树的存储结构</h4><ol><li><p>多叉链表表示法：将每一个结点的子结点都预设置为一个定值（树的最大度数）：浪费空间</p></li><li><p>孩子链表表示法：自顶向下存储边的信息</p> <figure class="highlight c++"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br></pre></td><td class="code"><pre><code class="hljs c++"><span class="hljs-keyword">template</span>&lt;<span class="hljs-keyword">class</span> <span class="hljs-title class_">T</span>&gt;<br>   <span class="hljs-keyword">struct</span> <span class="hljs-title class_">CTBox</span> {<br>       T data;<br>       CTNode* firstchild;<br>   };<br>   <span class="hljs-keyword">struct</span> <span class="hljs-title class_">CTNode</span> {<br>       <span class="hljs-type">int</span> child;<br>       CTNode* next;<br>   };<br></code></pre></td></tr></tbody></table></figure><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218537.png" alt="孩子链表表示法"></p></li><li><p>双亲表示法：自下向上存储边的信息</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218538.png" alt="双亲表示法"></p></li><li><p>孩子兄弟表示法：左结点存储孩子，右结点存储兄弟</p></li></ol><h4 id="7-7-2-树的操作算法">7.7.2 树的操作算法</h4><p>构造、计算树的高度、计算树中所有结点的度</p><h3 id="7-8-哈夫曼树与哈夫曼编码">7.8 哈夫曼树与哈夫曼编码</h3><h4 id="7-8-1-哈夫曼树的定义">7.8.1 哈夫曼树的定义</h4><p>树的路径长度：叶子结点到根结点的路径之和</p><ol><li>树的带权路径长度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>W</mi><mi>P</mi><mi>L</mi></mrow><annotation encoding="application/x-tex">WPL</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">W</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal">L</span></span></span></span> ：叶子结点到根结点的路径之和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>×</mo></mrow><annotation encoding="application/x-tex">\times</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">×</span></span></span></span> 叶子结点的权重，整体之和</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>W</mi><mi>P</mi><mi>L</mi></mrow><annotation encoding="application/x-tex">WPL</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">W</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal">L</span></span></span></span> 最小的树就叫做哈夫曼树：对于一个结点序列n，每次选择其中的两个权值最小的两个结点进行合并，在进行了n-1次以后，得到的二叉树就是哈夫曼树</li><li>哈夫曼编码：<ul><li>编码：利用二叉树进行前缀编码 - 避免解码时的二义性</li><li>解码：根据编码的二叉 trie 树，进行解码</li></ul></li></ol><h4 id="7-8-2-操作算法">7.8.2 操作算法</h4><p>构造 Huffman 树、编码、解码</p><h2 id="八、图">八、图</h2><h3 id="8-1-图的基本概念">8.1 图的基本概念</h3><h4 id="8-1-1-图的定义">8.1.1 图的定义</h4><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>G</mi><mi>r</mi><mi>a</mi><mi>p</mi><mi>h</mi><mo>=</mo><mo stretchy="false">(</mo><mi>V</mi><mo separator="true">,</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Graph = (V,E)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">G</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">a</span><span class="mord mathnormal">p</span><span class="mord mathnormal">h</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">E</span><span class="mclose">)</span></span></span></span></span></p><p>完全无向图：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi><mi>d</mi><mi>g</mi><mi>e</mi><mo>=</mo><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">edge=n(n-1)/2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">e</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord mathnormal">e</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mord">/2</span></span></span></span></p><p>完全有向图：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi><mi>d</mi><mi>g</mi><mi>e</mi><mo>=</mo><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">edge=n(n-1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">e</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord mathnormal">e</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></p><h4 id="8-1-2-图的基本术语">8.1.2 图的基本术语</h4><ul><li><p>带权图称为网</p></li><li><p>无向图：连通图和连通分量</p><ul><li>连通图：每一个顶点之间都有路径可达</li><li>连通分量：极大连通子图</li></ul></li><li><p>有向图：强连通图和强连通分量</p><ul><li>强连通图：每一个顶点之间都有路径可达</li></ul></li><li><p>强连通分量：极大强连通子图</p></li></ul><h3 id="8-2-图的存储结构">8.2 图的存储结构</h3><p>教材中的点编号统一从 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 开始</p><h4 id="8-2-1-邻接矩阵">8.2.1 邻接矩阵</h4><p>无向图的度：第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 行（列）的非标记数的个数</p><p>有向图的度：入度为第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 行的非标记数的个数；出度为第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 列的非标记数的个数</p><p>类定义：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218539.png" alt="邻接矩阵 - 1"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218540.png" alt="邻接矩阵 - 2"></p><h4 id="8-2-2-邻接表">8.2.2 邻接表</h4><p>存储出边表称为邻接表，存储入编表称为逆邻接表</p><p>类定义：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218541.png" alt="邻接表"></p><h3 id="8-3-图的遍历">8.3 图的遍历</h3><h4 id="8-3-1-图遍历的概念">8.3.1 图遍历的概念</h4><p>每个结点只能访问一次，故需要开启标记数组用来记录是否访问的情况</p><h4 id="8-3-2-深度优先搜索">8.3.2 深度优先搜索</h4><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218542.png" alt="深度优先搜索"></p><p>邻接矩阵：</p><ul><li><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></p></li><li><p>针对<strong>邻接矩阵</strong>的一个<strong>无向</strong>连通图的搜索代码示例</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218543.png" alt="邻接矩阵的一个无向连通图的搜索代码示例"></p></li></ul><p>邻接表：</p><ul><li><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n+e)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">e</span><span class="mclose">)</span></span></span></span></p></li><li><p>针对<strong>邻接表</strong>的一个<strong>无向</strong>连通图的搜索代码示例</p>  <figure class="highlight c++"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br></pre></td><td class="code"><pre><code class="hljs c++"><span class="hljs-function"><span class="hljs-keyword">template</span>&lt;<span class="hljs-keyword">class</span> T&gt;</span><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">ALGraph::DFS</span><span class="hljs-params">(<span class="hljs-type">int</span> v, <span class="hljs-type">bool</span>* visited)</span> </span>{<br>    cout &lt;&lt; vexs[v];<br>    visited[v] = <span class="hljs-literal">true</span>;<br>    <span class="hljs-comment">// 遍历所有的边</span><br>}<br></code></pre></td></tr></tbody></table></figure></li></ul><h4 id="8-3-3-广度优先搜索">8.3.3 广度优先搜索</h4><p>通过队列实现、时间复杂度与上述 DFS 算法类似</p><h4 id="8-3-4-图遍历算法的应用">8.3.4 图遍历算法的应用</h4><ol><li><p>求 <code>(u,v)</code> 的所有简单路径：dfs + 回溯法的简单应用</p></li><li><p>染色法求二部图：bfs 的简单应用。当然 dfs 也是可以的，只需要在染色之后判断是否有相同颜色的邻接点即可</p></li></ol><h3 id="8-4-最小生成树">8.4 最小生成树</h3><h4 id="8-4-1-最小生成树的概念及其性质">8.4.1 最小生成树的概念及其性质</h4><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mi>i</mi><mi>n</mi><mi>i</mi><mi>m</mi><mi>u</mi><mi>m</mi><mtext>&nbsp;</mtext><mi>S</mi><mi>p</mi><mi>a</mi><mi>n</mi><mi>n</mi><mi>i</mi><mi>n</mi><mi>g</mi><mtext>&nbsp;</mtext><mi>T</mi><mi>r</mi><mi>e</mi><mi>e</mi><mo stretchy="false">(</mo><mi>M</mi><mi>S</mi><mi>T</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Minimum\ Spanning\ Tree(MST)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mord mathnormal">inim</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mspace">&nbsp;</span><span class="mord mathnormal">Sp</span><span class="mord mathnormal">annin</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mord mathnormal">ree</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.13889em;">MST</span><span class="mclose">)</span></span></span></span></p><p>证明：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218544.png" alt="最小生成树性质证明 - 图例"></p><p>对于上述的一个割，选择其中权值最小的交叉边。从而对于所有的状态，每次选择最小交叉边即可。</p><h4 id="8-4-2-Prim算法">8.4.2 Prim算法</h4><p>算法标签：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mi>r</mi><mi>e</mi><mi>e</mi><mi>d</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">greedy</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord mathnormal">ree</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span></p><ul><li><p>构造 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 个割的状态</p></li><li><p>起始状态为：顶点集合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span></span></span></span> 含 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 个顶点，顶点集合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>−</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">V-U</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span></span></span></span> 含 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 个顶点</p></li><li><p>状态转移为：</p><ul><li>选完最小交叉边之后，将这条边在集合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>−</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">V-U</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span></span></span></span> 中的顶点加入到最小生成树集合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span></span></span></span> 中</li><li>更新最小交叉边数组 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mi>i</mi><mi>e</mi><mi>d</mi><mi>g</mi><mi>e</mi><mi>s</mi><mo stretchy="false">[</mo><mtext>&nbsp;</mtext><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">miniedges[\ ]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">mini</span><span class="mord mathnormal">e</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord mathnormal">es</span><span class="mopen">[</span><span class="mspace">&nbsp;</span><span class="mclose">]</span></span></span></span></li></ul></li><li><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></p></li></ul><h4 id="8-4-3-Kruskal算法">8.4.3 Kruskal算法</h4><p>算法流标签：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mi>r</mi><mi>e</mi><mi>e</mi><mi>d</mi><mi>y</mi><mo separator="true">,</mo><mi>d</mi><mi>s</mi><mi>u</mi></mrow><annotation encoding="application/x-tex">greedy,dsu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord mathnormal">ree</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">s</span><span class="mord mathnormal">u</span></span></span></span></p><ul><li>初始化 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个顶点作为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个连通分量</li><li>按照边的权值升序进行选择<ul><li>如果选出的边的两个顶点不在同一个集合，则加入最小生成树</li><li>如果选出的边的两个顶点在同一个集合，则不选择（如果选了就会使得生成树形成回路）</li></ul></li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>e</mi><mi>log</mi><mo>⁡</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(e\log e)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">e</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">e</span><span class="mclose">)</span></span></span></span></li></ul><h3 id="8-5-最短路径">8.5 最短路径</h3><h4 id="8-5-1-最短路径的概念">8.5.1 最短路径的概念</h4><p>单源最短路</p><blockquote><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mi>i</mi><mi>j</mi><mi>k</mi><mi>s</mi><mi>t</mi><mi>r</mi><mi>a</mi></mrow><annotation encoding="application/x-tex">Dijkstra</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.03148em;">ijk</span><span class="mord mathnormal">s</span><span class="mord mathnormal">t</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">a</span></span></span></span> 算法无法求解含负边权的单源最短路</p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mi>e</mi><mi>l</mi><mi>l</mi><mi>m</mi><mi>a</mi><mi>n</mi><mo>−</mo><mi>F</mi><mi>o</mi><mi>r</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">Bellman-Ford</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathnormal">e</span><span class="mord mathnormal" style="margin-right:0.01968em;">ll</span><span class="mord mathnormal">man</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mord mathnormal" style="margin-right:0.02778em;">or</span><span class="mord mathnormal">d</span></span></span></span> 算法支持负边权的单源最短路求解</p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mi>p</mi><mi>f</mi><mi>a</mi></mrow><annotation encoding="application/x-tex">Spfa</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Sp</span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mord mathnormal">a</span></span></span></span> 算法同样支持负边权的单元最短路，属于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><mi>e</mi><mi>l</mi><mi>l</mi><mi>m</mi><mi>a</mi><mi>n</mi><mo>−</mo><mi>F</mi><mi>o</mi><mi>r</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">Bellman-Ford</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord mathnormal">e</span><span class="mord mathnormal" style="margin-right:0.01968em;">ll</span><span class="mord mathnormal">man</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mord mathnormal" style="margin-right:0.02778em;">or</span><span class="mord mathnormal">d</span></span></span></span> 算法的优化</p></blockquote><p>多源最短路</p><blockquote><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mi>l</mi><mi>o</mi><mi>y</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">Floyd</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">Fl</span><span class="mord mathnormal" style="margin-right:0.03588em;">oy</span><span class="mord mathnormal">d</span></span></span></span> 适用于求解含负边权的多源最短路</p></blockquote><h4 id="8-5-2-单源最短路径-Dijkstra-算法">8.5.2 单源最短路径 - <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mi>i</mi><mi>j</mi><mi>k</mi><mi>s</mi><mi>t</mi><mi>r</mi><mi>a</mi></mrow><annotation encoding="application/x-tex">Dijkstra</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.03148em;">ijk</span><span class="mord mathnormal">s</span><span class="mord mathnormal">t</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">a</span></span></span></span> 算法</h4><p>算法标签：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>g</mi><mi>r</mi><mi>e</mi><mi>e</mi><mi>d</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">greedy</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mord mathnormal">ree</span><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>  <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mi>p</mi></mrow><annotation encoding="application/x-tex">dp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">p</span></span></span></span></p><blockquote><p>其实就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mi>r</mi><mi>i</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">Prim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">im</span></span></span></span> 的另一种应用</p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mi>r</mi><mi>i</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">Prim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">im</span></span></span></span> 是只存储交叉边的最小值</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mi>i</mi><mi>j</mi><mi>k</mi><mi>s</mi><mi>t</mi><mi>r</mi><mi>a</mi></mrow><annotation encoding="application/x-tex">Dijkstra</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal" style="margin-right:0.03148em;">ijk</span><span class="mord mathnormal">s</span><span class="mord mathnormal">t</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">a</span></span></span></span> 是存储交叉边的最小值 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>+</mo></mrow><annotation encoding="application/x-tex">+</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">+</span></span></span></span> 这条边在集合 S 中的点已经记录的值</li></ul></blockquote><ol><li><p>朴素版：</p><ul><li><p>邻接矩阵</p></li><li><p>定义 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">d[i]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">d</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span></span></span></span> 表示从起点到当前i号点的最短路径的长度</p></li><li><p>将顶点分为两个集合，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mo>−</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">V-S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span>，其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span> 表示已经更新了最短路径长度的顶点集合</p></li><li><p>迭代更新过程：依次更新每一个结点，对于当前结点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">v_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，在集合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span> 中的所有结点中，选择其中到当前结点路径最短的顶点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">v_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>，则 <code>d[i]=d[j]+edges[j][i]</code></p></li><li><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></p></li></ul></li><li><p>堆优化：</p><ul><li><p>邻接表</p></li><li><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>e</mi><mi>log</mi><mo>⁡</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(e \log e)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">e</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">e</span><span class="mclose">)</span></span></span></span></p></li></ul></li></ol><h4 id="8-5-3-多源最短路径-Floyd-算法">8.5.3 多源最短路径 - <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mi>l</mi><mi>o</mi><mi>y</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">Floyd</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">Fl</span><span class="mord mathnormal" style="margin-right:0.03588em;">oy</span><span class="mord mathnormal">d</span></span></span></span> 算法</h4><p>算法标签：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi><mi>p</mi></mrow><annotation encoding="application/x-tex">dp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">p</span></span></span></span></p><blockquote><p>多阶段决策共 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个阶段，<code>dp[i][j]</code> 表示每一个阶段 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span>，从 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> 的选择前 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个顶点后的最短路径的长度</p></blockquote><p>对于当前阶段 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span>，我们利用阶段 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 的状态进行转移更新，其实就是对于新增加的顶点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">v_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是否选择的过程</p><ul><li>选择 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">v_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，则 <code>dp[i][j] = dp[i][k] + dp[k][j]</code></li><li>不选 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>v</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">v_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，则 <code>dp[i][j]</code> 就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 状态下的 <code>dp[i][j]</code></li></ul><h3 id="8-6-AOV网与拓扑排序">8.6 AOV网与拓扑排序</h3><h4 id="8-6-1-有向无环图与AOV网的概念">8.6.1 有向无环图与AOV网的概念</h4><ul><li>有向无环图：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi><mi>A</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">DAG</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mord mathnormal">A</span><span class="mord mathnormal">G</span></span></span></span></li><li>AOV网： <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>y</mi><mtext>&nbsp;</mtext><mi>o</mi><mi>n</mi><mtext>&nbsp;</mtext><mi>v</mi><mi>e</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>x</mi><mtext>&nbsp;</mtext><mi>n</mi><mi>e</mi><mi>t</mi><mi>w</mi><mi>o</mi><mi>r</mi><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(activity\ on\ vertex\ network)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">a</span><span class="mord mathnormal">c</span><span class="mord mathnormal">t</span><span class="mord mathnormal">i</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mord mathnormal">i</span><span class="mord mathnormal">t</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace">&nbsp;</span><span class="mord mathnormal">o</span><span class="mord mathnormal">n</span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span><span class="mord mathnormal">t</span><span class="mord mathnormal">e</span><span class="mord mathnormal">x</span><span class="mspace">&nbsp;</span><span class="mord mathnormal">n</span><span class="mord mathnormal">e</span><span class="mord mathnormal" style="margin-right:0.02691em;">tw</span><span class="mord mathnormal" style="margin-right:0.02778em;">or</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span></li><li>应用场景：在时间先后上有约束关系的工程管理问题</li></ul><h4 id="8-6-2-拓扑排序">8.6.2 拓扑排序</h4><ul><li>定义：顶点线性化</li><li>应用：判环、判断一个图是否可以进行动态规划</li><li>算法设计：对于有向图，从所有的入度为 0 的点开始删点删边，到最后判断有多少点被删除即可</li><li>算法实现：可以采用 dfs 进行缩点删边，也可以采用 bfs 进行缩点删边</li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n+e)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">e</span><span class="mclose">)</span></span></span></span></li></ul><h2 id="九、查找">九、查找</h2><h3 id="9-1-静态查找表">9.1 静态查找表</h3><p>定义：只支持查询和修改，不支持删除与插入</p><h4 id="9-1-1-顺序查找">9.1.1 顺序查找</h4><h4 id="9-1-2-折半查找">9.1.2 折半查找</h4><h4 id="9-1-3-分块查找">9.1.3 分块查找</h4><p>结合顺序查找与分块查找的一种方法</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218545.png" alt="分块查找"></p><ul><li>索引表可以折半或者顺序查找</li><li>块内部只能顺序查找</li></ul><h3 id="9-2-动态查找表">9.2 动态查找表</h3><h4 id="9-2-1-二叉排序树">9.2.1 二叉排序树</h4><p>定义：根结点比左子树所有结点的值都大，比右子树所有结点的值都小。关键字唯一</p><p>操作：查找、插入、删除</p><p>判定：想要判定一棵二叉树是否为二叉排序树，只需要判断中序遍历的结果是不是递增的即可，可以采取中序遍历序列比对的方法，也可以在递归遍历二叉树的过程中通过记录前驱结点的值直接进行比较判断。时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p><h4 id="9-2-2-平衡二叉树">9.2.2 平衡二叉树</h4><p>定义：平衡因子为左子树的高度 - 右子树的高度，平衡二叉树的平衡因子绝对值 &lt;= 1</p><p>构建：当插入结点进行构建时出现了有结点平衡因子的绝对值超过了1，则进行“旋转”调整，旋转共分为4种</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218546.png" alt="旋转 - LL、LR"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218547.png" alt="旋转 - LR"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218548.png" alt="旋转 - RL"></p><p>尝试模拟一遍下列序列的构造过程就可以理解了</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406292218549.png" alt="例题"></p><h3 id="9-3-Hash-查找">9.3 Hash 查找</h3><p>定义：装填因子 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi><mo>=</mo><mfrac><mi>n</mi><mi>m</mi></mfrac></mrow><annotation encoding="application/x-tex">\alpha=\frac{n}{m}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.0037em;">α</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>，其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 表示待填入表中的结点数，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 表示哈希表的空间大小</p><p>哈希函数应该满足以下两点：第一、映射出来的地址不会越界；第二、映射出来的地址是唯一的</p><h4 id="9-3-1-构造表">9.3.1 构造表</h4><p><strong>常用的哈希函数</strong></p><ol><li><p>直接地址法 - 线性函数一对一映射</p><p>优点。计算简单且不可能产生冲突</p><p>缺点。对于空间的要求极高，如果数据过于离散，则会造成很大的空间浪费</p></li><li><p>数字分析法 - 按照数位中的数值分布情况进行哈希</p><p>缺点。需要预先知道数据的数字分布情况</p></li><li><p>平方取中法 - 对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mi>m</mi></msup></mrow><annotation encoding="application/x-tex">10^m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6644em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span></span></span></span></span></span></span></span> 的哈希空间，可以将数字平方后取中间 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 位进行哈希存储</p></li><li><p>折叠法</p><ul><li><p>移位法：将一个数字按照数位拆分为几个部分，然后将几个部分的数值累加出一个数即可，高位抹去不用</p></li><li><p>间隔法：与移位法几乎一致，只不过将其中的部分意义间隔的进行数值反转，最后累计即可，高位抹去不用</p></li></ul></li><li><p>除留余数法 - 按照数值 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>mod&nbsp;</mtext><mi>p</mi></mrow><annotation encoding="application/x-tex">\text{mod}\ p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">mod</span></span><span class="mspace">&nbsp;</span><span class="mord mathnormal">p</span></span></span></span> 后的数值进行哈希，假设哈希表空间大小为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> ，则 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">p</span></span></span></span> 一般取 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≤</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">\le m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 的质数</p></li></ol><p><strong>处理冲突</strong></p><ol><li>开放定址法 - 探测开放地址，一般有三种<ul><li>连续序列进行线性探测</li><li>左右倍增序列进行探测</li><li>伪随机序列进行探测</li><li>双 hash 探测法</li></ul></li><li>拉链法<ul><li>定义：将产生 hash 冲突的元素放入同一个子集，通过单链表进行存储</li><li>优点：没有堆积现象，从而减少了很多不必要的比价，提升比较效率；适合一开始不知道表长的情况；除结点更加容易。</li></ul></li></ol><h4 id="9-3-2-查找表">9.3.2 查找表</h4><p>按照构造相同的逻辑进行查找即可</p><h2 id="十、排序">十、排序</h2><h3 id="10-1-排序的基本概念">10.1 排序的基本概念</h3><p>关键字：</p><ul><li>主关键字：每一个待排序的该关键字是独一无二的</li><li>次关键字：每一个待排序的该关键字可能是重复的</li></ul><p>稳定性：</p><ul><li>场景：只针对次关键字的情况</li><li>稳定：按照次关键字排序后，原来相同关键字的顺序不变</li><li>不稳定：按照次关键字排序后，原来相同关键字的顺序可能会改变</li></ul><p>内外排序：</p><ul><li>内排序：数据全部存放在内存</li><li>外排序：数据量过大时，待排序的数据在内存与外存之间不断转换</li></ul><h3 id="10-2-冒泡排序">10.2 冒泡排序</h3><p><strong>稳定的</strong>。基于交换的思路进行</p><h3 id="10-3-选择排序">10.3 选择排序</h3><ul><li><p>选择第 1 小的数放在第一个位置，…，选择第 i 小的数放在第 i 个位置</p></li><li><p>共选择 n-1 次</p></li></ul><h3 id="10-4-插入排序">10.4 插入排序</h3><p><strong>稳定的</strong>。</p><ul><li>直接插入排序：依次向前缀已经排好序的序列中进行插入 - <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></li><li>折半插入排序：同上，只是选择插入位置的使用二分 - <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li><li>递归插入排序：排序 <code>[1,i]</code> 等价于先排好 <code>[1,i-1]</code>，然后插入当前 <code>num[i]</code> 即可</li></ul><h3 id="10-5-希尔排序">10.5 希尔排序</h3><p><strong>不稳定</strong>。</p><p>基于插入直接排序的优点：</p><ol><li>当序列基本有序时，效率很高</li><li>当待排序数很少时，效率很高</li></ol><p>于是希尔（Shell）就得出来以下的希尔排序算法：</p><ol><li>将序列划分一定次数，从 d&lt;n 到 1</li><li>每次划分都对组内的元素进行直接插入排序</li><li>最后分为 1 组时，直接排序一趟以后就可以得到 sortrd sequence</li></ol><h3 id="10-6-快速排序">10.6 快速排序</h3><p><strong>不稳定</strong>。</p><p>分治法三步骤：divide、conquer and combine</p><p>每次选择一个 pivot 进行 partition，递归两个 partition</p><figure class="highlight c++"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br></pre></td><td class="code"><pre><code class="hljs c++"><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">Sort</span><span class="hljs-params">(<span class="hljs-type">int</span> l, <span class="hljs-type">int</span> r)</span> </span>{<br>    <span class="hljs-keyword">if</span> (l &gt;= r) <span class="hljs-keyword">return</span>;<br><br>    <span class="hljs-type">int</span> i = l - <span class="hljs-number">1</span>, j = r + <span class="hljs-number">1</span>, x = a[l + r &gt;&gt; <span class="hljs-number">1</span>];<br>    <span class="hljs-keyword">while</span> (i &lt; j) {<br>        <span class="hljs-keyword">while</span> (a[++i] &lt; x);<br>        <span class="hljs-keyword">while</span> (a[--j] &gt; x);<br>        <span class="hljs-keyword">if</span> (i &lt; j) <span class="hljs-built_in">swap</span>(a[i], a[j]);            <br>    }<br>    <br>    <span class="hljs-built_in">Sort</span>(l, j), <span class="hljs-built_in">Sort</span>(j + <span class="hljs-number">1</span>, r);<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="10-7-堆排序">10.7 堆排序</h3><p><strong>不稳定</strong>。</p><p>堆与堆排序的定义：首先我们得知道什么是堆结构。堆是具有下面性质（对于任意的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">1\le i \le n/2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7955em;vertical-align:-0.136em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mord">/2</span></span></span></span> ）的完全二叉树</p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mi>i</mi></msub><mo>≤</mo><msub><mi>k</mi><mrow><mn>2</mn><mi>i</mi></mrow></msub><mo separator="true">,</mo><msub><mi>k</mi><mi>i</mi></msub><mo>≤</mo><msub><mi>k</mi><mrow><mn>2</mn><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">k_i \le k_{2i},k_i \le k_{2i+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9028em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight">i</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span> 叫做 小顶堆</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>k</mi><mi>i</mi></msub><mo>≥</mo><msub><mi>k</mi><mrow><mn>2</mn><mi>i</mi></mrow></msub><mo separator="true">,</mo><msub><mi>k</mi><mi>i</mi></msub><mo>≥</mo><msub><mi>k</mi><mrow><mn>2</mn><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">k_i \ge k_{2i},k_i \ge k_{2i+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9028em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathnormal mtight">i</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span> 叫做 大顶堆</li><li>因此一个堆结构可以采用线性的单元进行存储与维护。而堆排序利用堆顶是最值这一性质，通过不断的<strong>取堆顶，调整堆</strong>的方式获得最终的排好序的序列</li></ul><p>建立初始堆：由于完全二叉树中，每一个叶子结点都已经是堆结构，因此直接从第一个非叶子结点开始建堆即可。对每一个元素与左孩子、 右孩子进行比较</p><ul><li>如果当前结点的值比左右孩子都大，那么无需修改，当前位置就是堆顶</li><li>如果当前结点的值比左孩子或者右孩子中的最大值小，则将最大的孩子作为堆顶，并将当前值不断的“下沉”即可</li></ul><p>交换堆顶与记录位置后重新建堆：交换记录值获取当前堆中最值以后，需要将除了已记录的值的结点以外的所有结点重新调整为堆结构</p><ul><li>调整为堆结构的过程与上述初始建堆的过程完全一致，只是结点数每次 -1</li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p><h3 id="10-8-归并排序">10.8 归并排序</h3><p><strong>稳定的</strong>。</p><p>递归：同样采用分治法，我们按照分治法的三个步骤进行讨论</p><ul><li>divide：将当前序列划分为左右两部分</li><li>conquer：递归处理上述划分出来的两部分</li><li>combine：归并上述递归完的两部分</li></ul><p>非递归：就是模拟上述递归的过程，可以拆分为三步</p><ul><li>归并</li><li>按照指定的长度处理整个序列</li><li>划分局部排序的长度</li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo><mo>←</mo><mi>T</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mi>T</mi><mo stretchy="false">(</mo><mfrac><mi>n</mi><mn>2</mn></mfrac><mo stretchy="false">)</mo><mo>+</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)\leftarrow T(n)=2T(\frac{n}{2}) + O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">←</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.095em;vertical-align:-0.345em;"></span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mopen">(</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p>]]></content>
     
     
     <categories>
@@ -1115,11 +1115,11 @@
   
   
   <entry>
-    <title>DigitalLogicCircuit</title>
-    <link href="/GPA/3rd-term/DigitalLogicCircuit/"/>
-    <url>/GPA/3rd-term/DigitalLogicCircuit/</url>
+    <title>DataStructureClassDesign</title>
+    <link href="/GPA/3rd-term/DataStructureClassDesign/"/>
+    <url>/GPA/3rd-term/DataStructureClassDesign/</url>
     
-    <content type="html"><![CDATA[<blockquote><p>联系方式：</p><p><span class="github-emoji"><span>✉</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/2709.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span> <a href="mailto:lijuncst@njnu.edu.cn">lijuncst@njnu.edu.cn</a></p><p><span class="github-emoji"><span>☎</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/260e.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span> 13770610040</p></blockquote><h2 id="数字逻辑电路">数字逻辑电路</h2><h2 id="一、数字逻辑概论">一、数字逻辑概论</h2><h3 id="1-1-数字信号与数字电路">1.1 数字信号与数字电路</h3><h4 id="1-1-1-数字技术的发展及其应用">1.1.1 数字技术的发展及其应用</h4><ol><li><p>电流控制器件：电子管、晶体管（二极管、三极管）、半导体集成电路</p></li><li><p>EDA(Electronic Design Automation)技术（硬件设计软件化）：设计：EWB or Verilog、仿真、下载、验证结果</p></li></ol><h4 id="1-1-2-数字集成电路的分类及特点">1.1.2 数字集成电路的分类及特点</h4><ol><li><p>数字集成电路的分类</p><ol><li><p>从结构特点及其对输入信号的响应规则角度：组合逻辑电路、时序逻辑电路</p></li><li><p>从电路的形式角度：集成电路、分立电路</p></li><li><p>从器件的角度：TTL电路、CMOS电路</p></li><li><p>从集成度（每一个芯片所包含的门个数）的角度：小规模、中规模、大规模、超大规模、甚大规模</p></li></ol></li><li><p>数字集成电路的特点</p><ol><li>稳定性高：抗干扰能力强</li><li>易于设计：对0和1表示的信号进行逻辑运算和处理</li><li>便于集成：体积小、通用性好、成本低</li><li>可编程性：可实现硬件设计软件化</li><li>高速度、低功耗</li><li>便于存储、传输、处理</li></ol></li><li><p>数字电路的<strong>分析</strong>、<strong>设计</strong>与测试</p><ol><li>分析方法<ul><li>目标：确定输入与输出</li><li>工具：逻辑代数</li><li>方法：真值表、功能表、逻辑表达式和波形图</li></ul></li><li>设计方法<ul><li>从功能要求出发，选择合适的逻辑器件进行设计</li><li>设计方式：传统的设计方法 or 基于EDA的软件设计方法</li></ul></li><li>测试技术</li></ol></li></ol><h4 id="1-1-3-模拟信号与数字信号">1.1.3 模拟信号与数字信号</h4><ol><li><p>模拟信号：时间和数值均<strong>连续</strong>变化的信号</p></li><li><p>数字信号：时间和数值均<strong>离散</strong>变化的信号</p></li><li><p>模拟量的数字表示：模数转换，即将连续的模拟信号经过<strong>采样与编码</strong>转化为数字信号</p><ul><li>首先对时间离散</li><li>然后对幅值离散</li><li>最后对得到的数字量进行编码</li></ul></li></ol><h4 id="1-1-4-数字信号的描述方法">1.1.4 数字信号的描述方法</h4><ol><li><p>二值数字逻辑和逻辑电平</p><p>二值数字逻辑：0和1两种状态（定量）</p><p>逻辑电平：高电压和低电压（定性）</p><p>正逻辑关系表（负逻辑相反）：</p><table><thead><tr><th style="text-align:center">电压（V）</th><th style="text-align:center">二值逻辑</th><th style="text-align:center">电平</th></tr></thead><tbody><tr><td style="text-align:center">3.5~5</td><td style="text-align:center">1</td><td style="text-align:center">H（高电压）</td></tr><tr><td style="text-align:center">0~1.5</td><td style="text-align:center">0</td><td style="text-align:center">L（低电压）</td></tr></tbody></table></li><li><p>数字波形</p></li><li><p>实际数字信号波形</p></li><li><p>波形图、时序图或定时图</p></li></ol><h3 id="1-2-数制">1.2 数制</h3><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">(</mo><mi>N</mi><msub><mo stretchy="false">)</mo><mi>r</mi></msub><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mo>−</mo><mi mathvariant="normal">∞</mi></mrow><mi mathvariant="normal">∞</mi></munderover><msub><mi>K</mi><mi>i</mi></msub><mtext>&nbsp;</mtext><msup><mi>r</mi><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">(N)_r=\sum_{i=-\infty}^\infty K_i\ r^i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9874em;vertical-align:-1.336em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">−</span><span class="mord mtight">∞</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">∞</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.336em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace">&nbsp;</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8747em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span></span></span></p><h4 id="1-2-1-十进制">1.2.1 十进制</h4><h4 id="1-2-2-二进制">1.2.2 二进制</h4><p>优点：<strong>易于表达</strong>；二进制数字电路<strong>逻辑简单</strong>，所用元件少；基本运算<strong>规则简单</strong>，运算操作方便</p><p>波形表示：应用比如“计数器”</p><p>数据传输：应用比如“串行传输”</p><h4 id="1-2-3-十-二进制之间的转换">1.2.3 十-二进制之间的转换</h4><p>十进制小数转化为二进制：将小数部位不断×2，取整数，直到没有小数部分为止</p><h4 id="1-2-4-十六进制和八进制">1.2.4 十六进制和八进制</h4><p>二进制转十六进制：从右往左每四位换算成十六进制</p><p>二进制转八进制：从右往左每三位换算成八进制</p><h3 id="1-3-二进制数的算术运算">1.3 二进制数的算术运算</h3><h4 id="1-3-1-无符号二进制数的运算">1.3.1 无符号二进制数的运算</h4><h4 id="1-3-2-有符号二进制数的运算">1.3.2 有符号二进制数的运算</h4><p>定义：其实就是多了一个符号位，且不可以省略。其中0表示正数，1表示负数：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right" columnspacing=""><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mo stretchy="false">(</mo><mo>+</mo><mn>11</mn><msub><mo stretchy="false">)</mo><mi>D</mi></msub><mo>=</mo><mo stretchy="false">(</mo><mn>01011</mn><msub><mo stretchy="false">)</mo><mi>B</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>11</mn><msub><mo stretchy="false">)</mo><mi>D</mi></msub><mo>=</mo><mo stretchy="false">(</mo><mn>11011</mn><msub><mo stretchy="false">)</mo><mi>B</mi></msub></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}(+11)_D=(01011)_B \\(-11)_D=(11011)_B \\\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.75em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">+</span><span class="mord">11</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">D</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord">01011</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">−</span><span class="mord">11</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">D</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord">11011</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.25em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>补码、反码和原码：</p><ul><li><p>对于正数，补码反码原码全部一样</p></li><li><p>对于负数，反码为：符号位不动，原码按位取反；补码为：反码最低位+1即可</p></li></ul><p>加法：与十进制竖式计算类似</p><p>减法：与十进制竖式计算类似</p><p>溢出：是因为数值位不够了，解决方法是进行<strong>位扩展</strong></p><p>溢出的判别：</p><ul><li><p>两个正数的求和，得到的补码的最高位如果为 1，则溢出</p></li><li><p>两个负数的求和，得到的补码的最高位如果为 0，则溢出</p></li></ul><h3 id="1-4-二进制代码">1.4 二进制代码</h3><h4 id="1-4-1-二-十进制码">1.4.1 二-十进制码</h4><p>其实就是在表示 0-15 的十六个二进制里面，按照不同的规则选取 10 个二进制数来进行转换</p><ol><li>有权码 - 最接近逻辑的：8421BCD码</li><li>无权码</li></ol><h4 id="1-4-2-格雷码">1.4.2 格雷码</h4><h4 id="1-4-3-ASCII码">1.4.3 ASCII码</h4><h3 id="1-5-二值逻辑变量与基本逻辑运算">1.5 二值逻辑变量与基本逻辑运算</h3><h4 id="1-5-1-常见逻辑符号示例">1.5.1 常见逻辑符号示例</h4><table><thead><tr><th style="text-align:center">运算类型</th><th style="text-align:center">逻辑符号</th><th style="text-align:center">逻辑表达式</th></tr></thead><tbody><tr><td style="text-align:center">与</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245091.png" alt="image-20231008095031818" style="zoom:33%;"></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><mi>A</mi><mi>B</mi></mrow><annotation encoding="application/x-tex">Y=AB</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></td></tr><tr><td style="text-align:center">或</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245093.png" alt="image-20231008095047759" style="zoom:33%;"></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><mi>A</mi><mo>+</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">Y=A+B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></td></tr><tr><td style="text-align:center">非</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245094.png" alt="image-20231008094808523" style="zoom:33%;"></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><mover accent="true"><mi>A</mi><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">Y=\overline{A}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span></span></td></tr><tr><td style="text-align:center">与非</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245095.png" alt="image-20231008095111128" style="zoom:33%;"></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">Y=\overline{AB}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span></span></td></tr><tr><td style="text-align:center">或非</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245096.png" alt="image-20231008095809755" style="zoom:33%;"></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><mover accent="true"><mrow><mi>A</mi><mo>+</mo><mi>B</mi></mrow><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">Y=\overline{A+B}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9667em;vertical-align:-0.0833em;"></span><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.0833em;"><span></span></span></span></span></span></span></span></span></td></tr><tr><td style="text-align:center">异或</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245097.png" alt="image-20231008095902796" style="zoom:33%;"></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><mover accent="true"><mi>A</mi><mo stretchy="true">‾</mo></mover><mi>B</mi><mo>+</mo><mi>A</mi><mover accent="true"><mi>B</mi><mo stretchy="true">‾</mo></mover><mo>=</mo><mi>A</mi><mo>⊕</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">Y=\overline A B+A \overline B=A \oplus B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9667em;vertical-align:-0.0833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8833em;"></span><span class="mord mathnormal">A</span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊕</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></td></tr><tr><td style="text-align:center">同或</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245098.png" alt="image-20231008095922663" style="zoom:33%;"></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><mi>A</mi><mi>B</mi><mo>+</mo><mover accent="true"><mi>A</mi><mo>ˉ</mo></mover><mover accent="true"><mi>B</mi><mo>ˉ</mo></mover><mo>=</mo><mi>A</mi><mo>⊙</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">Y = AB + \bar A \bar B = A\odot B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8201em;"></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8201em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">A</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1111em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8201em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊙</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></td></tr><tr><td style="text-align:center">与或非</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245099.png" alt="image-20231008095947154" style="zoom:33%;"></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Y</mi><mo>=</mo><mover accent="true"><mrow><mi>A</mi><mi>B</mi><mo>+</mo><mi>C</mi><mi>D</mi></mrow><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">Y=\overline{AB+CD}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9667em;vertical-align:-0.0833em;"></span><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.0833em;"><span></span></span></span></span></span></span></span></span></td></tr></tbody></table><h4 id="1-5-2-使用逻辑函数表示实际问题">1.5.2 使用逻辑函数表示实际问题</h4><table><thead><tr><th style="text-align:center">实际问题</th><th style="text-align:center">图片示例</th><th style="text-align:center">变量表示</th><th style="text-align:center">列真值表</th><th style="text-align:center">逻辑函数</th></tr></thead><tbody><tr><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245100.png" alt="image-20231008102102546" style="zoom:25%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245101.png" alt="image-20231008102037276" style="zoom:25%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245102.png" alt="image-20231008101253710" style="zoom:50%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245104.png" alt="image-20231008101303842" style="zoom: 33%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245105.png" alt="image-20231008101318582" style="zoom:33%;"></td></tr></tbody></table><h3 id="1-6-逻辑函数及其表示方法">1.6 逻辑函数及其表示方法</h3><p>描述输入逻辑变量和输出逻辑变量之间的因果关系，称为逻辑函数</p><h4 id="1-6-1-逻辑函数的几种表示方法">1.6.1 逻辑函数的几种表示方法</h4><table><thead><tr><th style="text-align:center">方法</th><th style="text-align:center">示例</th></tr></thead><tbody><tr><td style="text-align:center">真值表</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245106.png" alt="image-20231008110656451" style="zoom: 33%;"></td></tr><tr><td style="text-align:center">逻辑函数表达式</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245107.png" alt="image-20231008110713908" style="zoom:25%;"></td></tr><tr><td style="text-align:center">逻辑图</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245108.png" alt="image-20240116131615935" style="zoom: 25%;"></td></tr><tr><td style="text-align:center">波形图</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245109.png" alt="image-20231008111013474" style="zoom:33%;"></td></tr></tbody></table><h4 id="1-6-2-逻辑函数表示方法之间的转换">1.6.2 逻辑函数表示方法之间的转换</h4><p><strong>真值表到逻辑图的转换</strong></p><ul><li><p>查看真值表</p>  <img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245110.png" alt="image-20231008112605100" style="zoom:33%;"></li><li><p>根据真值表写出逻辑表达式</p>  <img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245111.png" alt="image-20231008112628066" style="zoom: 50%;"></li><li><p>化简（上式不用化简）</p></li><li><p>绘制逻辑图</p>  <img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245112.png" alt="image-20231008112708161" style="zoom:50%;"></li></ul><p><strong>逻辑图到真值表的转换</strong></p><ul><li><p>根据逻辑图逐级写出表达式</p>  <img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245113.png" alt="image-20231008112800422" style="zoom:50%;"></li><li><p>化简</p></li><li><p>代入所有输入变量求真值表</p>  <img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245114.png" alt="image-20231008112839220" style="zoom:50%;"></li></ul><h2 id="二、逻辑代数与硬件描述语言基础">二、逻辑代数与硬件描述语言基础</h2><h3 id="2-1-逻辑代数的基本定律和规则">2.1 逻辑代数的基本定律和规则</h3><h4 id="2-1-1-逻辑代数的基本定律和恒等式">2.1.1 逻辑代数的基本定律和恒等式</h4><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245115.png" alt="image-20231008114849627"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245116.png" alt="image-20231008114943429"></p><h4 id="2-1-2-逻辑代数的基本规则">2.1.2 逻辑代数的基本规则</h4><ol><li><p><strong>代入规则</strong> - 类似于换元</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245117.png" alt="image-20231008115435355"></p></li><li><p><strong>反演规则</strong>（获得反函数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mi>Y</mi><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\overline Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span></span> ）</p><blockquote><p>觉得烦可以直接进行取反运算，简单明了不会错</p></blockquote><ul><li>对于任意一个逻辑表达式L，与门 &amp; 或门取反，变量取反，0 &amp; 1取反</li><li>保持原来的运算优先顺序（即如果在原函数表达式中，AB之间先运算，再和其他变量进行运算，那么非函数的表达式中，仍然是AB之间先运算）</li><li>对于反变量以外的非号应保留不变</li></ul><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245118.png" alt="image-20231008115328932"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245119.png" alt="image-20231008115337125"></p></li><li><p><strong>对偶规则</strong>（获得对偶式 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>L</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">L'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> ）</p><ul><li>对于任何逻辑函数式：与门、或门取反，0、1 取反</li></ul></li></ol><h3 id="2-2-逻辑函数表达式的形式">2.2 逻辑函数表达式的形式</h3><h4 id="2-2-1-逻辑函数表达式的基本形式">2.2.1 逻辑函数表达式的基本形式</h4><ol><li><p>与或表达式：若干个与项相或</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245120.png" alt="image-20231013094928938"></p></li><li><p>或与表达式：若干个或项相与</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245121.png" alt="image-20231013094938441"></p></li></ol><h4 id="2-2-2-最小项与最小项表达式">2.2.2 最小项与最小项表达式</h4><ol><li><p>最小项的定义和性质：n 个变量的最小项一共有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">2^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6644em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> 个</p></li><li><p>最小项表达式：所有的最小项相或</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245122.png" alt="image-20231013100105922"></p></li></ol><h4 id="2-2-3-最大项与最大项表达式">2.2.3 最大项与最大项表达式</h4><ol><li><p>最大项的定义和性质：n 个变量的最大项一共有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">2^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6644em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> 个</p></li><li><p>最大项表达式：所有的最大项相与</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245123.png" alt="image-20240116134303065"></p></li></ol><h4 id="2-2-4-最大项和最小项的关系">2.2.4 最大项和最小项的关系</h4><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>m</mi><mi>i</mi></msub><mo>=</mo><mover accent="true"><msub><mi>M</mi><mi>i</mi></msub><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">m_i=\overline {M_i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0333em;vertical-align:-0.15em;"></span><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p><h3 id="2-3-逻辑函数的代数化简法">2.3 逻辑函数的代数化简法</h3><p>为什么要学化简？因为化简之后可以减少门的使用，从而增强电路可靠性、降低成本</p><h4 id="2-3-1-逻辑函数的最简形式">2.3.1 逻辑函数的最简形式</h4><p>最简与或表达式：包含的与项数最少，且每个与项中变量数最少的与或表达式</p><h4 id="2-3-2-逻辑函数的代数化简法">2.3.2 逻辑函数的代数化简法</h4><ol><li><p>逻辑函数的<strong>化简</strong></p><table><thead><tr><th style="text-align:center">方法</th><th style="text-align:center">逻辑函数</th><th style="text-align:center">证明</th></tr></thead><tbody><tr><td style="text-align:center">并项法</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>+</mo><mover accent="true"><mi>A</mi><mo stretchy="true">‾</mo></mover><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">A+\overline A = 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></td><td style="text-align:center">显然</td></tr><tr><td style="text-align:center">吸收法</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>+</mo><mi>A</mi><mi>B</mi><mo>=</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">A+AB = A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span></td><td style="text-align:center">提取公因子</td></tr><tr><td style="text-align:center">消去法</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>+</mo><mover accent="true"><mi>A</mi><mo stretchy="true">‾</mo></mover><mi>B</mi><mo>=</mo><mi>A</mi><mo>+</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A+\overline A B = A+B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></td><td style="text-align:center">摩根定律使用两次</td></tr><tr><td style="text-align:center">配项法</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>=</mo><mi>A</mi><mo stretchy="false">(</mo><mi>B</mi><mo>+</mo><mover accent="true"><mi>B</mi><mo stretchy="true">‾</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A = A(B + \overline B)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">A</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1333em;vertical-align:-0.25em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></td><td style="text-align:center">显然</td></tr></tbody></table></li><li><p>逻辑函数形式的<strong>变换</strong></p><blockquote><p>使用场景：通常在一片集成电路芯片中只有一种门电路，为了减少门电路的种类，需要对逻辑函数表达式进行变换</p><p>变换方法：常常使用两次取反的套路进行变换</p></blockquote> <img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245124.png" alt="image-20231013112834593" style="zoom:50%;"></li></ol><h3 id="2-4-逻辑函数的卡诺图化简法">2.4 逻辑函数的卡诺图化简法</h3><h4 id="2-4-1-用卡诺图表示逻辑函数">2.4.1 用卡诺图表示逻辑函数</h4><p>首先写出逻辑函数的表达式并且转化为最小项表达式，最后将最小项填入相应的矩阵中即可</p><h4 id="2-4-2-用卡诺图化简逻辑函数">2.4.2 用卡诺图化简逻辑函数</h4><p>尽可能使得圈出来的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>k</mi></msup></mrow><annotation encoding="application/x-tex">2^k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span></span></span> 圈中包含的数尽可能的多，即让 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 尽可能的大。注意：圈中的数全部都得是最小项的数</p><h3 id="2-5-硬件描述语言-Verilog-HDL-基础">2.5 硬件描述语言 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>l</mi><mi>o</mi><mi>g</mi><mtext>&nbsp;</mtext><mi>H</mi><mi>D</mi><mi>L</mi></mrow><annotation encoding="application/x-tex">Verilog\ HDL</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span><span class="mord mathnormal">i</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right:0.02778em;">HD</span><span class="mord mathnormal">L</span></span></span></span> 基础</h3><p>为了从软件代码的角度描述电路，从下面三个方面介绍如何用相关的方式进行描述</p><h4 id="2-5-1-门级描述">2.5.1 门级描述</h4><p>门级元件中，第一个位置是输出变量，之后的都是输入变量，可解释为：多输入门</p><table><thead><tr><th style="text-align:center">门级元件</th><th style="text-align:center">元件符号</th></tr></thead><tbody><tr><td style="text-align:center">与</td><td style="text-align:center">and</td></tr><tr><td style="text-align:center">或</td><td style="text-align:center">or</td></tr><tr><td style="text-align:center">非</td><td style="text-align:center">not</td></tr><tr><td style="text-align:center">与非</td><td style="text-align:center">nand</td></tr><tr><td style="text-align:center">或非</td><td style="text-align:center">nor</td></tr><tr><td style="text-align:center">异或</td><td style="text-align:center">xor</td></tr><tr><td style="text-align:center">同或</td><td style="text-align:center">xnor</td></tr></tbody></table><h4 id="2-5-2-数据流描述">2.5.2 数据流描述</h4><p>简单的概括就是使用相关的位运算进行表述，因为电路逻辑本就是二元逻辑，因此位运算就刚好匹配。在使用数据流进行电路描述时，采用的语句都是连续赋值语句，由 assign 关键词开始，多条 assign 语句是<strong>并行</strong>运行的</p><p>需要注意的是，在连续赋值语句中，被赋值的变量一定是 wire 的<strong>线网</strong>类型的变量，示例如下</p><figure class="highlight verilog"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><code class="hljs verilog"><span class="hljs-comment">// 其中 Y 为 wire 类型的变量</span><br><span class="hljs-keyword">assign</span> Y = (~S &amp; D0) | (S &amp; D1)<br></code></pre></td></tr></tbody></table></figure><h4 id="2-5-3-行为描述">2.5.3 行为描述</h4><p>简单的概括就是使用底层语言进行编程，类似于最开始的 C 语言。使用行为描述语句进行描述时，使用 always 关键字开始变量赋值逻辑，多条 always 语句是<strong>串行</strong>运行的</p><p>需要注意的是，在行为描述语句中，被赋值的变量一定是 reg 等<strong>寄存器</strong>类型的变量，这与上述数据流描述的方式不同，示例如下</p><figure class="highlight verilog"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br></pre></td><td class="code"><pre><code class="hljs verilog"><span class="hljs-comment">// 其中 Y 为 reg 类型的变量</span><br><span class="hljs-keyword">always</span>@(*)<span class="hljs-comment">// * 为敏感变量，对于组合电路而言，所有的输入都是敏感变量</span><br>    <span class="hljs-keyword">if</span> (S) Y = D1;<br><span class="hljs-keyword">else</span>   Y = D0;<br></code></pre></td></tr></tbody></table></figure><h2 id="三、逻辑门电路">三、逻辑门电路</h2><h3 id="3-1-逻辑门电路简介">3.1 逻辑门电路简介</h3><p>MOS 管含有 NMOS 管和 PMOS 管，NMOS 管与 PMOS 管的组合称为互补 MOS，或称为 CMOS 电路</p><h3 id="3-2-基本CMOS逻辑门电路">3.2 基本CMOS逻辑门电路</h3><p>附上启蒙的博客：<a href="https://zhuanlan.zhihu.com/p/258852620">MOS管及简单CMOS逻辑门电路原理图解析！</a></p><table><thead><tr><th style="text-align:center">器件</th><th style="text-align:center">电路</th></tr></thead><tbody><tr><td style="text-align:center">开关</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245125.png" alt="image-20231201090156879" style="zoom:50%;"></td></tr><tr><td style="text-align:center">反相器（非门）</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245126.png" alt="image-20231201090250860" style="zoom:33%;"></td></tr><tr><td style="text-align:center">与非门</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245127.png" alt="image-20231201090233084" style="zoom: 33%;"></td></tr><tr><td style="text-align:center">或非门</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245128.png" alt="image-20231201090319423" style="zoom:33%;"></td></tr><tr><td style="text-align:center">传输门（开关）</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245129.png" alt="image-20231222104533397" style="zoom:33%;"></td></tr><tr><td style="text-align:center">与门</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245130.png" alt="image-20240117003028351" style="zoom:33%;"></td></tr><tr><td style="text-align:center">或门</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245131.png" alt="image-20240117003058029" style="zoom:33%;"></td></tr></tbody></table><p><strong>应用示例</strong></p><p>解读的逻辑其实很简单，在理解之前，应该首先观看上面给出的连接中的 MOS 电路的简化版，从而理解电路的正确结构！即，每一个 MOS 管都理解为一个开关，何时闭合与断开完全取决于相应的 MOS 管的种类与电平，如果是 NMOS 管，即箭头指向左边的，为高电平导通，PMOS 管则相反，只需要知道此电路基本逻辑，那么接下来的分析结果就是水到渠成的事</p><p>需要知道一个理念就是，两个电路如果是并联的存在，那么逻辑表达式就是或，简称为<strong>并联相或</strong>；对应的，两个电路如果是串联的存在，那么逻辑表达式就是与，简称为<strong>串联相与</strong>。最后需要补充一点的就是关于取反的辨识，我们知道一个反相器 MOS 管的逻辑是非常简单的，就是一个 NMOS 管和一个 PMOS 管的组合，那么只需要在分析多个线路是串联还是并联的关系之后，最后经过一个反相器就是一个<strong>取反</strong>逻辑</p><table><thead><tr><th style="text-align:center">电路</th><th style="text-align:center">逻辑表达式</th><th style="text-align:center">功能描述</th></tr></thead><tbody><tr><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245132.png" alt="image-20231201090643503" style="zoom:50%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245133.png" alt="image-20231201090727291" style="zoom:50%;"></td><td style="text-align:center">异或门</td></tr><tr><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245134.png" alt="image-20240117103214449"></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi><mo>=</mo><mover accent="true"><mrow><mo stretchy="false">(</mo><mi>B</mi><mi>C</mi><mo>+</mo><mi>D</mi><mo stretchy="false">)</mo><mi>A</mi></mrow><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">L=\overline{(BC+D)A}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.25em;"></span><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07153em;">BC</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="mclose">)</span><span class="mord mathnormal">A</span></span></span><span style="top:-3.87em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.25em;"><span></span></span></span></span></span></span></span></span></td><td style="text-align:center"></td></tr><tr><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245135.png" alt="image-20240117103238963"></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi><mo>=</mo><mover accent="true"><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>+</mo><mi>B</mi><mo stretchy="false">)</mo><mi>X</mi></mrow><mo stretchy="true">‾</mo></mover><mo>=</mo><mover accent="true"><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>+</mo><mi>B</mi><mo stretchy="false">)</mo><mover accent="true"><mrow><mi>A</mi><mi>B</mi></mrow><mo stretchy="true">‾</mo></mover></mrow><mo stretchy="true">‾</mo></mover><mo>=</mo><mi>A</mi><mo>⊙</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">L=\overline{(A+B)X}=\overline{(A+B)\overline{AB}}=A\odot B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.25em;"></span><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span></span></span><span style="top:-3.87em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.25em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.3333em;vertical-align:-0.25em;"></span><span class="mord overline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span><span style="top:-4.0033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.25em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊙</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span></td><td style="text-align:center">同或门</td></tr><tr><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245136.png" alt="image-20231201091736402" style="zoom:50%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245137.png" alt="image-20231201091750341" style="zoom:50%;"></td><td style="text-align:center">异或门</td></tr><tr><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245138.png" alt="image-20231201091831389" style="zoom:50%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245139.png" alt="image-20231201091858929" style="zoom:50%;"></td><td style="text-align:center">2选1数据选择器</td></tr></tbody></table><h2 id="四、组合逻辑电路">四、组合逻辑电路</h2><h3 id="4-1-组合逻辑电路的分析">4.1 组合逻辑电路的分析</h3><h4 id="4-1-1-定义">4.1.1 定义</h4><p>只取决于实时输入，从而给出相应的输出，与之前的运行结果无关。<strong>没有反馈</strong>，<strong>没有记忆单元</strong></p><h4 id="4-1-2-分析步骤与方法">4.1.2 分析步骤与方法</h4><ol><li>由逻辑图得到<strong>逻辑表达式</strong></li><li><strong>化简和变换</strong></li><li>列<strong>真值表</strong></li><li>根据真值表（或者波形图）<strong>分析电路功能</strong></li></ol><h3 id="4-2-组合逻辑电路的设计">4.2 组合逻辑电路的设计</h3><h4 id="4-2-1-设计过程">4.2.1 设计过程</h4><ol><li>明确逻辑含义：确定输入输出并定义逻辑状态的含义</li><li>列出真值表：根据逻辑描述写出真值表</li><li>写出逻辑表达式：由真值表写出逻辑表达式，真值取原、假值取反</li><li>化简逻辑表达式：代数化简法 or 卡诺图化简法</li><li>画出逻辑图：使用相应的门级元件进行组合连接</li></ol><h4 id="4-2-2-优化实现">4.2.2 优化实现</h4><table><thead><tr><th style="text-align:center">电路类型</th><th style="text-align:center">优化策略</th><th style="text-align:center">电路图</th><th style="text-align:center">优化结果</th></tr></thead><tbody><tr><td style="text-align:center">单输出电路</td><td style="text-align:center">统一元件类型</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245140.png" alt="image-20231117112320588" style="zoom: 33%;"></td><td style="text-align:center">见左图文字</td></tr><tr><td style="text-align:center">多输出电路</td><td style="text-align:center">共享相同逻辑项</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245141.png" alt="image-20231117112426358" style="zoom:33%;"></td><td style="text-align:center">见左图文字</td></tr><tr><td style="text-align:center">多级逻辑电路（限定入数）</td><td style="text-align:center">提取公因项</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245142.png" alt="image-20231117112559081" style="zoom:33%;"></td><td style="text-align:center">见左图文字</td></tr><tr><td style="text-align:center">多级逻辑电路（限定入数）</td><td style="text-align:center">提取公因项</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245143.png" alt="image-20231117112618856" style="zoom: 33%;"></td><td style="text-align:center">见左图文字</td></tr></tbody></table><h3 id="4-3-组合逻辑电路中的竞争与冒险">4.3 组合逻辑电路中的竞争与冒险</h3><h4 id="4-3-1-产生的原因">4.3.1 产生的原因</h4><p>门级元件的延时效应</p><h4 id="4-3-2-消去的方法">4.3.2 消去的方法</h4><ol><li><p>消除互补变量</p></li><li><p>增加乘积项，避免互补项相加</p></li><li><p>输出端并联电容器</p> <img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245144.png" alt="image-20231117115217185" style="zoom: 50%;"></li></ol><h3 id="4-4-若干典型的组合逻辑电路">4.4 若干典型的组合逻辑电路</h3><h4 id="4-4-1-编码器">4.4.1 编码器</h4><p>普通编码器：只允许有一个输入，从而进行编码，一旦出现多输入就会发生错误</p><p>优先编码器：无论多少输入，都会按照一开始设定的优先级进行最高等级的那一个信号位的编码</p><table><thead><tr><th style="text-align:center">名称</th><th style="text-align:center">型号</th><th style="text-align:center">逻辑符号</th><th style="text-align:center">功能分析</th><th style="text-align:center">逻辑图</th></tr></thead><tbody><tr><td style="text-align:center">4-2优先编码器</td><td style="text-align:center">74LS00</td><td style="text-align:center">2片7400（4个2输入与非门）实现</td><td style="text-align:center">需要将4-2优先编码器的两个逻辑函数转化为与非式，从而进行电路逻辑的搭建。化简后发现需要7个2输入与非门，故需要2片7400才能实现4-2线优先编码器</td><td style="text-align:center"></td></tr><tr><td style="text-align:center">8-3优先编码器</td><td style="text-align:center">CD4532</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245145.png" alt="image-20231208104642447" style="zoom: 50%;"></td><td style="text-align:center">除了8个输入端与3个输出端，还有EI、EO与GS端。其中<strong>GS</strong>是用来标明当前电路是否处于工作状态的，即如果没有输入端为有效信号，GS就是低电平，反之则是高电平。而<strong>EI与EO</strong>是为了电路扩展而诞生的，当EI为高电平且没有任何输入的情况下，EO也是1，此时的4532就相当于一根导线，从而可以进行片子的扩展</td><td style="text-align:center">由于有现成的集成电路板，故就是逻辑符号</td></tr><tr><td style="text-align:center">16-4优先编码器</td><td style="text-align:center">CD4532</td><td style="text-align:center">2片4532实现</td><td style="text-align:center">首先确保EI始终为高电平。<strong>输出后三位</strong>就是两个4532片子的3输出分别或的结果，<strong>最高位的输出</strong>是高位片的GS端的结果</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245146.png" alt="image-20240117120736794" style="zoom:200%;"></td></tr><tr><td style="text-align:center">32-5优先编码器</td><td style="text-align:center">74LS00+CD4532</td><td style="text-align:center">1片7400+4片4532实现</td><td style="text-align:center">首先确保EI始终为高电平。<strong>输出后三位</strong>就是四个4532片子的3输出分别或的结果，<strong>最高位的两个输出</strong>取决于4个片子GS端4-2优先编码的结果。</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245147.png" alt="image-20240117121434992" style="zoom:200%;"></td></tr></tbody></table><h4 id="4-4-2-译码器-数据分配器">4.4.2 译码器/数据分配器</h4><table><thead><tr><th style="text-align:center">名称</th><th style="text-align:center">型号</th><th style="text-align:center">逻辑符号</th><th style="text-align:center">功能分析</th><th style="text-align:center">逻辑图</th></tr></thead><tbody><tr><td style="text-align:center">2-4译码器</td><td style="text-align:center">74X139</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245148.png" alt="image-20231208101820876" style="zoom: 33%;"></td><td style="text-align:center">使能端有效时。按照对应的输出给出相应输出的低电平信号</td><td style="text-align:center">由于有现成的集成电路板，故就是逻辑符号</td></tr><tr><td style="text-align:center">3-8译码器</td><td style="text-align:center">74X138</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245149.png" alt="image-20231208101658175" style="zoom:25%;"></td><td style="text-align:center">使能端有效时。按照对应的输出给出相应输出的低电平信号</td><td style="text-align:center">由于有现成的集成电路板，故就是逻辑符号</td></tr><tr><td style="text-align:center">4-16译码器</td><td style="text-align:center">74X138或74X139</td><td style="text-align:center">2片74X138或5片74X139</td><td style="text-align:center">使能端有效时。<strong>输入的前三位</strong>分别接入两片3-8译码器的输入端，<strong>输入的最后一位</strong>接入两片3-8译码器的高电平使能端即可；如果用2-4译码器来实现，<strong>输入的前两位</strong>分别接入四片2-4译码器的输入端，<strong>输入的后两位</strong>通过一个2-4译码器的四个输入分别接入4片2-4译码器的低电平使能端即可</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245150.png" alt="image-20240117125311036"></td></tr><tr><td style="text-align:center">5-32译码器</td><td style="text-align:center">74X139+74X138</td><td style="text-align:center">1片74X139+4片74X138</td><td style="text-align:center">使能端有效时。<strong>输入的前三位</strong>分别接入四片3-8译码器的输入端，<strong>输入的后两位</strong>通过2-4译码的4个结果分别接入四片3-8译码器的低电平使能端，从而决定是哪一个3-8译码器在工作</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245151.png" alt="image-20231208104231966" style="zoom: 200%;"></td></tr></tbody></table><p><strong>使用译码器实现逻辑函数</strong></p><p>我们知道译码器的每一个输出代表一个最小项，那么对于一个 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 变量的逻辑函数，可以通过以下步骤用 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>−</mo><msup><mn>2</mn><mi>x</mi></msup></mrow><annotation encoding="application/x-tex">x-2^x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6644em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span></span> 译码器实现任意 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 变量的逻辑函数</p><ol><li>将逻辑函数转化为最小项表达式（大量使用摩根定律）</li><li>转化为译码器的输出（写成 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>∑</mo><msub><mi>m</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\sum m_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 的形式）</li><li>在译码器的输出端加一个多输入与非门即可（对结果进行与非）</li></ol><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245152.png" alt="image-20231208110328594" style="zoom:50%;"><p><strong>数据分配器</strong></p><p>功能：相当于多输出的单刀多掷开关，是将公共数据线上的数据按需要送到不同的通道上去的逻辑电路。</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245153.png" alt="image-20231208114700981"></p><center>图一：示意图</center><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245154.png" alt="image-20231208114749275"></p><center>图二：功能仿真图</center><h4 id="4-4-3-数据选择器">4.4.3 数据选择器</h4><table><thead><tr><th style="text-align:center">名称</th><th style="text-align:center">型号</th><th style="text-align:center">逻辑符号</th><th style="text-align:center">功能分析</th><th style="text-align:center">逻辑图</th></tr></thead><tbody><tr><td style="text-align:center">2选1</td><td style="text-align:center">—</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245155.png" alt="image-20240117135219030" style="zoom:33%;"></td><td style="text-align:center">通过控制端 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span> 来选择 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>D</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">D_0,D_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245156.png" alt="image-20240117135236405" style="zoom:33%;"></td></tr><tr><td style="text-align:center">4选1</td><td style="text-align:center">—</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245157.png" alt="image-20240117135258894" style="zoom:33%;"></td><td style="text-align:center">通过控制端 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>S</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>S</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">S_0,S_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 来选择 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>D</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>D</mi><mn>2</mn></msub><mo separator="true">,</mo><msub><mi>D</mi><mn>3</mn></msub></mrow><annotation encoding="application/x-tex">D_0,D_1,D_2,D_3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245158.png" alt="image-20240117135326247" style="zoom:33%;"></td></tr><tr><td style="text-align:center">8选1</td><td style="text-align:center">74HC151</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245159.png" alt="image-20240117135401170" style="zoom: 25%;"></td><td style="text-align:center">通过控制端 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>S</mi><mn>0</mn></msub><mo>−</mo><msub><mi>S</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">S_0-S_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 来选择 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mn>0</mn></msub><mo>−</mo><msub><mi>D</mi><mn>7</mn></msub></mrow><annotation encoding="application/x-tex">D_0-D_7</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">7</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></td><td style="text-align:center">由于有现成的集成电路板，故就是逻辑符号</td></tr><tr><td style="text-align:center">16选1</td><td style="text-align:center">2片74HC151</td><td style="text-align:center">—</td><td style="text-align:center">通过控制端 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>S</mi><mn>0</mn></msub><mo>−</mo><msub><mi>S</mi><mn>3</mn></msub></mrow><annotation encoding="application/x-tex">S_0-S_3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 来选择 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mn>0</mn></msub><mo>−</mo><msub><mi>D</mi><mn>15</mn></msub></mrow><annotation encoding="application/x-tex">D_0-D_{15}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">15</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，<strong>输入的前三位</strong>连接三个控制端，<strong>输入的最后一位</strong>连接两片74151的使能端，其实就是译码器的魔改版，让输出为相应的译码结果的高电平而已</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245160.png" alt="image-20240117135555070"></td></tr></tbody></table><p><strong>使用数据选择器实现逻辑函数</strong></p><ul><li>变量个数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>&lt;</mo></mrow><annotation encoding="application/x-tex">&lt;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">&lt;</span></span></span></span> 数据选择端个数：变量直接对应数据选择端，多余的选择端置0，最后相应的信号输入端进行赋1或赋0的操作即可</li><li>变量个数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>=</mo></mrow><annotation encoding="application/x-tex">=</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">=</span></span></span></span> 数据选择端个数：本质上就是将逻辑函数转化为最小项表达式，然后与标准与或式进行比对，已出现的最小项与1，未出现的最小项与0，从而配凑产生了数据选择器最开始的式子。落到逻辑图上就是，数据选择端接入函数变量，信号输入端接入相应的高低电平，出现的最小项就输入1，未出现的就输入0即可</li><li>变量个数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>&gt;</mo></mrow><annotation encoding="application/x-tex">&gt;</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mrel">&gt;</span></span></span></span> 数据选择端个数：<ul><li>刚好多 1 个：变量 or 变量的非接入信号输入端</li><li>不止多 1 个：同样采用将变量作为数据信号输入端，此外可能需要借助相关的门电路辅助进行</li></ul></li></ul><h4 id="4-4-4-数值比较器">4.4.4 数值比较器</h4><p>略</p><h4 id="4-4-5-算术运算电路">4.4.5 算术运算电路</h4><p>半加器：即不考虑低位进位的一位二进制加法器。其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span> 为输出位，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> 为进位，没有考虑低位的进位</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>S</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">=</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>A</mi><mo>⊕</mo><mi>B</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>C</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">=</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>A</mi><mi>B</mi></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases}S &amp;=&amp; A \oplus B \\C &amp;=&amp; AB\end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mrel">=</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mrel">=</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊕</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><p>全加器：即考虑低位进位的一位二进制加法器。其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span> 为输出位，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>C</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">C_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 为低位的进位，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>C</mi><mi>o</mi></msub></mrow><annotation encoding="application/x-tex">C_o</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">o</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 为进位</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.36em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>S</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">=</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>A</mi><mo>⊕</mo><mi>B</mi><mo>⊕</mo><msub><mi>C</mi><mi>i</mi></msub></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>C</mi><mi>o</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo lspace="0em" rspace="0em">=</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>A</mi><mi>B</mi><mo>+</mo><mo stretchy="false">(</mo><mi>A</mi><mo>⊕</mo><mi>B</mi><mo stretchy="false">)</mo><msub><mi>C</mi><mi>i</mi></msub><mo>=</mo><mi>A</mi><mi>B</mi><mo>+</mo><mo stretchy="false">(</mo><mi>A</mi><mo>+</mo><mi>B</mi><mo stretchy="false">)</mo><msub><mi>C</mi><mi>i</mi></msub></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\begin{cases}S &amp;=&amp; A \oplus B \oplus C_i \\C_o &amp;=&amp; AB + (A \oplus B)C_i = AB+(A+B)C_i\end{cases}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3em;vertical-align:-1.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">o</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mrel">=</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mrel">=</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.69em;"><span style="top:-3.69em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊕</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊕</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊕</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord mathnormal">A</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.19em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><h3 id="4-5-用-verilog-描述组合逻辑电路">4.5 用 verilog 描述组合逻辑电路</h3><h2 id="五、锁存器和触发器">五、锁存器和触发器</h2><p>本章介绍时序逻辑电路的存储单元，分别为锁存器和触发器。其中锁存器对电平敏感，触发器对边沿敏感</p><h3 id="5-1-基本双稳态电路">5.1 基本双稳态电路</h3><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245162.png" alt="image-20231222110345818" style="zoom:80%;"><h3 id="5-2-SR-锁存器">5.2 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">SR</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">SR</span></span></span></span> 锁存器</h3><table><thead><tr><th style="text-align:center">门级元件组成</th><th style="text-align:center">电路图</th><th style="text-align:center">功能分析</th></tr></thead><tbody><tr><td style="text-align:center">或非门实现</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245163.png" alt="image-20231222100822028" style="zoom:50%;"></td><td style="text-align:center">高电平有效。全0不变，谁1谁有效，都1不确定状态</td></tr><tr><td style="text-align:center">与非门实现</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245164.png" alt="image-20231222100904959" style="zoom:50%;"></td><td style="text-align:center">低电平有效。全1不变，谁0谁有效，都0不确定状态</td></tr></tbody></table><table><thead><tr><th style="text-align:center">应用</th><th style="text-align:center">电路图</th><th style="text-align:center">功能分析</th></tr></thead><tbody><tr><td style="text-align:center">开关电路</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245165.png" alt="image-20231222101855246" style="zoom:50%;"><img src="https://s2.loli.net/2023/12/22/tzFvJp8ulGnRsj5.png" alt="image-20231222102251343" style="zoom:35%;"></td><td style="text-align:center">无论开关如何震动，输出始终正常</td></tr><tr><td style="text-align:center">门控 SR 锁存器</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245166.png" alt="image-20231222103758704" style="zoom:33%;"></td><td style="text-align:center">就是加了一个使能端 E，如果 E 为 1，则就是一个基本的 SR 锁存器，如果 E 为 0，则保持</td></tr></tbody></table><h3 id="5-3-D-锁存器">5.3 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span> 锁存器</h3><table><thead><tr><th style="text-align:center">电路名称</th><th style="text-align:center">逻辑电路图</th><th style="text-align:center">功能分析</th></tr></thead><tbody><tr><td style="text-align:center">传输门控制的D锁存器</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245167.png" alt="image-20231222105139215" style="zoom: 50%;"></td><td style="text-align:center">E=0, Q=不变；E=1, Q=D</td></tr><tr><td style="text-align:center">逻辑门控制的D锁存器</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245168.png" alt="image-20240117191849592" style="zoom:67%;"></td><td style="text-align:center">E=0, Q=不变；E=1, Q=D</td></tr></tbody></table><h3 id="5-4-触发器">5.4 触发器</h3><h4 id="5-4-1-主从-D-触发器的电路结构和工作原理">5.4.1 主从 D 触发器的电路结构和工作原理</h4><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245169.png" alt="image-20231222112319354" style="zoom:80%;"><h4 id="5-4-2-典型的主从-D-触发器集成电路">5.4.2 典型的主从 D 触发器集成电路</h4><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245170.png" alt="image-20231222113334822" style="zoom:50%;"><center>    <img src="https://s2.loli.net/2024/01/17/qzpDMR7BXtQETbr.png" alt="image-20231222113438371" style="zoom:67%;">    <img src="https://s2.loli.net/2024/01/17/GY3msiIEMtXr1kU.png" alt="image-20231222113452256" style="zoom:50%;"></center>### 5.5 触发器的逻辑功能<p>本目需要掌握有关触发器的<strong>特性表</strong>、<strong>特性方程</strong>、<strong>状态图</strong>三者的单独书写以及相互转化的逻辑过程，还需要掌握不同的触发器之间的相互<strong>替换实现</strong></p><table><thead><tr><th style="text-align:center">类型</th><th style="text-align:center">逻辑符号</th><th style="text-align:center">特性表</th><th style="text-align:center">特性方程</th><th style="text-align:center">状态图</th></tr></thead><tbody><tr><td style="text-align:center">D 触发器</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245171.png" alt="image-20240117190140089" style="zoom: 33%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245172.png" alt="image-20240117190521349" style="zoom:25%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245173.png" alt="image-20240117190538232" style="zoom:25%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245174.png" alt="image-20240117190553444" style="zoom:33%;"></td></tr><tr><td style="text-align:center">JK 触发器</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245175.png" alt="image-20240117190400613" style="zoom:33%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245176.png" alt="image-20240117190647444" style="zoom:25%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245177.png" alt="image-20240117190703599" style="zoom:25%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245178.png" alt="image-20240117190716843" style="zoom: 67%;"></td></tr><tr><td style="text-align:center">T 触发器</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245179.png" alt="image-20240117190415482" style="zoom:33%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245180.png" alt="image-20240117190800382" style="zoom:33%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245181.png" alt="image-20240117190818921" style="zoom:25%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245182.png" alt="image-20240117190838344" style="zoom:67%;"></td></tr><tr><td style="text-align:center">T’ 触发器</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245183.png" alt="image-20240117190915612" style="zoom:33%;"></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi><mo>≡</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">T\equiv 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≡</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245184.png" alt="image-20240117190926990" style="zoom:33%;"></td><td style="text-align:center"></td></tr><tr><td style="text-align:center">SR 触发器</td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245185.png" alt="image-20240117190440059" style="zoom:33%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245186.png" alt="image-20240117191017989" style="zoom:25%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245187.png" alt="image-20240117191040611" style="zoom:33%;"></td><td style="text-align:center"><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245188.png" alt="image-20240117191057024" style="zoom:33%;"></td></tr></tbody></table><h2 id="六、时序逻辑电路-star">六、时序逻辑电路<span class="github-emoji"><span>⭐</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/2b50.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span></h2><h3 id="6-1-同步时序逻辑电路的分析">6.1 同步时序逻辑电路的分析</h3><p>本部分只需要掌握同步时序逻辑电路的分析即可，具体直接从例题出发。三道同步时序逻辑电路分析的例题见教材 P282 ~ P286，分别为</p><ul><li>例一：可控二进制计数器</li><li>例二：可控双向二进制计数器</li><li>例三：脉冲分配器</li></ul><p>下面介绍同步时序逻辑电路分析的五个步骤。在分析之前我们要知道我们的最终目标是什么，可以知道，我们分析电路的最终目标是想要量化的确定电路的物理实现的功能，至于如何设计，此处不予讨论。现在给定了一个同步时序逻辑电路的<strong>逻辑电路图</strong>，接下来我们应该：</p><ol><li><p>了解电路组成：同步 or 异步？穆尔型输出（与输入无关） or 米利型输出（与输入有关） or 都有？由什么触发器组成的？触发器类型是上升沿出发 or 下降沿触发？</p></li><li><p>列出三个方程：</p><ul><li><p>输出方程：电路的最终输出</p></li><li><p>激励方程：触发器的输入</p></li><li><p>状态方程：触发器的输出（将触发器的输入也就是激励方程代入触发器的特性方程即可）</p></li></ul></li><li><p>写出转换表（分析功能用）</p></li><li><p>写出状态图（分析功能用）</p></li><li><p>写出时序图（分析功能用）默认状态的初值设置为 0</p></li></ol><h3 id="6-2-典型的时序逻辑电路-计数器">6.2 典型的时序逻辑电路 - 计数器</h3><p>主要讲 N 位二进制计数器中，利用集成电路板 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>74</mn><mi>L</mi><mi>V</mi><mi>C</mi><mn>161</mn></mrow><annotation encoding="application/x-tex">74LVC161</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">74</span><span class="mord mathnormal">L</span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord">161</span></span></span></span> 实现的 <strong>4 位同步二进制递增计数器</strong>。进而引出利用该 4 位计数器<strong>实现模 N 计数器</strong>的分析与设计思路。同时补充 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>74</mn><mi>L</mi><mi>V</mi><mi>C</mi><mn>162</mn></mrow><annotation encoding="application/x-tex">74LVC162</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord">74</span><span class="mord mathnormal">L</span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord">162</span></span></span></span> 实现的 <strong>4 位同步十进制递增计时器</strong>，进而引出相关的模 N 设计思路。下面分析 74LVC161 4 位同步二进制递增计数器集成板</p><center><img src="https://s2.loli.net/2024/01/11/Ivm2g5NrxeCL7Du.png" alt="image-20240111233651905"></center><center>图1 74LVC161集成板</center><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404101245189.png" alt="image-20240111234456513" style="zoom: 50%;"><center>图2 74LVC161逻辑功能表</center><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>C</mi><mi>R</mi></mrow><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\overline{CR}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">CR</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span></span>: <strong>异步清零</strong>。即无视时钟脉冲信号，直接清零</p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mover accent="true"><mrow><mi>P</mi><mi>E</mi></mrow><mo stretchy="true">‾</mo></mover></mrow><annotation encoding="application/x-tex">\overline{PE}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8833em;"></span><span class="mord overline"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8833em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">PE</span></span></span><span style="top:-3.8033em;"><span class="pstrut" style="height:3em;"></span><span class="overline-line" style="border-bottom-width:0.04em;"></span></span></span></span></span></span></span></span></span>: <strong>同步预置</strong>。即当有效始终脉冲沿到来时，实现 4 个预置位的输出，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mn>3</mn></msub><mo separator="true">,</mo><msub><mi>D</mi><mn>2</mn></msub><mo separator="true">,</mo><msub><mi>D</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>D</mi><mn>0</mn></msub></mrow><annotation encoding="application/x-tex">D_3,D_2,D_1,D_0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi><mi>E</mi><mi>P</mi><mo separator="true">,</mo><mi>C</mi><mi>E</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">CEP,CET</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">CEP</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">CET</span></span></span></span>: 使能端。同时为高电平电路才能正常工作</p><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>T</mi><mi>C</mi></mrow><annotation encoding="application/x-tex">TC</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">TC</span></span></span></span>: 进位输出</p><p><strong>本目小结</strong></p><p>多个集成板进行计数</p><p>考虑如何通信：低位进位作为高位使能？</p><p>考虑如何清零：同步（异步）清零？同步（异步）置数？</p><p>学会利用 74LVC161 的“反馈清零法”实现模 N 计数器</p><p>学会利用 74LVC161 的”反馈置数法“实现模 N 计数器</p><p>学会实现 74LVC162 十进制递增功能功能（<strong>同步清零</strong>、<strong>同步置数</strong>）</p><p>学会利用 74LVC162 的同步清零的特性实现模 9 的九进制计数器功能</p><p>学会利用 74LVC162 实现模 24 的二十四进制计数器功能</p><blockquote><p>00-09与10-19的计数：通过低位片的进位端，作为高位片的使能端即可</p><p>20-23与23-00的计数：通过将低位片的两个低位与高位片通过 4023 三输入与非门连接起来，当全为1时，就是计数到23的状态，此时对高低片进行同步清零即可</p></blockquote><h3 id="6-3-用-verilog-描述同步时序逻辑电路">6.3 用 verilog 描述同步时序逻辑电路</h3>]]></content>
+    <content type="html"><![CDATA[<h2 id="数据结构课程设计">数据结构课程设计</h2><h2 id="效果演示">效果演示</h2><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403021802720.gif" alt="sort 窗口程序"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403021802061.gif" alt="hash 窗口程序"></p><blockquote><p>[!note]</p><p>以下为课设报告内容</p></blockquote><h2 id="一、必做题">一、必做题</h2><p>题目：编程实现希尔、快速、堆排序、归并排序算法。要求随机产生 10000 个数据存入磁盘文件，然后读入数据文件，分别采用不同的排序方法进行排序，并将结果存入文件中。</p><h3 id="1-1-数据结构设计与算法思想">1.1 数据结构设计与算法思想</h3><p>本程序涉及到四种排序算法，其中：</p><ul><li>希尔排序：通过倍增方法逐渐扩大选择插入数据量的规模，而达到减小数据比较的次数，时间复杂度优化到近似 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>1.65</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^{1.65})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1.65</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></li><li>快速排序：通过分治的算法策略，不断确定每一个数的最终位置，从而达到近似 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 的时间复杂度</li><li>堆排序：通过堆的树形结构，减小两数的比较次数，最终通过不断维护堆结构来获得有序序列，时间复杂度为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li><li>归并排序：通过分治排序回溯时左右两支有序的特点，进行归并，使算法整体的时间复杂度为固定的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li></ul><h3 id="1-2-程序结构">1.2 程序结构</h3><p>为了更好的了解排序算法内部的机制，我统计了每一种排序算法内部的比较次数,并结合了 Qt 的可视化框架进行编写。将四种排序作为算法内核嵌入 GUI 界面，程序结构如下：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403011712981.png" alt="sort 程序结构图"></p><h4 id="UI">UI</h4><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-comment">// 窗口 继承窗口库组件</span><br>&lt;widget <span class="hljs-keyword">class</span>=<span class="hljs-string">"QWidget"</span> name=<span class="hljs-string">"SortWidget"</span>&gt;<br>    <span class="hljs-comment">// 标题 标签组件</span><br>    &lt;widget <span class="hljs-keyword">class</span>=<span class="hljs-string">"QLabel"</span> name=<span class="hljs-string">"titleLabel"</span>&gt;&lt;/widget&gt;<br>    <span class="hljs-comment">// 输入 网格布局组件</span><br>    &lt;widget <span class="hljs-keyword">class</span>=<span class="hljs-string">"QGridLayout"</span> name=<span class="hljs-string">"inputGridLayout"</span>&gt;&lt;/widget&gt;<br>    <span class="hljs-comment">// 输出 网格布局组件</span><br>    &lt;widget <span class="hljs-keyword">class</span>=<span class="hljs-string">"QGridLayout"</span> name=<span class="hljs-string">"outputGridLayout"</span>&gt;&lt;/widget&gt;  <br>    <span class="hljs-comment">// 交互 水平布局组件</span><br>    &lt;widget <span class="hljs-keyword">class</span>=<span class="hljs-string">"QHBoxLayout"</span> name=<span class="hljs-string">"buttonHLayout"</span>&gt;&lt;/widget&gt;     <br>&lt;/widget&gt;<br></code></pre></td></tr></tbody></table></figure><h4 id="Qt-Application">Qt Application</h4><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">ifndef</span> DATASTRUCTURECLASSDESIGN_SORTWIDGET_H</span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> DATASTRUCTURECLASSDESIGN_SORTWIDGET_H</span><br><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;QWidget&gt;</span></span><br><br><br>QT_BEGIN_NAMESPACE<br><span class="hljs-keyword">namespace</span> Ui {<br>    <span class="hljs-keyword">class</span> <span class="hljs-title class_">SortWidget</span>;<br>}<br>QT_END_NAMESPACE<br><br><span class="hljs-keyword">class</span> <span class="hljs-title class_">SortWidget</span> : <span class="hljs-keyword">public</span> QWidget {<br>Q_OBJECT<br><br><span class="hljs-keyword">private</span>:<br>    Ui::SortWidget* ui;         <span class="hljs-comment">// 窗口对象指针</span><br>    QString path;               <span class="hljs-comment">// 文件存储路径</span><br><br><span class="hljs-keyword">private</span> slots:<br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">pushFolderButton</span><span class="hljs-params">()</span></span>;    <span class="hljs-comment">// 槽函数 - 触发事件：获取存储路径的窗口对话</span><br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">pushCommitButton</span><span class="hljs-params">()</span></span>;    <span class="hljs-comment">// 槽函数 - 触发事件：根据输入数据量执行算法</span><br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">pushCancelButton</span><span class="hljs-params">()</span></span>;    <span class="hljs-comment">// 槽函数 - 触发事件：清空窗口所有标签的数据</span><br><br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-function"><span class="hljs-keyword">explicit</span> <span class="hljs-title">SortWidget</span><span class="hljs-params">(QWidget* parent = <span class="hljs-literal">nullptr</span>)</span></span>; <span class="hljs-comment">// 窗口构造函数</span><br><br>    ~<span class="hljs-built_in">SortWidget</span>() <span class="hljs-keyword">override</span>;                         <span class="hljs-comment">// 窗口析构函数</span><br>};<br></code></pre></td></tr></tbody></table></figure><h4 id="Algorithm">Algorithm</h4><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;vector&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;fstream&gt;</span></span><br><br><span class="hljs-keyword">class</span> <span class="hljs-title class_">sortAlgorithm</span> {<br><span class="hljs-keyword">private</span>:<br>    <span class="hljs-type">int</span> Size, Range;<br>    std::string Path;<br>    std::vector&lt;<span class="hljs-type">int</span>&gt; arr;<br><br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">Generate</span><span class="hljs-params">(<span class="hljs-type">int</span> num, <span class="hljs-type">int</span> range)</span></span>;                 <span class="hljs-comment">// 数据生成</span><br>    <br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">ShellSort</span><span class="hljs-params">(std::vector&lt;<span class="hljs-type">int</span>&gt; a)</span></span>;                 <span class="hljs-comment">// 希尔排序</span><br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">QuickSort</span><span class="hljs-params">(std::vector&lt;<span class="hljs-type">int</span>&gt; a)</span></span>;                 <span class="hljs-comment">// 快速排序</span><br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">HeapSort</span><span class="hljs-params">(std::vector&lt;<span class="hljs-type">int</span>&gt; a)</span></span>;                  <span class="hljs-comment">// 堆排序</span><br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">MergeSort</span><span class="hljs-params">(std::vector&lt;<span class="hljs-type">int</span>&gt; a)</span></span>;                 <span class="hljs-comment">// 归并排序</span><br>    <br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">WriteToFile</span><span class="hljs-params">(std::string path, std::vector&lt;<span class="hljs-type">int</span>&gt;&amp; a)</span></span>;   <span class="hljs-comment">// 写入文件</span><br><br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-built_in">sortAlgorithm</span>(<span class="hljs-type">int</span> _Size, <span class="hljs-type">int</span> _Range, std::string _Path);   <span class="hljs-comment">// 构造函数</span><br><br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">ShellSort</span><span class="hljs-params">()</span></span>;                                   <span class="hljs-comment">// 用户调用希尔排序</span><br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">QuickSort</span><span class="hljs-params">()</span></span>;                                   <span class="hljs-comment">// 用户调用快速排序</span><br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">HeapSort</span><span class="hljs-params">()</span></span>;                                    <span class="hljs-comment">// 用户调用堆排序</span><br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">MergeSort</span><span class="hljs-params">()</span></span>;                                   <span class="hljs-comment">// 用户调用归并排序</span><br>};<br></code></pre></td></tr></tbody></table></figure><h3 id="1-3-实验数据与测试结果分析">1.3 实验数据与测试结果分析</h3><p>原始数据序列生成算法使用时间种子的除留余数法，在数据规模为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10000</mn></mrow><annotation encoding="application/x-tex">10000</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10000</span></span></span></span> 的情况下，四种算法的比较次数如下：</p><table><thead><tr><th style="text-align:center"></th><th style="text-align:center">shell</th><th style="text-align:center">heap</th><th style="text-align:center">quick</th><th style="text-align:center">merge</th></tr></thead><tbody><tr><td style="text-align:center">1</td><td style="text-align:center">129419</td><td style="text-align:center">231824</td><td style="text-align:center">39061</td><td style="text-align:center">120488</td></tr><tr><td style="text-align:center">2</td><td style="text-align:center">128051</td><td style="text-align:center">232204</td><td style="text-align:center">38723</td><td style="text-align:center">120487</td></tr><tr><td style="text-align:center">3</td><td style="text-align:center">134873</td><td style="text-align:center">231945</td><td style="text-align:center">39052</td><td style="text-align:center">120251</td></tr><tr><td style="text-align:center">4</td><td style="text-align:center">135719</td><td style="text-align:center">232096</td><td style="text-align:center">39158</td><td style="text-align:center">120510</td></tr><tr><td style="text-align:center">5</td><td style="text-align:center">128821</td><td style="text-align:center">232051</td><td style="text-align:center">38747</td><td style="text-align:center">120434</td></tr><tr><td style="text-align:center">average</td><td style="text-align:center">131376</td><td style="text-align:center">232024</td><td style="text-align:center">38948</td><td style="text-align:center">120434</td></tr></tbody></table><p>可以发现：在数据规模在  <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mi>e</mi><mn>4</mn></mrow><annotation encoding="application/x-tex">1e4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span><span class="mord mathnormal">e</span><span class="mord">4</span></span></span></span> 的情况下，堆排序的比较次数最多，而快速排序的比较次数最少</p><h3 id="1-4-程序清单">1.4 程序清单</h3><ul><li><code>/Src/Algorithm/sortAlgorithm.cpp</code>：内核排序算法</li><li><code>/Src/Forms/SortWidget.ui</code>：ui 界面</li><li><code>/Src/Headers/SortWidget.h</code>：排序组件逻辑头文件</li><li><code>/Src/Post/SortWidget.cpp</code>：排序组件逻辑源文件</li><li><code>/Src/Test/TestSort.cpp</code>：程序测试文件</li></ul><p>其中，除了 ui 界面使用 Qt Designer 设计师工具进行编写，其余文件均为与文件名同名的 C++ 类，便于维护与扩展</p><h2 id="二、选做题">二、选做题</h2><p>题三：扫描一个 C 源程序，利用两种方法统计该源程序中的关键字出现的频度，并比较各自查找的比较次数。</p><ol><li><p>用 Hash 表存储源程序中出现的关键字，利用 Hash 查找技术统计该程序中的关键字出现的频度。用线性探测法解决 Hash 冲突。设 Hash 函数为 Hash(key) = [(key 的第一个字母序号) × 100 + (key 的最后一个字母序号)] % 41</p></li><li><p>用顺序表存储源程序中出现的关键字，利用二分查找技术统计该程序中的关键字出现的频度。</p></li></ol><h3 id="2-1-数据结构设计与算法思想">2.1 数据结构设计与算法思想</h3><ul><li>第一问：<ul><li><p>首先用关键字序列构造哈希表，哈希函数使用题中所给，容易算出哈希表的最大容量为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>25</mn><mo>×</mo><mn>100</mn><mo>+</mo><mn>40</mn></mrow><annotation encoding="application/x-tex">25 \times 100+40</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">25</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">100</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">40</span></span></span></span> ，我们设置为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3000</mn></mrow><annotation encoding="application/x-tex">3000</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3000</span></span></span></span> ，哈希表数据结构为：<code>int: &lt;string, int&gt;</code> 的键值对形式，可以使用动态数组容器 <code>std::vector&lt;std::pair&gt;</code> 进行存储。其中键设为 <code>int</code> 可以使得对于一个 <code>int</code> 类型的哈希值进行 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 的存储与查找，获得 <code>&lt;word, cnt&gt;</code>，即 <code>&lt;std::string, int&gt;</code></p></li><li><p>然后对于文件中的每一个单词采用哈希搜索。利用哈希函数计算出每一个单词的哈希值，然后通过线性探测法进行搜索比对。关键在于如何解析出一个完整的单词，对于流读入的字符串（不包含空格、换行符、制表符等空白符），我们删除其中的符号后将剩余部分进行合并，之后进行异常处理，排除空串后进行哈希查找。查找次数包含成功和失败的比较次数</p></li></ul></li><li>第二问：将关键字存储于顺序表中，排序后，利用二分查找技术进行搜索统计。查找次数包含成功和失败的比较次数</li></ul><h3 id="2-2-程序结构">2.2 程序结构</h3><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403011712321.png" alt="hash 程序结构图"></p><h4 id="UI-2">UI</h4><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-comment">// 窗口 继承窗口库组件</span><br>&lt;widget <span class="hljs-keyword">class</span>=<span class="hljs-string">"QWidget"</span> name=<span class="hljs-string">"SortWidget"</span>&gt;                         <br>    <span class="hljs-comment">// 标题 标签组件</span><br>    &lt;widget <span class="hljs-keyword">class</span>=<span class="hljs-string">"QLabel"</span> name=<span class="hljs-string">"titleLabel"</span>&gt;&lt;/widget&gt;             <br>    <span class="hljs-comment">// 输入 网格布局组件</span><br>    &lt;widget <span class="hljs-keyword">class</span>=<span class="hljs-string">"QGridLayout"</span> name=<span class="hljs-string">"inputGridLayout"</span>&gt;&lt;/widget&gt;   <br>    <span class="hljs-comment">// 输出 网格布局组件</span><br>    &lt;widget <span class="hljs-keyword">class</span>=<span class="hljs-string">"QGridLayout"</span> name=<span class="hljs-string">"outputGridLayout"</span>&gt;&lt;/widget&gt;<br>    <span class="hljs-comment">// 交互 水平布局组件</span><br>    &lt;widget <span class="hljs-keyword">class</span>=<span class="hljs-string">"QHBoxLayout"</span> name=<span class="hljs-string">"buttonHLayout"</span>&gt;&lt;/widget&gt;     <br>&lt;/widget&gt;<br></code></pre></td></tr></tbody></table></figure><h4 id="Qt-Application-2">Qt Application</h4><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">ifndef</span> DATASTRUCTURECLASSDESIGN_HASHWIDGET_H</span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> DATASTRUCTURECLASSDESIGN_HASHWIDGET_H</span><br><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;QWidget&gt;</span></span><br><br><br>QT_BEGIN_NAMESPACE<br><span class="hljs-keyword">namespace</span> Ui {<br>    <span class="hljs-keyword">class</span> <span class="hljs-title class_">HashWidget</span>;<br>}<br>QT_END_NAMESPACE<br><br><span class="hljs-keyword">class</span> <span class="hljs-title class_">HashWidget</span> : <span class="hljs-keyword">public</span> QWidget {<br>Q_OBJECT<br><br><span class="hljs-keyword">private</span>:<br>    Ui::HashWidget* ui;                       <span class="hljs-comment">// 窗口对象指针</span><br>    QString inputFilePath, outputFolderPath;  <span class="hljs-comment">// 输入输出路径</span><br><br><span class="hljs-keyword">private</span> slots:<br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">pushInputFileButton</span><span class="hljs-params">()</span></span>;     <span class="hljs-comment">// 槽函数 - 触发事件：获取输入文件路径的窗口对话</span><br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">pushOutputFolderButton</span><span class="hljs-params">()</span></span>;  <span class="hljs-comment">// 槽函数 - 触发事件：获取存储文件路径的窗口对话</span><br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">pushCommitButton</span><span class="hljs-params">()</span></span>;        <span class="hljs-comment">// 槽函数 - 触发事件：根据输入的文件开始执行算法</span><br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">pushCancelButton</span><span class="hljs-params">()</span></span>;        <span class="hljs-comment">// 槽函数 - 触发事件：清除当前窗口所有标签的内容</span><br><br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-function"><span class="hljs-keyword">explicit</span> <span class="hljs-title">HashWidget</span><span class="hljs-params">(QWidget* parent = <span class="hljs-literal">nullptr</span>)</span></span>; <span class="hljs-comment">// 窗口构造函数</span><br><br>    ~<span class="hljs-built_in">HashWidget</span>() <span class="hljs-keyword">override</span>;                         <span class="hljs-comment">// 窗口析构函数</span><br><br>};<br><br><span class="hljs-meta">#<span class="hljs-keyword">endif</span> <span class="hljs-comment">//DATASTRUCTURECLASSDESIGN_HASHWIDGET_H</span></span><br></code></pre></td></tr></tbody></table></figure><h4 id="Algorithm-2">Algorithm</h4><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;unordered_map&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;fstream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;vector&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;algorithm&gt;</span></span><br><br><br><span class="hljs-keyword">class</span> <span class="hljs-title class_">hashAlgorithm</span> {<br><span class="hljs-keyword">private</span>:<br>    std::vector&lt;std::string&gt; keywords = {<br>            <span class="hljs-string">"auto"</span>, <span class="hljs-string">"break"</span>, <span class="hljs-string">"case"</span>, <span class="hljs-string">"char"</span>, <span class="hljs-string">"const"</span>,<br>            <span class="hljs-string">"continue"</span>, <span class="hljs-string">"default"</span>, <span class="hljs-string">"do"</span>, <span class="hljs-string">"double"</span>, <span class="hljs-string">"else"</span>,<br>            <span class="hljs-string">"enum"</span>, <span class="hljs-string">"extern"</span>, <span class="hljs-string">"float"</span>, <span class="hljs-string">"for"</span>, <span class="hljs-string">"goto"</span>,<br>            <span class="hljs-string">"if"</span>, <span class="hljs-string">"int"</span>, <span class="hljs-string">"long"</span>, <span class="hljs-string">"register"</span>, <span class="hljs-string">"return"</span>,<br>            <span class="hljs-string">"short"</span>, <span class="hljs-string">"signed"</span>, <span class="hljs-string">"sizeof"</span>, <span class="hljs-string">"static"</span>, <span class="hljs-string">"struct"</span>,<br>            <span class="hljs-string">"switch"</span>, <span class="hljs-string">"typedef"</span>, <span class="hljs-string">"union"</span>, <span class="hljs-string">"unsigned"</span>, <span class="hljs-string">"void"</span>,<br>            <span class="hljs-string">"volatile"</span>, <span class="hljs-string">"while"</span><br>    };<br><br>    std::string inputPath, outputPath;<br>    <span class="hljs-type">int</span> hashSize;<br><br>    <span class="hljs-comment">// std 检验</span><br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">writeToFile</span><span class="hljs-params">(std::unordered_map&lt;std::string, <span class="hljs-type">int</span>&gt;&amp; keywordCount, std::string path)</span></span>;<br><br>    <span class="hljs-comment">// 自定义哈希表检验</span><br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">writeToFile</span><span class="hljs-params">(std::vector&lt;std::pair&lt;std::string, <span class="hljs-type">int</span>&gt;&gt;&amp; keywordCount, std::string path)</span></span>;<br><br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-comment">// 构造函数</span><br>    <span class="hljs-built_in">hashAlgorithm</span>(std::string _inputPath, std::string _outputPath);   <br><br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">stdHash</span><span class="hljs-params">()</span></span>;                 <span class="hljs-comment">// 标准哈希 - 用于测试检验</span><br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">selfHash</span><span class="hljs-params">()</span></span>;                 <span class="hljs-comment">// 自定义哈希</span><br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">binaryHash</span><span class="hljs-params">()</span></span>;               <span class="hljs-comment">// 二分哈希</span><br>};<br></code></pre></td></tr></tbody></table></figure><h3 id="2-3-实验数据与测试结果分析">2.3 实验数据与测试结果分析</h3><p>代码文件从 github 上下载而来，其中 100 行与 500 行的代码文件为完整程序，2500 行代码文件为手动构造程序，三种文件数据规模的查找比较次数如下：</p><table><thead><tr><th style="text-align:center"></th><th style="text-align:center">self hash</th><th style="text-align:center">binary hash</th></tr></thead><tbody><tr><td style="text-align:center">100 lines</td><td style="text-align:center">120</td><td style="text-align:center">775</td></tr><tr><td style="text-align:center">500 lines</td><td style="text-align:center">1546</td><td style="text-align:center">10300</td></tr><tr><td style="text-align:center">2500 lines</td><td style="text-align:center">5561</td><td style="text-align:center">39097</td></tr></tbody></table><p>可以发现：使用线性探测法进行哈希的比较次数远小于二分搜索的比较次数</p><h3 id="2-4-程序清单">2.4 程序清单</h3><ul><li><code>/Src/Algorithm/hashAlgorithm.cpp</code>：内核哈希算法</li><li><code>/Src/Forms/HashWidget.ui</code>：ui 界面</li><li><code>/Src/Headers/HashWidget.h</code>：哈希组件逻辑头文件</li><li><code>/Src/Post/HashWidget.cpp</code>：哈希组件逻辑源文件</li><li><code>/Src/Test/TestHash.cpp</code>：程序测试文件</li></ul><p>其中，除了 ui 界面使用 Qt Designer 设计师工具进行编写，其余文件均为与文件名同名的 C++ 类，便于维护与扩展</p><h2 id="三、收获与体会">三、收获与体会</h2><p><strong>算法思维</strong>。四种排序算法总共也就 100 行，加上堆排序采用树形迭代，快排与归并排序采用递归，整体编码难度并不算高。即使加上了文件 I/O 操作也不过 150 行。而第二道题也是 hash 查找与 binary 查找加上文件 I/O 操作，整体算法难度也十分有限。但是对于我来说也加强了 hash 构建与查找的逻辑。</p><p><strong>前后端开发</strong>。为了锻炼设计模式思维与软件工程思维，我加入了 Qt 集成的 GUI 界面，不仅体会到了算法对于软件的核心作用所在，也体会到了窗口程序的数据交互与数据接口的开发过程。比如 Qt 中信号与槽的概念，其实就是一种数据交互的 API 格式。同时在体验开发过程的同时，也感受到了用户体验的前端板块，通过使用 Qt Designer 的设计师开发界面，可以通过拖动组件的形式进行可视化设计界面，提高了开发效率，如果可以多人组队，也实现了前后端分离的同步开发模式。</p><p><strong>异常处理</strong>。与传统 OJ 不同的是，一个软件还需要考虑更多的异常处理，没有标准的 std 数据，加上用户输入、操作的无穷性，注定了软件的异常处理不是一个简单的事，比如在解析用户输入的字符串以及解析用户按钮操作的过程中，需要添加很多的字符串处理结构以及事件触发处理结构，这对于我强化软件开发的健壮性很有帮助。</p><p><strong>版本管理</strong>。通过 Git 版本管理与 Github 的云端同步功能，也更能体会到开发留痕与 bug 检测的优势。</p><p>当然美中不足的也有很多，比如单人全栈开发并不利于锻炼团队协作的能力，同时对于程序的架构设计也没有经过深度考虑，仅仅有界面，数据正常交互就戛然而止。希望在未来的算法与开发路上可以走的更远、更坚定。</p><h2 id="仓库地址">仓库地址</h2><p>github: <a href="https://github.com/Explorer-Dong/DataStructureClassDesign">https://github.com/Explorer-Dong/DataStructureClassDesign</a></p><p>gitee: <a href="https://gitee.com/explorer-dong/DataStructureClassDesign">https://gitee.com/explorer-dong/DataStructureClassDesign</a></p>]]></content>
     
     
     <categories>
@@ -1199,11 +1199,11 @@
   
   
   <entry>
-    <title>hexo-migrate</title>
-    <link href="/FrontEnd/Hexo/hexo-migrate/"/>
-    <url>/FrontEnd/Hexo/hexo-migrate/</url>
+    <title>hexo-learning-record</title>
+    <link href="/FrontEnd/Hexo/hexo-learning-record/"/>
+    <url>/FrontEnd/Hexo/hexo-learning-record/</url>
     
-    <content type="html"><![CDATA[<h1>Hexo 迁移指南</h1><h2 id="前言">前言</h2><p>为了改变本地项目名称（强迫症），同时为了练习迁移编辑环境，特此记录。</p><h2 id="一、克隆仓库">一、克隆仓库</h2><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">git <span class="hljs-built_in">clone</span> https://github.com/Explorer-Dong/explorer-dong.github.io.git HexoBlog<br></code></pre></td></tr></tbody></table></figure><h2 id="二、安装依赖">二、安装依赖</h2><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">npm install<br></code></pre></td></tr></tbody></table></figure><h2 id="三、编辑推送">三、编辑推送</h2><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><code class="hljs bash">git add .<br>git commit -m <span class="hljs-string">'add: xxx'</span><br>git push -u origin main<br></code></pre></td></tr></tbody></table></figure><h2 id="四、个性化配置">四、个性化配置</h2><p>修改本地的远程仓库名称</p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">git remote rename origin github<br></code></pre></td></tr></tbody></table></figure><p>增加同步 push 备份仓库</p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">git remote set-url --add github https://gitee.com/idwj/idwj.git<br></code></pre></td></tr></tbody></table></figure><p>修改推送分支源</p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">git push -u github main<br></code></pre></td></tr></tbody></table></figure><h2 id="缺点">缺点</h2><p>修改编辑路径以后，会导致所有的文章的<strong>编写</strong>日期刷新为当前时间</p>]]></content>
+    <content type="html"><![CDATA[<h1>Hexo 学习记录</h1><h2 id="前言">前言</h2><p>操作系统：Windows 11 OS</p><p>建站第一枪，使用 Hexo 静态站点生成器搭建自己的博客网站。首先解释一下什么叫静态站点生成器，其实就是将自己写好的 Markdown   文件自动构建为 HTML 文件，然后进行浏览器渲染，而不需要自己编写前端样式文件。在开始介绍 Hexo 建站步骤之前，最好需要掌握一下技能（不掌握也没事，边建边学也行，我就是这样，不过可能会比较浪费时间）：</p><ul><li>git 与 github：进行项目版本管理，并且在后期自动化打包部署上有很大的用</li><li>nodejs 与 npm：可以将其粗略的类比为 python 与 pip 的关系与作用</li><li>html、css、javascrip：大概看得懂会 Ctrl C+V 即可，可以后期修改主题样式</li></ul><p>为了防止某些同学还没开始就结束，先补充一下 nodejs 与 npm 的说明与使用、避坑指南，本地已经有可运行 node 环境的同学可以直接跳到 <strong>#步骤一：创建本地环境</strong>。下面开始介绍：</p><blockquote><p>概念介绍：</p><ul><li><p>nodejs：就是 JavaScript 语言的运行环境，可以将其粗略的类比于 python 解释器（实质上功能远不止解释器），你需要将其下载到本地，推荐长期支持版（LTS）。</p></li><li><p>npm：npm 是 JavaScript 第三方包的管理工具，可以将其粗略的类比于 pip 包管理工具。本篇博客使用的 Hexo 其实就是一个 JavaScrip 第三方包。在下载 nodejs 时，npm 就会自动打包一起下载在同一个目录下。</p></li></ul><p>注意点：</p><ul><li>第一个就是需要将下载解压后的 nodejs 文件夹的权限设置为最高，即所有的权限全部赋予。防止后期出现安装第三方包失败的情况（因为需要对该文件夹中的内容进行读写）</li><li>第二个就是需要设定环境变量，不光设定 npm 的环境变量，还需要设定第三方包的环境变量。前者是为了使用 npm 命令行工具，后者是为了使用第三方包携带的自己的命令行工具，比如此处涉及到的 hexo 命令行工具</li><li>第三个就是需要设定 npm 的代理。因为 node 的第三方库下载地址墙的严重，不设置代理会很慢。具体设定语法自行搜索</li></ul></blockquote><p>下面开始正式介绍基于 hexo 搭建个人博客网站的步骤</p><h2 id="一、创建本地环境">一、创建本地环境</h2><p>就像一个项目一样，一个 hexo 网站也就是一个项目。接下来进入该项目后，打开命令行工具进行操作。目前主流的几乎都可以：bash、cmd、当然由于本地环境时 Windows OS，故对于 Linux、MacOS 的用户就不用说了。</p><ul><li><p>创建 git 仓库进行版本管理</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">git init<br></code></pre></td></tr></tbody></table></figure></li><li><p>安装 JavaScrip 第三方包 Hexo</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">npm install -g hexo-cli<br></code></pre></td></tr></tbody></table></figure></li><li><p>检查是否安装成功</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">hexo -v<br></code></pre></td></tr></tbody></table></figure></li><li><p>初始化一个静态站点并刷新</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><code class="hljs bash">hexo init<br>npm install<br></code></pre></td></tr></tbody></table></figure></li></ul><h2 id="二、本地运行测试">二、本地运行测试</h2><ul><li><p>生成静态文件，可简写为 <code>hexo g -d</code>（就是最终部署的文件，会打包在一个叫 public 的文件夹中）不要少 -d 参数</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">hexo generate -d<br></code></pre></td></tr></tbody></table></figure></li><li><p>启动本地服务，可简写为 <code>hexo s</code></p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">hexo server<br></code></pre></td></tr></tbody></table></figure></li></ul><p>hexo 服务默认启动 4000 端口，现在浏览器访问 localhost:4000 即可看到 hexo 的模板博客网站</p><h2 id="三、手动部署云端">三、手动部署云端</h2><p>现在我们需要将其部署到 [username].github.io 上。一般博客该路径进行部署，如果想要展示其他的网站 demo，一般路径设置为 [username].github.io/[repo]。</p><ul><li><p>安装 hexo 部署插件 hexo-deployer-git，也是 nodejs 第三方包（一定不要漏掉 --save 参数，否则卸载重装该包）</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">npm install -g hexo-deployer-git --save<br></code></pre></td></tr></tbody></table></figure></li><li><p>创建一个 github 仓库，仓库名就设置为 [username].github.io，然后在本地进行远程绑定，即</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">git remote add origin &lt;https <span class="hljs-built_in">link</span> or ssh <span class="hljs-built_in">link</span>&gt;<br></code></pre></td></tr></tbody></table></figure></li><li><p>如果没有执行生成静态文件的命令，需要执行一下（一定不要漏掉 -d 参数，否则重新执行）</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">hexo generate -d<br></code></pre></td></tr></tbody></table></figure></li><li><p>执行部署命令，下面两条均可</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">npm run deploy<br></code></pre></td></tr></tbody></table></figure>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">hexo deploy<br></code></pre></td></tr></tbody></table></figure></li></ul><p>现在访问 <code>https://[username].github.io</code> 如果可以看到与之前本地测试一样的界面的，恭喜你，现在已经可以进行快乐的博客编写并且链接分享进行技术交流~~（装b）~~了</p><h2 id="四、自动部署云端">四、自动部署云端</h2><p>这一步个人认为属于锦上添花的一步。其实进行自动化部署相比手动部署只少了两步，分别为生成静态文件 <code>hexo generate -d</code> 与部署 <code>hexo deploy</code> 的步骤。那么我们为什么还要弄自动化部署呢？愚以为：是为了源文件备份。我们知道，当前仓库里的博客文件格式都是 .html 但是我们博客的源码都是 .md，前者不便于修改，后者十分便于修改。如果本地数据丢失，那么对于之前写的博客就丢失了所有的 md 源码。但是自动化部署的逻辑就可以规避掉这个问题，可以实现 md 文件的云端备份，当然也可以自动化部署啦~。下面介绍一下如何使用 Github Action 进行自动化部署的逻辑：</p><p>其实就是将本地源码修改完后 push 到该仓库的另一个分支下，然后 github 自带的 Github Action 会监听到该仓库的 push 操作，然后进行工作流脚本文件的执行，实现在本地部署的一系列操作。是不是很简单的逻辑！写到这其实可以发现，自动化部署还有一个好处就是：博客编写者只需要熟悉 git 的相关指令与逻辑即可，不需要关心 hexo 的指令，提高了编写的效率。接下来进行详细步骤的讲解：</p><ul><li><p>创建工作流文件。在根目录下创建以下文件关系：</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">touch</span> .github/workflows/deploy.yml<br></code></pre></td></tr></tbody></table></figure></li><li><p>编辑 deploy.yml 文件。模板如下：</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br></pre></td><td class="code"><pre><code class="hljs bash">name: Build and Deploy<br>on: [push]<br><span class="hljs-built_in">jobs</span>:<br>  build-and-deploy:<br>    runs-on: ubuntu-latest<br>    steps:<br>      <span class="hljs-comment"># 相当于 git clone 到服务器</span><br>      - name: Checkout 🛎️<br>        uses: actions/checkout@v3<br>        with:<br>          persist-credentials: <span class="hljs-literal">false</span><br><br>      <span class="hljs-comment"># 安装依赖并生成页面</span><br>      - name: Install and Build 🔧<br>        run: |<br>          npm install<br>          npm run build<br>        <span class="hljs-built_in">env</span>:<br>          CI: <span class="hljs-literal">false</span><br><br>      <span class="hljs-comment"># 部署</span><br>      - name: Deploy 🚀<br>        uses: JamesIves/github-pages-deploy-action@v4<br>        with:<br>          GITHUB_TOKEN: <span class="hljs-variable">${{ secrets.GITHUB_TOKEN }</span>}<br>          BRANCH: main   <span class="hljs-comment"># 修改为你的静态文件分支</span><br>          FOLDER: public <span class="hljs-comment"># 修改为你的静态文件生成存放的文件夹</span><br></code></pre></td></tr></tbody></table></figure></li><li><p>将本地相关文件进行 git 管理后，创建分支（假设叫 auto-deploy）</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">git branch auto-deploy<br></code></pre></td></tr></tbody></table></figure></li></ul><p>push 到仓库，等待 github 自动部署后，重新加载 <code>https://[username].github.io</code> 就可以发现站点内容已经更新了！</p><h2 id="参考">参考</h2><p>搭建：<a href="https://www.bilibili.com/video/BV1dt4y1Q7UE">Luke教你20分钟快速搭建个人博客系列(hexo篇)</a></p><p>个性化定制：<a href="https://blog.csdn.net/yaorongke/article/details/119089190">GitHub Pages + Hexo搭建个人博客网站，史上最全教程</a></p><p>原理：<a href="https://blog.csdn.net/weixin_42596246/article/details/129586384">npm install是什么意思</a></p>]]></content>
     
     
     <categories>
@@ -1220,11 +1220,11 @@
   
   
   <entry>
-    <title>hexo-learning-record</title>
-    <link href="/FrontEnd/Hexo/hexo-learning-record/"/>
-    <url>/FrontEnd/Hexo/hexo-learning-record/</url>
+    <title>hexo-migrate</title>
+    <link href="/FrontEnd/Hexo/hexo-migrate/"/>
+    <url>/FrontEnd/Hexo/hexo-migrate/</url>
     
-    <content type="html"><![CDATA[<h1>Hexo 学习记录</h1><h2 id="前言">前言</h2><p>操作系统：Windows 11 OS</p><p>建站第一枪，使用 Hexo 静态站点生成器搭建自己的博客网站。首先解释一下什么叫静态站点生成器，其实就是将自己写好的 Markdown   文件自动构建为 HTML 文件，然后进行浏览器渲染，而不需要自己编写前端样式文件。在开始介绍 Hexo 建站步骤之前，最好需要掌握一下技能（不掌握也没事，边建边学也行，我就是这样，不过可能会比较浪费时间）：</p><ul><li>git 与 github：进行项目版本管理，并且在后期自动化打包部署上有很大的用</li><li>nodejs 与 npm：可以将其粗略的类比为 python 与 pip 的关系与作用</li><li>html、css、javascrip：大概看得懂会 Ctrl C+V 即可，可以后期修改主题样式</li></ul><p>为了防止某些同学还没开始就结束，先补充一下 nodejs 与 npm 的说明与使用、避坑指南，本地已经有可运行 node 环境的同学可以直接跳到 <strong>#步骤一：创建本地环境</strong>。下面开始介绍：</p><blockquote><p>概念介绍：</p><ul><li><p>nodejs：就是 JavaScript 语言的运行环境，可以将其粗略的类比于 python 解释器（实质上功能远不止解释器），你需要将其下载到本地，推荐长期支持版（LTS）。</p></li><li><p>npm：npm 是 JavaScript 第三方包的管理工具，可以将其粗略的类比于 pip 包管理工具。本篇博客使用的 Hexo 其实就是一个 JavaScrip 第三方包。在下载 nodejs 时，npm 就会自动打包一起下载在同一个目录下。</p></li></ul><p>注意点：</p><ul><li>第一个就是需要将下载解压后的 nodejs 文件夹的权限设置为最高，即所有的权限全部赋予。防止后期出现安装第三方包失败的情况（因为需要对该文件夹中的内容进行读写）</li><li>第二个就是需要设定环境变量，不光设定 npm 的环境变量，还需要设定第三方包的环境变量。前者是为了使用 npm 命令行工具，后者是为了使用第三方包携带的自己的命令行工具，比如此处涉及到的 hexo 命令行工具</li><li>第三个就是需要设定 npm 的代理。因为 node 的第三方库下载地址墙的严重，不设置代理会很慢。具体设定语法自行搜索</li></ul></blockquote><p>下面开始正式介绍基于 hexo 搭建个人博客网站的步骤</p><h2 id="一、创建本地环境">一、创建本地环境</h2><p>就像一个项目一样，一个 hexo 网站也就是一个项目。接下来进入该项目后，打开命令行工具进行操作。目前主流的几乎都可以：bash、cmd、当然由于本地环境时 Windows OS，故对于 Linux、MacOS 的用户就不用说了。</p><ul><li><p>创建 git 仓库进行版本管理</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">git init<br></code></pre></td></tr></tbody></table></figure></li><li><p>安装 JavaScrip 第三方包 Hexo</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">npm install -g hexo-cli<br></code></pre></td></tr></tbody></table></figure></li><li><p>检查是否安装成功</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">hexo -v<br></code></pre></td></tr></tbody></table></figure></li><li><p>初始化一个静态站点并刷新</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><code class="hljs bash">hexo init<br>npm install<br></code></pre></td></tr></tbody></table></figure></li></ul><h2 id="二、本地运行测试">二、本地运行测试</h2><ul><li><p>生成静态文件，可简写为 <code>hexo g -d</code>（就是最终部署的文件，会打包在一个叫 public 的文件夹中）不要少 -d 参数</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">hexo generate -d<br></code></pre></td></tr></tbody></table></figure></li><li><p>启动本地服务，可简写为 <code>hexo s</code></p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">hexo server<br></code></pre></td></tr></tbody></table></figure></li></ul><p>hexo 服务默认启动 4000 端口，现在浏览器访问 localhost:4000 即可看到 hexo 的模板博客网站</p><h2 id="三、手动部署云端">三、手动部署云端</h2><p>现在我们需要将其部署到 [username].github.io 上。一般博客该路径进行部署，如果想要展示其他的网站 demo，一般路径设置为 [username].github.io/[repo]。</p><ul><li><p>安装 hexo 部署插件 hexo-deployer-git，也是 nodejs 第三方包（一定不要漏掉 --save 参数，否则卸载重装该包）</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">npm install -g hexo-deployer-git --save<br></code></pre></td></tr></tbody></table></figure></li><li><p>创建一个 github 仓库，仓库名就设置为 [username].github.io，然后在本地进行远程绑定，即</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">git remote add origin &lt;https <span class="hljs-built_in">link</span> or ssh <span class="hljs-built_in">link</span>&gt;<br></code></pre></td></tr></tbody></table></figure></li><li><p>如果没有执行生成静态文件的命令，需要执行一下（一定不要漏掉 -d 参数，否则重新执行）</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">hexo generate -d<br></code></pre></td></tr></tbody></table></figure></li><li><p>执行部署命令，下面两条均可</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">npm run deploy<br></code></pre></td></tr></tbody></table></figure>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">hexo deploy<br></code></pre></td></tr></tbody></table></figure></li></ul><p>现在访问 <code>https://[username].github.io</code> 如果可以看到与之前本地测试一样的界面的，恭喜你，现在已经可以进行快乐的博客编写并且链接分享进行技术交流~~（装b）~~了</p><h2 id="四、自动部署云端">四、自动部署云端</h2><p>这一步个人认为属于锦上添花的一步。其实进行自动化部署相比手动部署只少了两步，分别为生成静态文件 <code>hexo generate -d</code> 与部署 <code>hexo deploy</code> 的步骤。那么我们为什么还要弄自动化部署呢？愚以为：是为了源文件备份。我们知道，当前仓库里的博客文件格式都是 .html 但是我们博客的源码都是 .md，前者不便于修改，后者十分便于修改。如果本地数据丢失，那么对于之前写的博客就丢失了所有的 md 源码。但是自动化部署的逻辑就可以规避掉这个问题，可以实现 md 文件的云端备份，当然也可以自动化部署啦~。下面介绍一下如何使用 Github Action 进行自动化部署的逻辑：</p><p>其实就是将本地源码修改完后 push 到该仓库的另一个分支下，然后 github 自带的 Github Action 会监听到该仓库的 push 操作，然后进行工作流脚本文件的执行，实现在本地部署的一系列操作。是不是很简单的逻辑！写到这其实可以发现，自动化部署还有一个好处就是：博客编写者只需要熟悉 git 的相关指令与逻辑即可，不需要关心 hexo 的指令，提高了编写的效率。接下来进行详细步骤的讲解：</p><ul><li><p>创建工作流文件。在根目录下创建以下文件关系：</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash"><span class="hljs-built_in">touch</span> .github/workflows/deploy.yml<br></code></pre></td></tr></tbody></table></figure></li><li><p>编辑 deploy.yml 文件。模板如下：</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br></pre></td><td class="code"><pre><code class="hljs bash">name: Build and Deploy<br>on: [push]<br><span class="hljs-built_in">jobs</span>:<br>  build-and-deploy:<br>    runs-on: ubuntu-latest<br>    steps:<br>      <span class="hljs-comment"># 相当于 git clone 到服务器</span><br>      - name: Checkout 🛎️<br>        uses: actions/checkout@v3<br>        with:<br>          persist-credentials: <span class="hljs-literal">false</span><br><br>      <span class="hljs-comment"># 安装依赖并生成页面</span><br>      - name: Install and Build 🔧<br>        run: |<br>          npm install<br>          npm run build<br>        <span class="hljs-built_in">env</span>:<br>          CI: <span class="hljs-literal">false</span><br><br>      <span class="hljs-comment"># 部署</span><br>      - name: Deploy 🚀<br>        uses: JamesIves/github-pages-deploy-action@v4<br>        with:<br>          GITHUB_TOKEN: <span class="hljs-variable">${{ secrets.GITHUB_TOKEN }</span>}<br>          BRANCH: main   <span class="hljs-comment"># 修改为你的静态文件分支</span><br>          FOLDER: public <span class="hljs-comment"># 修改为你的静态文件生成存放的文件夹</span><br></code></pre></td></tr></tbody></table></figure></li><li><p>将本地相关文件进行 git 管理后，创建分支（假设叫 auto-deploy）</p>  <figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">git branch auto-deploy<br></code></pre></td></tr></tbody></table></figure></li></ul><p>push 到仓库，等待 github 自动部署后，重新加载 <code>https://[username].github.io</code> 就可以发现站点内容已经更新了！</p><h2 id="参考">参考</h2><p>搭建：<a href="https://www.bilibili.com/video/BV1dt4y1Q7UE">Luke教你20分钟快速搭建个人博客系列(hexo篇)</a></p><p>个性化定制：<a href="https://blog.csdn.net/yaorongke/article/details/119089190">GitHub Pages + Hexo搭建个人博客网站，史上最全教程</a></p><p>原理：<a href="https://blog.csdn.net/weixin_42596246/article/details/129586384">npm install是什么意思</a></p>]]></content>
+    <content type="html"><![CDATA[<h1>Hexo 迁移指南</h1><h2 id="前言">前言</h2><p>为了改变本地项目名称（强迫症），同时为了练习迁移编辑环境，特此记录。</p><h2 id="一、克隆仓库">一、克隆仓库</h2><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">git <span class="hljs-built_in">clone</span> https://github.com/Explorer-Dong/explorer-dong.github.io.git HexoBlog<br></code></pre></td></tr></tbody></table></figure><h2 id="二、安装依赖">二、安装依赖</h2><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">npm install<br></code></pre></td></tr></tbody></table></figure><h2 id="三、编辑推送">三、编辑推送</h2><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><code class="hljs bash">git add .<br>git commit -m <span class="hljs-string">'add: xxx'</span><br>git push -u origin main<br></code></pre></td></tr></tbody></table></figure><h2 id="四、个性化配置">四、个性化配置</h2><p>修改本地的远程仓库名称</p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">git remote rename origin github<br></code></pre></td></tr></tbody></table></figure><p>增加同步 push 备份仓库</p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">git remote set-url --add github https://gitee.com/idwj/idwj.git<br></code></pre></td></tr></tbody></table></figure><p>修改推送分支源</p><figure class="highlight bash"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs bash">git push -u github main<br></code></pre></td></tr></tbody></table></figure><h2 id="缺点">缺点</h2><p>修改编辑路径以后，会导致所有的文章的<strong>编写</strong>日期刷新为当前时间</p>]]></content>
     
     
     <categories>
@@ -1302,18 +1302,18 @@
   
   
   <entry>
-    <title>self-config</title>
-    <link href="/DevTools/DevCpp/devc-self-config/"/>
-    <url>/DevTools/DevCpp/devc-self-config/</url>
+    <title>solve-clion-decoding-error</title>
+    <link href="/DevTools/CLion/solve-clion-decoding-error/"/>
+    <url>/DevTools/CLion/solve-clion-decoding-error/</url>
     
-    <content type="html"><![CDATA[<h1>DevCpp 偏好配置</h1><h2 id="前言">前言</h2><p>为了在比赛中不因为编辑器而拖后腿，写一个配置说明。编辑器为 <code>Dev-C++ 5.11</code></p><h2 id="一、编译标准">一、编译标准</h2><p>进入以下目录：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403210937178.png" alt="进入以下目录"></p><p>添加以下编译命令：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403210937916.png" alt="添加以下编译命令"></p><h2 id="二、环境选项">二、环境选项</h2><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403210954362.png" alt="进入以下目录"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403210955883.png" alt="注意其中的主题和 UI font"></p><h2 id="三、编辑器选项">三、编辑器选项</h2><p>进入以下目录：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403210953731.png" alt="进入以下目录"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403210956025.png" alt="取消中括号的配对"></p><h2 id="四、快捷键选项">四、快捷键选项</h2><p>进入以下目录：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403210959850.png" alt="进入以下目录"></p><p>设置注释配置：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403210959208.png" alt="设置注释配置"></p>]]></content>
+    <content type="html"><![CDATA[<h1>CLion 解决中文输出乱码的问题</h1><h2 id="问题介绍">问题介绍</h2><p>在 <code>Clion</code> 的默认设置下，输出中文会出现乱码，如下</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    cout &lt;&lt; <span class="hljs-string">"你好"</span> &lt;&lt; endl;<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><p>输出</p><figure class="highlight awk"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><code class="hljs awk">浣犲ソ<br><br>Process finished with <span class="hljs-keyword">exit</span> code <span class="hljs-number">0</span><br></code></pre></td></tr></tbody></table></figure><h2 id="解决方案">解决方案</h2><p>编码问题就需要修改编码方式，按照下面的流程进行操作即可</p><p>进入设置<br><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403011101299.png" alt="进入设置"></p><p>选择 <code>Editor</code> 中的 <code>File Encodings</code><br><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403011101301.png" alt="选择 Editor 中的 File Encodings"></p><p>将这两个下拉框中的选项全部选择为 <code>UTF-8</code>，点击 <code>OK</code><br><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403011101302.png" alt="将这两个下拉框中的选项全部选择为 UTF-8"></p><p>在主页面的右下角，将这个选项设置为 <code>GBK</code><br><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403011101303.png" alt="设置为 GBK"></p><p>选择 <code>Convert</code><br><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403011101304.png" alt="选择 Convert"></p><p>重新运行即可<span class="github-emoji"><span>😎</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/1f60e.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span></p><figure class="highlight c"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><code class="hljs c">你好<br>Process finished with <span class="hljs-built_in">exit</span> code <span class="hljs-number">0</span><br></code></pre></td></tr></tbody></table></figure><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mi>N</mi><mi>D</mi></mrow><annotation encoding="application/x-tex">END</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">EN</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span></p>]]></content>
     
     
     <categories>
       
       <category>DevTools</category>
       
-      <category>DevCpp</category>
+      <category>CLion</category>
       
     </categories>
     
@@ -1323,18 +1323,18 @@
   
   
   <entry>
-    <title>solve-clion-decoding-error</title>
-    <link href="/DevTools/CLion/solve-clion-decoding-error/"/>
-    <url>/DevTools/CLion/solve-clion-decoding-error/</url>
+    <title>self-config</title>
+    <link href="/DevTools/DevCpp/devc-self-config/"/>
+    <url>/DevTools/DevCpp/devc-self-config/</url>
     
-    <content type="html"><![CDATA[<h1>CLion 解决中文输出乱码的问题</h1><h2 id="问题介绍">问题介绍</h2><p>在 <code>Clion</code> 的默认设置下，输出中文会出现乱码，如下</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    cout &lt;&lt; <span class="hljs-string">"你好"</span> &lt;&lt; endl;<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><p>输出</p><figure class="highlight awk"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><code class="hljs awk">浣犲ソ<br><br>Process finished with <span class="hljs-keyword">exit</span> code <span class="hljs-number">0</span><br></code></pre></td></tr></tbody></table></figure><h2 id="解决方案">解决方案</h2><p>编码问题就需要修改编码方式，按照下面的流程进行操作即可</p><p>进入设置<br><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403011101299.png" alt="进入设置"></p><p>选择 <code>Editor</code> 中的 <code>File Encodings</code><br><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403011101301.png" alt="选择 Editor 中的 File Encodings"></p><p>将这两个下拉框中的选项全部选择为 <code>UTF-8</code>，点击 <code>OK</code><br><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403011101302.png" alt="将这两个下拉框中的选项全部选择为 UTF-8"></p><p>在主页面的右下角，将这个选项设置为 <code>GBK</code><br><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403011101303.png" alt="设置为 GBK"></p><p>选择 <code>Convert</code><br><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403011101304.png" alt="选择 Convert"></p><p>重新运行即可<span class="github-emoji"><span>😎</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/1f60e.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span></p><figure class="highlight c"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><code class="hljs c">你好<br>Process finished with <span class="hljs-built_in">exit</span> code <span class="hljs-number">0</span><br></code></pre></td></tr></tbody></table></figure><p><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>E</mi><mi>N</mi><mi>D</mi></mrow><annotation encoding="application/x-tex">END</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">EN</span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span></span></span></p>]]></content>
+    <content type="html"><![CDATA[<h1>DevCpp 偏好配置</h1><h2 id="前言">前言</h2><p>为了在比赛中不因为编辑器而拖后腿，写一个配置说明。编辑器为 <code>Dev-C++ 5.11</code></p><h2 id="一、编译标准">一、编译标准</h2><p>进入以下目录：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403210937178.png" alt="进入以下目录"></p><p>添加以下编译命令：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403210937916.png" alt="添加以下编译命令"></p><h2 id="二、环境选项">二、环境选项</h2><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403210954362.png" alt="进入以下目录"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403210955883.png" alt="注意其中的主题和 UI font"></p><h2 id="三、编辑器选项">三、编辑器选项</h2><p>进入以下目录：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403210953731.png" alt="进入以下目录"></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403210956025.png" alt="取消中括号的配对"></p><h2 id="四、快捷键选项">四、快捷键选项</h2><p>进入以下目录：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403210959850.png" alt="进入以下目录"></p><p>设置注释配置：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403210959208.png" alt="设置注释配置"></p>]]></content>
     
     
     <categories>
       
       <category>DevTools</category>
       
-      <category>CLion</category>
+      <category>DevCpp</category>
       
     </categories>
     
@@ -1384,16 +1384,16 @@
   
   
   <entry>
-    <title>back-end-guide</title>
-    <link href="/BackEnd/back-end-guide/"/>
-    <url>/BackEnd/back-end-guide/</url>
+    <title>z_logic</title>
+    <link href="/Algorithm/z_logic/"/>
+    <url>/Algorithm/z_logic/</url>
     
-    <content type="html"><![CDATA[<h2 id="后端开发指南">后端开发指南</h2><h2 id="前言">前言</h2><p>本文旨在介绍后端开发的基本理念和路径规划。正在不断完善中。</p><p>对于后端开发来说，语言只是一个工具和基础。除了<strong>语言本身</strong>和对应的<strong>开发框架</strong>外，其他要学的技术在后端开发中往往是通用的，比如：数据库、缓存、消息队列、搜索引擎、Linux、分布式、高并发、设计模式、架构设计等等。</p><h2 id="参考">参考</h2><p><a href="https://roadmap.sh/backend?r=backend-beginner">Backend Beginner Roadmap</a></p><p><a href="https://roadmap.sh/backend">Backend Roadmap</a></p>]]></content>
+    <content type="html"><![CDATA[<h3 id="思维题">思维题</h3><p>多角度切入。</p><h3 id="【思维-哈希-排序】最长严格递增子序列">【思维/哈希/排序】最长严格递增子序列</h3><p><a href="https://www.acwing.com/problem/content/5273/">https://www.acwing.com/problem/content/5273/</a></p><blockquote><p>题意：给定长度为 n 的序列，问将这个序列拼接 n 次后，最长严格递增子序列的长度为多少？</p><p>思路：其实最终的思路很简单，将题目转化为在每个序列中选一个数，一共可以选出多少个不同的数。但是在产生这样的想法之前，先讲一下我的思考过程。我将序列脑补出一幅散点折线图，然后将这些点投影到y轴上，最终投影点的个数就是答案的数量，但是投影会有重合，因此答案最多就是 n 个数，最少 1 个数，从而想到就是在 n 个序列中选数，选的数依次增大即可，相应的就是一个求序列不重复数的个数的过程。去重即可。</p><p>注意点：由于 C++ 的 STL 的 unique 函数的前提是一个有序的序列，因此在unique之前需要将序列进行排序。</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span></span><br><span class="hljs-function"></span>{<br>    <span class="hljs-type">int</span> T;<br>    cin &gt;&gt; T;<br><br>    <span class="hljs-keyword">while</span> (T--)<br>    {<br>        <span class="hljs-type">int</span> n;<br>        cin &gt;&gt; n;<br><br>        vector&lt;<span class="hljs-type">int</span>&gt; a;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++)<br>        {<br>            <span class="hljs-type">int</span> x;<br>            cin &gt;&gt; x;<br>            a.<span class="hljs-built_in">emplace_back</span>(x);<br>        }<br><br>        <span class="hljs-built_in">sort</span>(a.<span class="hljs-built_in">begin</span>(), a.<span class="hljs-built_in">end</span>());<br>        a.<span class="hljs-built_in">erase</span>(<span class="hljs-built_in">unique</span>(a.<span class="hljs-built_in">begin</span>(), a.<span class="hljs-built_in">end</span>()), a.<span class="hljs-built_in">end</span>());<br><br>        cout &lt;&lt; a.<span class="hljs-built_in">size</span>() &lt;&lt; endl;<br>    }<br><br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【分讨-组合数学】三元组">【分讨/组合数学】三元组</h3><p><a href="https://www.acwing.com/problem/content/5280/">https://www.acwing.com/problem/content/5280/</a></p><blockquote><p>题意：在一个序列中如何选择一个三元组，使得三个数之积最小，给出情况数</p><p>思路：首先很容易得知，这三个数一定是序列排序后的前三个数，那么就是对这三个数可能的情况进行讨论</p><ul><li>情况1：<strong>前三个数都相等</strong> <code>（2 2 2 2 2 4 5）</code>，则就是在所有的 <code>a[0]</code> 中选3个，情况数就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>C</mi><mrow><mi>c</mi><mi>n</mi><mi>t</mi><mi mathvariant="normal">_</mi><mi>a</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo></mrow><mn>3</mn></msubsup></mrow><annotation encoding="application/x-tex">C_{cnt\_a[0]}^{3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3531em;vertical-align:-0.539em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.378em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">c</span><span class="mord mathnormal mtight">n</span><span class="mord mathnormal mtight">t</span><span class="mord mtight" style="margin-right:0.02778em;">_</span><span class="mord mathnormal mtight">a</span><span class="mopen mtight">[</span><span class="mord mtight">0</span><span class="mclose mtight">]</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.539em;"><span></span></span></span></span></span></span></span></span></span></li><li>情况2：<strong>前三个数中有两个数相等</strong>，由于数组经过了排序，因此情况2分为两种情况<ul><li>前两个数相等 <code>（1 1 3 3 3 5）</code>，则就是在所有的 <code>a[2]</code> 中选1个，情况数就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>C</mi><mrow><mi>c</mi><mi>n</mi><mi>t</mi><mi mathvariant="normal">_</mi><mi>a</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow><mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">C_{cnt\_a[2]}^{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3531em;vertical-align:-0.539em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.378em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">c</span><span class="mord mathnormal mtight">n</span><span class="mord mathnormal mtight">t</span><span class="mord mtight" style="margin-right:0.02778em;">_</span><span class="mord mathnormal mtight">a</span><span class="mopen mtight">[</span><span class="mord mtight">2</span><span class="mclose mtight">]</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.539em;"><span></span></span></span></span></span></span></span></span></span></li><li>后两个数相等 <code>（1 2 2 2 4 6）</code>，则就是在所有的 <code>a[1]</code> 中选2个，情况数就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>C</mi><mrow><mi>c</mi><mi>n</mi><mi>t</mi><mi mathvariant="normal">_</mi><mi>a</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">C_{cnt\_a[1]}^{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3531em;vertical-align:-0.539em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.378em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">c</span><span class="mord mathnormal mtight">n</span><span class="mord mathnormal mtight">t</span><span class="mord mtight" style="margin-right:0.02778em;">_</span><span class="mord mathnormal mtight">a</span><span class="mopen mtight">[</span><span class="mord mtight">1</span><span class="mclose mtight">]</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.539em;"><span></span></span></span></span></span></span></span></span></span></li></ul></li><li>情况3：<strong>前三个数中全都不相等</strong> <code>（1 2 4 4 4 5 7）</code>，这就是在所有的 <code>a[2]</code> 中选1个，情况数就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>C</mi><mrow><mi>c</mi><mi>n</mi><mi>t</mi><mi mathvariant="normal">_</mi><mi>a</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow><mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">C_{cnt\_a[2]}^{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3531em;vertical-align:-0.539em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.378em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">c</span><span class="mord mathnormal mtight">n</span><span class="mord mathnormal mtight">t</span><span class="mord mtight" style="margin-right:0.02778em;">_</span><span class="mord mathnormal mtight">a</span><span class="mopen mtight">[</span><span class="mord mtight">2</span><span class="mclose mtight">]</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.539em;"><span></span></span></span></span></span></span></span></span></span></li></ul><p>可以发现上述情况2的第一种和情况3的答案相等，故分为三种情况即可。</p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-keyword">typedef</span> <span class="hljs-type">long</span> <span class="hljs-type">long</span> ll;<br><br><span class="hljs-comment">// 计算组合数 C_{a}^{b}</span><br><span class="hljs-function">ll <span class="hljs-title">calc</span><span class="hljs-params">(ll a, ll b)</span> </span>{<br>    ll fz = <span class="hljs-number">1</span>, fm = <span class="hljs-number">1</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>, num = a; i &lt; b; i++, num--) {<br>        fz *= num;<br>    }<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= b; i++) {<br>        fm *= i;<br>    }<br>    <span class="hljs-keyword">return</span> fz / fm;<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-type">int</span> n; cin &gt;&gt; n;<br>    <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">a</span><span class="hljs-params">(n)</span></span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        cin &gt;&gt; a[i];<br>    }<br>    <br>    <span class="hljs-built_in">sort</span>(a.<span class="hljs-built_in">begin</span>(), a.<span class="hljs-built_in">begin</span>() + n);<br>    <br>    ll res = <span class="hljs-number">0</span>;<br>    <br>    <span class="hljs-keyword">if</span> (a[<span class="hljs-number">0</span>] == a[<span class="hljs-number">1</span>] &amp;&amp; a[<span class="hljs-number">1</span>] == a[<span class="hljs-number">2</span>]) {<br>        <span class="hljs-comment">// 情况1：前三个数都相等（2 2 2 2 2 4 5），则就是在所有的a[0]中选3个</span><br>        ll cnt = <span class="hljs-built_in">count</span>(a.<span class="hljs-built_in">begin</span>(), a.<span class="hljs-built_in">begin</span>() + n, a[<span class="hljs-number">0</span>]);<br>        res = <span class="hljs-built_in">calc</span>(cnt, <span class="hljs-number">3</span>);<br>    } <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (a[<span class="hljs-number">0</span>] == a[<span class="hljs-number">1</span>] || (a[<span class="hljs-number">0</span>] != a[<span class="hljs-number">1</span>] &amp;&amp; a[<span class="hljs-number">1</span>] != a[<span class="hljs-number">2</span>])) {<br>        <span class="hljs-comment">// 情况2：前两个数相等（1 1 3 3 3 4 5） or 三个数都不相等（1 2 4 4 4 5 7），则就是在所有的a[2]中选2个</span><br>        res = <span class="hljs-built_in">count</span>(a.<span class="hljs-built_in">begin</span>(), a.<span class="hljs-built_in">begin</span>() + n, a[<span class="hljs-number">2</span>]);<br>    } <span class="hljs-keyword">else</span> {<br>        <span class="hljs-comment">// 情况3：后两个数相等（1 2 2 2 4 6），则就是在所有的a[1]中选2个</span><br>        ll cnt = <span class="hljs-built_in">count</span>(a.<span class="hljs-built_in">begin</span>(), a.<span class="hljs-built_in">begin</span>() + n, a[<span class="hljs-number">1</span>]);<br>        res = <span class="hljs-built_in">calc</span>(cnt, <span class="hljs-number">2</span>);<br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>    <br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【分讨】删除元素">【分讨】删除元素</h3><p><a href="https://www.acwing.com/problem/content/5281/">https://www.acwing.com/problem/content/5281/</a></p><blockquote><p>题意：给定一个排列为1~n，定义一个数“有价值”为当前数的前面没有数比当前的数大。现在需要删除一个数，使得序列中增加尽可能多的“有价值”的数，如果这个数有多个，则删除最小的那个数</p><p>最开始想到的思路：枚举每一个数，如果删除，则序列中会增加多少个“有价值”的数，算法设计如下：</p><ol><li>首先判断每一个数是否是有价值的数<ul><li>创建一个变量来记录当前数的前面序列的最大值</li><li>比较判断当前数和前方最大值的关系，如果小于，则无价值，反之有价值</li></ul></li><li>接着枚举每一个数，如果删除该数，则序列会损失多少个有价值的数<ul><li>首先判断自己是不是有价值的数，如果是，则当前损失值 -1</li><li>接着判断删除当前数对后续的影响 <span class="github-emoji"><span>⭐</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/2b50.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span> ：我们在枚举后续数的时候，起始的newd（前驱除掉a[i]的最大值）应该就是当前的d，只不过需要使用新变量newd而非d是因为这一步与整体无关，不可以改变整体的d。否则会出现错误，比如对于<code>4 3 5 1 2</code>，如果我们在枚举后续数的时候直接对d进行迭代，那么在第一轮d就会被更新为5，就再也无法更新了</li></ul></li><li>最终根据维护的损失值数组cnt即可求解</li><li>时间复杂度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></li></ol><p>优化的思路：</p><ol><li><p>同样是枚举每一个数，如果当前数字是有价值的数，那么cnt[a[i]]就 -1</p><ul><li>取决于当前数字前面是否有比它大的数，如果没有，那么就是有价值的</li></ul></li><li><p>接下来我们不需要遍历j=i+1到j=n-1，而是讨论当前数a[i]不是有价值数时的所有情况。现在我们想要维护cnt数组。不难发现，对于某个位置上的数a[k]，能否成为有价值的数只取决于前排序列是否有比他大的数以及大的数的个数。那么理论上我们在枚举a[i]的时候维护cnt[1~i-1]就可以确保cnt数组的正确性了。</p><ul><li>假设a[k]的前排有一个比它大的数d：那么把d去掉之后，a[k]就会从无价值变为有价值</li><li>假设a[k]的前排有至少两个比它大的数d1,d2,…dj…：那么不管去掉哪一个 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">d_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>，我们都没法让a[k]变得有价值</li></ul><p>如此一来，cnt数组就可以正确维护了</p></li><li><p>最终根据维护的损失值数组cnt即可求解</p></li><li><p>时间复杂度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></li></ol></blockquote><p>暴力：</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-type">int</span> n;<br>    cin &gt;&gt; n;<br>    <br>    <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">a</span><span class="hljs-params">(n)</span></span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        cin &gt;&gt; a[i];<br>    }<br>    <br>    <span class="hljs-comment">// 判断每个数是否是有价值的数</span><br>    <span class="hljs-type">int</span> d = <span class="hljs-number">0</span>; <span class="hljs-comment">// 前方序列中的最大值</span><br>    <span class="hljs-function">vector&lt;<span class="hljs-type">bool</span>&gt; <span class="hljs-title">is_val</span><span class="hljs-params">(n + <span class="hljs-number">1</span>)</span></span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        <span class="hljs-keyword">if</span> (a[i] &gt; d) {<br>            is_val[a[i]] = <span class="hljs-literal">true</span>;<br>            d = a[i];<br>        }<br>    }<br>    <br>    <span class="hljs-comment">// 枚举每一个数，计算删除该数之后会增加的有价值的数的个数 </span><br>    d = <span class="hljs-number">0</span>; <span class="hljs-comment">// 同上</span><br>    <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">cnt</span><span class="hljs-params">(n + <span class="hljs-number">1</span>)</span></span>; <span class="hljs-comment">// cnt[x]表示删除数x之后可以增加的有价值的数的个数</span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        <span class="hljs-keyword">if</span> (is_val[a[i]]) {<br>            cnt[a[i]]--;<br>        }<br>        <br>        <span class="hljs-type">int</span> newd = d; <span class="hljs-comment">// 去掉a[i]后的前方序列的最大值 newd</span><br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = i + <span class="hljs-number">1</span>; j &lt; n; j++) {<br>            <span class="hljs-keyword">if</span> (is_val[a[j]]) {<br>                <span class="hljs-keyword">if</span> (a[j] &lt; newd) {<br>                    cnt[a[i]]--;<br>                }<br>            } <span class="hljs-keyword">else</span> {<br>                <span class="hljs-keyword">if</span> (a[j] &gt; newd) {<br>                    cnt[a[i]]++;<br>                }<br>            }<br>            <span class="hljs-keyword">if</span> (a[j] &gt; newd) {<br>                newd = a[j];<br>            }<br>        }<br>        <br>        <span class="hljs-comment">// 更新前方序列的最大值 d</span><br>        <span class="hljs-keyword">if</span> (a[i] &gt; d) {<br>            d = a[i];<br>        }<br>    }<br><br>    <span class="hljs-comment">// 找出可以增加的有价值的数的最大个数 ma</span><br>    <span class="hljs-type">int</span> ma = -n - <span class="hljs-number">1</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        <span class="hljs-keyword">if</span> (cnt[a[i]] &gt; ma) {<br>            ma = <span class="hljs-built_in">max</span>(ma, cnt[a[i]]);<br>        }<br>    }<br><br>    <span class="hljs-comment">// 答案是满足个数的最小的数字</span><br>    <span class="hljs-type">int</span> res = <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-keyword">if</span> (cnt[i] == ma) {<br>            res = i;<br>            <span class="hljs-keyword">break</span>;<br>        }<br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>    <br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><p>优化：</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-type">int</span> n; <br>    cin &gt;&gt; n;<br>    <br>    <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">a</span><span class="hljs-params">(n)</span></span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        cin &gt;&gt; a[i];<br>    }<br>     <br>    <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">cnt</span><span class="hljs-params">(n + <span class="hljs-number">1</span>)</span></span>;     <span class="hljs-comment">// cnt[x]表示删除数x之后可以增加的有价值的数的个数</span><br>    <span class="hljs-type">int</span> d1 = <span class="hljs-number">0</span>, d2 = <span class="hljs-number">0</span>;         <span class="hljs-comment">// d1: 最大值, d2: 次大值</span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        <span class="hljs-keyword">if</span> (a[i] &gt; d1) {<br>            <span class="hljs-comment">// 前方序列 没有数比当前的数大</span><br>            cnt[a[i]]--;<br>        } <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (a[i] &gt; d2) {<br>            <span class="hljs-comment">// 前方序列 只有一个数比当前大</span><br>            cnt[d1]++;<br>        }<br>        <br>        <span class="hljs-comment">// 更新最大值d1和次大值d2</span><br>        <span class="hljs-keyword">if</span> (a[i] &gt; d1) d2 = d1, d1 = a[i];<br>        <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (a[i] &gt; d2) d2 = a[i];<br>    }<br>    <br>    <span class="hljs-comment">// 找到可以增加的最大值</span><br>    <span class="hljs-type">int</span> ma = <span class="hljs-number">-1</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-keyword">if</span> (cnt[i] &gt; ma) {<br>            ma = cnt[i];<br>        }<br>    }<br>    <br>    <span class="hljs-comment">// 取元素的最小值</span><br>    <span class="hljs-type">int</span> res = n + <span class="hljs-number">1</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-keyword">if</span> (cnt[i] == ma) {<br>            res = i;<br>            <span class="hljs-keyword">break</span>;<br>        }<br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>    <br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【构造】Sorting-with-Twos">【构造】Sorting with Twos</h3><p><a href="https://codeforces.com/contest/1891/problem/A">https://codeforces.com/contest/1891/problem/A</a></p><blockquote><p>题意：给定一个序列，现在需要通过以下方法对序列进行升序排序</p><ul><li>可以选择前 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>m</mi></msup></mrow><annotation encoding="application/x-tex">2^m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6644em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span></span></span></span></span></span></span></span> 个数执行 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span> 的操作（保证不会越界的情况下）</li></ul><p>现在需要确定给定的序列经过k次上述后能否变为升序序列</p><p>思路：我们可以将每 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>k</mi></msup></mrow><annotation encoding="application/x-tex">2^k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span></span></span> 个数看成一个数，可以不断的-1。那么可以发现执行一定的次数后，一定可以实现升序序列。但是现在是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>k</mi></msup></mrow><annotation encoding="application/x-tex">2^k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span></span></span> 个数，那么只需要这相邻区段的数是升序即可，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>→</mo><msup><mn>2</mn><mi>k</mi></msup></mrow><annotation encoding="application/x-tex">2^{k-1} \to 2^k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span></span></span> 之间的数是升序即可。按区间进行判断即可</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">judge</span><span class="hljs-params">(vector&lt;<span class="hljs-type">int</span>&gt;&amp; a, <span class="hljs-type">int</span> l, <span class="hljs-type">int</span> r, <span class="hljs-type">bool</span>&amp; ok, <span class="hljs-type">int</span> n)</span> </span>{<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = l + <span class="hljs-number">1</span>; i &lt;= r &amp;&amp; i &lt;= n; i++) {<br>        <span class="hljs-keyword">if</span> (a[i] &lt; a[i - <span class="hljs-number">1</span>]) {<br>            ok = <span class="hljs-literal">false</span>;<br>            <span class="hljs-keyword">return</span>;<br>        }<br>    }<br>}<br> <br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-type">int</span> n;<br>    cin &gt;&gt; n;<br> <br>    <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">a</span><span class="hljs-params">(n + <span class="hljs-number">1</span>)</span></span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        cin &gt;&gt; a[i];<br>    }<br> <br>    <span class="hljs-type">bool</span> ok = <span class="hljs-literal">true</span>;<br>    <span class="hljs-built_in">judge</span>(a, <span class="hljs-number">3</span>, <span class="hljs-number">4</span>, ok, n);<br>    <span class="hljs-built_in">judge</span>(a, <span class="hljs-number">5</span>, <span class="hljs-number">8</span>, ok, n);<br>    <span class="hljs-built_in">judge</span>(a, <span class="hljs-number">9</span>, <span class="hljs-number">16</span>, ok, n);<br>    <span class="hljs-built_in">judge</span>(a, <span class="hljs-number">17</span>, <span class="hljs-number">20</span>, ok, n);<br> <br>    <span class="hljs-keyword">if</span> (ok) {<br>        cout &lt;&lt; <span class="hljs-string">"YES\n"</span>;<br>    } <span class="hljs-keyword">else</span> {<br>        cout &lt;&lt; <span class="hljs-string">"NO\n"</span>;<br>    }<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【模拟】指针运动​-fire">【模拟】指针运动​ <span class="github-emoji"><span>🔥</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/1f525.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span></h3><p><a href="https://www.acwing.com/problem/content/description/5468/">https://www.acwing.com/problem/content/description/5468/</a></p><blockquote><ul><li>题意：给定一个序列，和一个从第一个数开始的指针，若当前指的数为 0 则输出当前数的下标结束程序，若不为 0 则全体正数减一并将指针后移，若指针越界则重新指向第一个数。问最终指向第几个数</li><li>思路：<ul><li>第一层思路：纯暴力。即按照题意进行模拟，时间复杂度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>×</mo><mi>max</mi><mo>⁡</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \times \max (a_i))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">max</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">))</span></span></span></span></li><li>第二层思路：枚举优化。假设当前枚举到了第 k 次，若第一次遇到当前的数小于 k，则该数就是终止指向的数，时间复杂度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>×</mo><mi>max</mi><mo>⁡</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \times \max(a_i))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">max</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">))</span></span></span></span></li><li>第三层思路：数列优化。</li></ul></li><li>时间复杂度：</li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><br></code></pre></td></tr></tbody></table></figure><h3 id="【哈希】套餐设计">【哈希】套餐设计</h3><p><a href="https://www.acwing.com/problem/content/5480/">https://www.acwing.com/problem/content/5480/</a></p><blockquote><p>题意：给定 n 个食物，其中可能有相同种类的，现在需要设计一个食物套餐包含 k 个食物，问如何设计套餐可以使得产生的套餐数最多，给出最多数量的套餐数</p><p>思路：如果不限定套餐中食物的种类数，那么我们可以从最多可生产的套餐数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">⌊</mo><mfrac><mi>n</mi><mi>k</mi></mfrac><mo fence="true">⌋</mo></mrow><annotation encoding="application/x-tex">\left \lfloor \frac{n}{k} \right \rfloor</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">⌊</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">⌋</span></span></span></span></span></span> 开始降序检查，直到一套都无法生产为止（即一套需要的食物数超过了总的食物数）。检查的逻辑很简单，我们知道对于当前的套餐数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>，每一种食物的数量如果超过了 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>，就至少可以对套餐贡献 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 个食物的数量。我们统计当前需要套餐数的局面下，所有食物可以贡献的食物数量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi><mi>n</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">cnt</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span></span></span></span>，如果超过了需要的套餐数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>，显然是可以满足需求的套餐数的。为了求得最大的套餐数，从降序开始检查直到可以满足即可。</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;unordered_map&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><span class="hljs-keyword">typedef</span> <span class="hljs-type">long</span> <span class="hljs-type">long</span> ll;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">110</span>;<br><br><span class="hljs-type">int</span> n, k, a[N];<br>unordered_map&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt; v;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; k &gt;&gt; n;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        cin &gt;&gt; a[i];<br>        v[a[i]]++;<br>    }<br>    <br>    <span class="hljs-type">int</span> res = <span class="hljs-number">0</span>;<br>    <br>    <span class="hljs-type">int</span> i; <span class="hljs-comment">// 当前需要生产的套餐数</span><br>    <span class="hljs-keyword">for</span> (i = n / k; i &gt;= <span class="hljs-number">1</span>; i--) {<br>        <span class="hljs-comment">// 所有食物对于当前套餐需要的食物数量可以贡献的最大食物数量</span><br>        <span class="hljs-type">int</span> cnt = <span class="hljs-number">0</span>; <br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; x: v) {<br>            <span class="hljs-comment">// 每种食物对于当前套餐需要的食物数量可以贡献的最大食物数量</span><br>            cnt += x.second / i;<br>        }<br>        <span class="hljs-keyword">if</span> (cnt &gt;= k) {<br>            res = i;<br>            <span class="hljs-keyword">break</span>;<br>        }<br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>), cin.<span class="hljs-built_in">tie</span>(<span class="hljs-number">0</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-number">0</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span>(T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【贪心-分讨】分班">【贪心/分讨】分班</h3><p><a href="https://www.acwing.com/problem/content/5481/">https://www.acwing.com/problem/content/5481/</a></p><blockquote><p>题意：给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>×</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">n \times k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个学生，需要将这些学生分成 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 组，每组 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个人，第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 个学生有一个属性值 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">a_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 。问如何分组可以使得在满足任意两组学生中最小值相差不超过 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>i</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">lim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">im</span></span></span></span> 的情况下，所有组最小值之和最大，给出这个最大值</p><p>思路：一开始觉得是二分答案，但是不知道在已知答案的情况下如何检查，因为我不知道应该如何分组最优，未遂。重读题目发现题中的 lim 约束不是相邻两组，而是任意两组，思路打开，那我直接根据这个条件将满足约束的所有学生枚举出来（很显然是相对于数值最小的学生而言），然后设计如何将这些学生安排到 n 组且保证他们是组里最小的即可。先看下面的学生编号 - 学生属性示意图：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403030040949.png" alt="学生编号 - 学生属性示意图"></p><p>很显然我们需要将这 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>×</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">n \times k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个学生按数值升序排序，为了满足任意两组的最小值不超过 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>i</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">lim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">im</span></span></span></span>，我们找到前 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span> 个学生满足比最小的学生超过的数不超过 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>i</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">lim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">im</span></span></span></span> 即可。接下来就是从这 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span> 个学生中选择出 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个安排到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个组中，即可满足题中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>i</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">lim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">im</span></span></span></span> 的限制，现在就是考虑如何分组可以使得答案最大了。不难发现一共有三种情况</p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mo>&lt;</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">r&lt;n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> ：即可选择人数都没有组数多。那么分完组以后肯定无法满足任意两组的最小值不超过 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>i</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">lim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">im</span></span></span></span>，直接输出 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 即可</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mo>=</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">r=n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> ：即可选择人数和组数相等。这种情况最好理解，就是将这 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 个人安排到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 组，答案就是这 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 个人数值之和</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mo>&gt;</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">r&gt;n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> ：即可选择人数多于组数。如果纯粹的贪心，将第一个人放在第一组，然后从第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span> 个人开始将 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 个人安排在剩余的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 组。这样的确可以满足答案最大，但是忽略了一个点就是从第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span> 个人开始的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mo>−</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">r-n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个人何去何从？如果全部安排到了第一组那万事大吉，如果第一组塞不进去就不得不安排到别的组，这样答案就不是上面计算的结果了，应该更小才对！那么应该如何解决这个问题呢？我们可以从往第一组塞剩余的人得到启发：如果第一组可以容纳剩余的人活着还有空位就已经满足了第二种情况，就可以直接计算结果，如果第一组无法容纳从第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span> 个人开始的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mo>−</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">r-n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个人，就将第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>+</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">1+k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个人作为一组的最小值，继续贪心的容纳紧跟着的人，从而确保剩下的人数值尽可能的大。如此循环知道出现 <strong>剩余的人的数量=还未安排最小值的组的数量</strong> 为止，计算答案总和即可！</li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><span class="hljs-keyword">typedef</span> <span class="hljs-type">long</span> <span class="hljs-type">long</span> ll;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">1e5</span> + <span class="hljs-number">10</span>;<br><br>ll n, k, lim, a[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; k &gt;&gt; lim;<br><br>    <span class="hljs-type">int</span> num = n * k;<br><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= num; i++) cin &gt;&gt; a[i];<br><br>    <span class="hljs-built_in">sort</span>(a + <span class="hljs-number">1</span>, a + num + <span class="hljs-number">1</span>);<br><br>    <span class="hljs-comment">// 找到最大的满足 lim 限制的人的下标 r - 线性或二分均可</span><br>    <span class="hljs-type">int</span> l = <span class="hljs-number">1</span>, r = num;<br>    <span class="hljs-keyword">while</span> (l &lt; r) {<br>        <span class="hljs-type">int</span> mid = (l + r + <span class="hljs-number">1</span>) &gt;&gt; <span class="hljs-number">1</span>;<br>          <span class="hljs-keyword">if</span> (a[mid] &lt;= a[<span class="hljs-number">1</span>] + lim) l = mid;<br>          <span class="hljs-keyword">else</span> r = mid - <span class="hljs-number">1</span>;<br>    }<br><br>    ll res = <span class="hljs-number">0</span>;<br><br>    <span class="hljs-keyword">if</span> (r &lt; n) res = <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">else</span> {<br>        res += a[<span class="hljs-number">1</span>];<br>        <span class="hljs-type">int</span> idx = <span class="hljs-number">2</span>;   <span class="hljs-comment">// 指针</span><br>        <span class="hljs-type">int</span> in = <span class="hljs-number">1</span>;    <span class="hljs-comment">// 已安排作为组内最小值的人员数</span><br>        <span class="hljs-type">int</span> team = <span class="hljs-number">1</span>;  <span class="hljs-comment">// 当前组内的人员数</span><br>        <span class="hljs-keyword">while</span> (r - idx + <span class="hljs-number">1</span> &gt; n - in) { <span class="hljs-comment">// 当可选择人数多于组数时</span><br>            <span class="hljs-keyword">if</span> (team &lt; k) team++, idx++; <span class="hljs-comment">// 塞得下就继续塞</span><br>            <span class="hljs-keyword">else</span> team = <span class="hljs-number">1</span>, res += a[idx], idx++, in++; <span class="hljs-comment">// 塞不下就重开一组</span><br>        }<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = idx; i &lt;= r; i++) {<br>            res += a[i];<br>        }<br>    }<br><br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>), cin.<span class="hljs-built_in">tie</span>(<span class="hljs-number">0</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-number">0</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span>(T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【贪心-分讨】双端队列">【贪心/分讨】双端队列</h3><p><a href="https://www.acwing.com/problem/content/5484/">https://www.acwing.com/problem/content/5484/</a></p><blockquote><p>题意：给定一个序列，每次可以从序列头或尾取出一个数，在满足：按取出数的顺序是升序序列的条件下，给定的序列最多可以取出多少数？</p><p>思路：首先很容易想到的就是哪头小取哪头，因为取了较小者后较大者可以后续继续被选择，反之则不能。当然，还需要考虑的一个点就是如果较小者不符合题意，即较小者比上一次选出来的数更小从而无法组成严格升序序列时，还是得考虑较大值的那一头数。接下来考虑相等的情况，若两头数值相等，那一定只能选择一端且后续都只能从那端继续选择，我们直接两种方法都计算统计一遍可选择的数的数量，取较大者即可</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;queue&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;cstring&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;vector&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;unordered_map&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;algorithm&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;stack&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">200010</span>;<br><br><span class="hljs-type">int</span> n, a[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) cin &gt;&gt; a[i];<br>    <br>    <span class="hljs-type">int</span> res = <span class="hljs-number">0</span>, top = <span class="hljs-number">-1</span>, l = <span class="hljs-number">1</span>, r = n;<br>    <span class="hljs-keyword">while</span> (l &lt;= r) {<br>        <span class="hljs-comment">// 左端是较小值</span><br>        <span class="hljs-keyword">if</span> (a[l] &lt; a[r]) {<br>            <span class="hljs-keyword">if</span> (a[l] &gt; top) {<br>                res++;<br>                top = a[l++];<br>            } <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (a[r] &gt; top) {<br>                res++;<br>                top = a[r--];<br>            } <span class="hljs-keyword">else</span> {<br>                <span class="hljs-keyword">break</span>;<br>            }<br>        <span class="hljs-comment">// 右端是较小值</span><br>        } <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (a[l] &gt; a[r]) {<br>            <span class="hljs-keyword">if</span> (a[r] &gt; top) {<br>                res++;<br>                top = a[r--];<br>            } <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (a[l] &gt; top) {<br>                res++;<br>                top = a[l++];<br>            } <span class="hljs-keyword">else</span> {<br>                <span class="hljs-keyword">break</span>;<br>            }<br>        <span class="hljs-comment">// 两端数值相等</span><br>        } <span class="hljs-keyword">else</span> { <span class="hljs-comment">// a[l] == a[r]</span><br>            <span class="hljs-keyword">if</span> (a[l] &gt; top) {<br>                <span class="hljs-type">int</span> lcnt = <span class="hljs-number">1</span>, rcnt = <span class="hljs-number">1</span>;<br>                <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = l + <span class="hljs-number">1</span>; i &lt;= r; i++) {<br>                    <span class="hljs-keyword">if</span> (a[i] &gt; a[i - <span class="hljs-number">1</span>]) lcnt++;<br>                    <span class="hljs-keyword">else</span> <span class="hljs-keyword">break</span>;<br>                }<br>                <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = r - <span class="hljs-number">1</span>; i &gt;= l; i--) {<br>                    <span class="hljs-keyword">if</span> (a[i] &gt; a[i + <span class="hljs-number">1</span>]) rcnt++;<br>                    <span class="hljs-keyword">else</span> <span class="hljs-keyword">break</span>;<br>                }<br>                res += <span class="hljs-built_in">max</span>(lcnt, rcnt);<br>                <span class="hljs-keyword">break</span>;<br>            } <span class="hljs-keyword">else</span> {<br>                <span class="hljs-keyword">break</span>;<br>            }<br>        }<br>    }<br><br>    cout &lt;&lt; res;<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【思维】你说的对，但是这是签到-I">【思维】你说的对，但是这是签到 I</h3><p><a href="https://hydro.ac/d/nnu_contest/p/P1201">https://hydro.ac/d/nnu_contest/p/P1201</a></p><blockquote><p>题意：给定两个单调不减且只含有小写字母的字符串，求解其中最长公共子串长度</p><p>思路：</p><ul><li>一眼 dp？错误的，那时间复杂度就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>，显然不可能。可以发现与常规的最长公共子串的题目不同，本题中两个字符串是单调的，这个性质一定很有用。事实的确如此。</li><li>对于单调的两个字符串而言，如果<strong>连续的</strong>字符数量之和相等，即对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>c</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">c_i,c_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 之间的所有的字符对应的数量都相等 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo stretchy="false">[</mo><msub><mi>c</mi><mi>k</mi></msub><mo stretchy="false">]</mo><mo>=</mo><mi>b</mi><mo stretchy="false">[</mo><msub><mi>c</mi><mi>k</mi></msub><mo stretchy="false">]</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mi>k</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo separator="true">,</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mi>j</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a[c_k]=b[c_k],(k=i+1,i+1, \cdots,j-1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">b</span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7429em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span>，则该段字符串就一定是公共子串。答案就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></munderover><mi>a</mi><mo stretchy="false">[</mo><msub><mi>c</mi><mi>k</mi></msub><mo stretchy="false">]</mo></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle  \sum_{k=i+1}^{j-1}a[c_k]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.2192em;vertical-align:-1.3604em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8588em;"><span style="top:-1.8479em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mathnormal mtight">i</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3471em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3604em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span> 或 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></munderover><mi>b</mi><mo stretchy="false">[</mo><msub><mi>c</mi><mi>k</mi></msub><mo stretchy="false">]</mo></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle  \sum_{k=i+1}^{j-1}b[c_k]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.2192em;vertical-align:-1.3604em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8588em;"><span style="top:-1.8479em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mathnormal mtight">i</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3471em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3604em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span>。在枚举字符时，需要特判一下只有一种字符的情况。</li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Θ</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>2</mn><msup><mn>6</mn><mn>3</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta(n + 26^3)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Θ</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;cstring&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;vector&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;queue&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;stack&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;algorithm&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;unordered_map&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;set&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">26</span>;<br><br>string s, t;<br><span class="hljs-type">int</span> a[N], b[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; s &gt;&gt; t;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span> c: s) a[c - <span class="hljs-string">'a'</span>]++;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span> c: t) b[c - <span class="hljs-string">'a'</span>]++;<br>    <br>    <span class="hljs-type">int</span> res = <span class="hljs-number">-1</span>;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; N; i++) {<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = i; j &lt; N; j++) {<br>            <span class="hljs-keyword">if</span> (j == i) res = <span class="hljs-built_in">max</span>(res, <span class="hljs-built_in">min</span>(a[i], b[j]));<br>            <span class="hljs-keyword">else</span> {<br>                <span class="hljs-type">bool</span> ok = <span class="hljs-literal">true</span>;<br>                <span class="hljs-type">int</span> mid = <span class="hljs-number">0</span>;<br>                <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> k = i + <span class="hljs-number">1</span>; k &lt;= j - <span class="hljs-number">1</span>; k++) {<br>                    <span class="hljs-keyword">if</span> (a[k] != b[k]) ok = <span class="hljs-literal">false</span>;<br>                    mid += a[k];<br>                }<br>                <br>                <span class="hljs-keyword">if</span> (ok) {<br>                    res = <span class="hljs-built_in">max</span>(res, <span class="hljs-built_in">min</span>(a[i], b[i]) + mid + <span class="hljs-built_in">min</span>(a[j], b[j]));<br>                }<br>            }<br>        }<br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【贪心】从双倍数组中还原原数组">【贪心】从双倍数组中还原原数组</h3><p><a href="https://leetcode.cn/problems/find-original-array-from-doubled-array/description/">https://leetcode.cn/problems/find-original-array-from-doubled-array/description/</a></p><blockquote><p>题意：给定一个改变后数组，问其原始数组是什么？我们定义改变后数组为原始数组每一个元素乘以2以后加入原始数组，并随机打乱得到的数组</p><p>思路：答案与下标无关，我们考虑对原始数组升序排序。在哈希统计每一个元素出现的频率后，可以枚举每一个元素，如果当前元素与当前元素的二倍都存在，则说明当前元素一定存在于原始数组中，我们将其加入答案并对两个数的频率统计减一。</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span> {<br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">findOriginalArray</span><span class="hljs-params">(vector&lt;<span class="hljs-type">int</span>&gt;&amp; changed)</span> </span>{<br>        vector&lt;<span class="hljs-type">int</span>&gt; res;<br><br>        <span class="hljs-comment">// 剪枝：合法原始数组一定是偶数个</span><br>        <span class="hljs-type">int</span> n = changed.<span class="hljs-built_in">size</span>();<br>        <span class="hljs-keyword">if</span> (n &amp; <span class="hljs-number">1</span>) <span class="hljs-keyword">return</span> {};<br><br>        <span class="hljs-built_in">sort</span>(changed.<span class="hljs-built_in">begin</span>(), changed.<span class="hljs-built_in">end</span>());<br>        unordered_map&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt; cnt;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> x: changed) cnt[x]++;<br><br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> x: changed) {<br>            <span class="hljs-keyword">if</span> (cnt[x] == <span class="hljs-number">0</span>) <span class="hljs-keyword">continue</span>;<br><br>            cnt[x]--;<br>            <span class="hljs-keyword">if</span> (cnt[x * <span class="hljs-number">2</span>]) {<br>                cnt[x * <span class="hljs-number">2</span>]--;<br>                res.<span class="hljs-built_in">push_back</span>(x);<br>            }<br>            <span class="hljs-keyword">else</span> {<br>                <span class="hljs-keyword">return</span> {};<br>            }<br>        }<br><br>        <span class="hljs-keyword">return</span> res;<br>    }<br>};<br></code></pre></td></tr></tbody></table></figure><h3 id="【前缀和-双指针】可获得的最大点数">【前缀和/双指针】可获得的最大点数</h3><p><a href="https://leetcode.cn/problems/maximum-points-you-can-obtain-from-cards/description/">https://leetcode.cn/problems/maximum-points-you-can-obtain-from-cards/description/</a></p><blockquote><p>题意：给定一个序列，每次可以从头或尾取一个元素并弹出该元素，问如何取数可以使得取到的数总和最大</p><p>思路：<strong>前缀和、双指针</strong>。很显然，我们选到的数的组成一定是原数组的头部一部分和尾部一部分，从反面思考，其实就是让中间一部分的数值总和最小！因此我们用前缀和记录一下然后双指针枚举序列即可。之所以记录本地是因为双指针好久没写了，调了好久 : (</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span> {<br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">maxScore</span><span class="hljs-params">(vector&lt;<span class="hljs-type">int</span>&gt;&amp; cardPoints, <span class="hljs-type">int</span> k)</span> </span>{<br>        <span class="hljs-type">int</span> n = cardPoints.<span class="hljs-built_in">size</span>();<br>        <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">s</span><span class="hljs-params">(n + <span class="hljs-number">1</span>, <span class="hljs-number">0</span>)</span></span>;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>            s[i] = s[i - <span class="hljs-number">1</span>] + cardPoints[i - <span class="hljs-number">1</span>];<br>        }<br><br>        <span class="hljs-type">int</span> res = <span class="hljs-number">-1</span>;<br>        <span class="hljs-type">int</span> l = <span class="hljs-number">1</span>, r = l + n - k - <span class="hljs-number">1</span>;<br>        <span class="hljs-keyword">while</span> (r &lt;= n) {<br>            res = <span class="hljs-built_in">max</span>(res, s[n] - (s[r] - s[l - <span class="hljs-number">1</span>]));<br>            l++, r++;<br>        }<br><br>        <span class="hljs-keyword">return</span> res;<br>    }<br>};<br></code></pre></td></tr></tbody></table></figure><h3 id="【思维】确定两个字符串是否接近">【思维】确定两个字符串是否接近</h3><p><a href="https://leetcode.cn/problems/determine-if-two-strings-are-close/description/">https://leetcode.cn/problems/determine-if-two-strings-are-close/description/</a></p><blockquote><p>题意：给定两个字符串，可以对两个字符串进行任意次下面的操作：问在操作后使得两个字符串完全相等</p><ol><li>交换一个字符串中的任意两个字符</li><li>交换一个字符串中的任意两个相同字符的集合。比如 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder accentunder="true"><mrow><mi>a</mi><mi>a</mi></mrow><mo stretchy="true">‾</mo></munder><mi>c</mi><munder accentunder="true"><mrow><mi>a</mi><mi>b</mi><mi>b</mi></mrow><mo stretchy="true">‾</mo></munder><mo>→</mo><munder accentunder="true"><mrow><mi>b</mi><mi>b</mi></mrow><mo stretchy="true">‾</mo></munder><mi>c</mi><munder accentunder="true"><mrow><mi>b</mi><mi>a</mi><mi>a</mi></mrow><mo stretchy="true">‾</mo></munder></mrow><annotation encoding="application/x-tex">\underline {aa} c \underline{abb} \to \underline{bb}c\underline{baa}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8944em;vertical-align:-0.2em;"></span><span class="mord underline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.4306em;"><span style="top:-2.84em;"><span class="pstrut" style="height:3em;"></span><span class="underline-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">aa</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2em;"><span></span></span></span></span></span><span class="mord mathnormal">c</span><span class="mord underline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.84em;"><span class="pstrut" style="height:3em;"></span><span class="underline-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">abb</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8944em;vertical-align:-0.2em;"></span><span class="mord underline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.84em;"><span class="pstrut" style="height:3em;"></span><span class="underline-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">bb</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2em;"><span></span></span></span></span></span><span class="mord mathnormal">c</span><span class="mord underline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.84em;"><span class="pstrut" style="height:3em;"></span><span class="underline-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">baa</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2em;"><span></span></span></span></span></span></span></span></span></li></ol><p>思路：一道纯纯思维题。首先对于第一个操作，我们可以直接对两个字符串排序即可，至于第二个操作，我们需要确保两个字符串哈希值中，所有的键全部相等，即两个字符串排序去重后应该完全相等，同时还需要确保两个字符串哈希的结果中值排序的结果完全相等，这样就可以通过操作二使得两个字符串完全相等</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span> {<br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-function"><span class="hljs-type">bool</span> <span class="hljs-title">closeStrings</span><span class="hljs-params">(string word1, string word2)</span> </span>{<br>        <span class="hljs-comment">// 剪枝：两个字符串长度不相等，则操作后一定不可能相等</span><br>        <span class="hljs-keyword">if</span> (word<span class="hljs-number">1.</span><span class="hljs-built_in">size</span>() != word<span class="hljs-number">2.</span><span class="hljs-built_in">size</span>()) <span class="hljs-keyword">return</span> <span class="hljs-literal">false</span>;<br><br>        <span class="hljs-comment">// 执行操作1</span><br>        <span class="hljs-built_in">sort</span>(word<span class="hljs-number">1.</span><span class="hljs-built_in">begin</span>(), word<span class="hljs-number">1.</span><span class="hljs-built_in">end</span>());<br>        <span class="hljs-built_in">sort</span>(word<span class="hljs-number">2.</span><span class="hljs-built_in">begin</span>(), word<span class="hljs-number">2.</span><span class="hljs-built_in">end</span>());<br>        <br>        <span class="hljs-comment">// 剪枝：若两个字符串排序后相等就不需要执行操作2了，已经相等了</span><br>        <span class="hljs-keyword">if</span> (word1 == word2) <span class="hljs-keyword">return</span> <span class="hljs-literal">true</span>;<br><br>        <span class="hljs-comment">// 哈希两个字符串的字符与对应字符数量</span><br>        unordered_map&lt;<span class="hljs-type">char</span>, <span class="hljs-type">int</span>&gt; a, b;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; c: word1) a[c]++;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; c: word2) b[c]++;<br>        <br>        <span class="hljs-comment">// 剪枝：字符种类数不一样，则操作2后一定不可能相等</span><br>        <span class="hljs-keyword">if</span> (a.<span class="hljs-built_in">size</span>() != b.<span class="hljs-built_in">size</span>()) <span class="hljs-keyword">return</span> <span class="hljs-literal">false</span>;<br>        <br>        <span class="hljs-comment">// 剪枝：字符种类数一样但是种类不一样，则操作2后一定不可能相等</span><br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span> it: a) {<br>            <span class="hljs-keyword">if</span> (b[it.first] == <span class="hljs-number">0</span>) {<br>                <span class="hljs-keyword">return</span> <span class="hljs-literal">false</span>;<br>            }<br>        }<br>        <br>        <span class="hljs-comment">// 执行操作2：比较所有种类字符的数量</span><br>        vector&lt;<span class="hljs-type">int</span>&gt; va, vb;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; it: a) va.<span class="hljs-built_in">push_back</span>(it.second);<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; it: b) vb.<span class="hljs-built_in">push_back</span>(it.second);<br>        <span class="hljs-built_in">sort</span>(va.<span class="hljs-built_in">begin</span>(), va.<span class="hljs-built_in">end</span>());<br>        <span class="hljs-built_in">sort</span>(vb.<span class="hljs-built_in">begin</span>(), vb.<span class="hljs-built_in">end</span>());<br>        <span class="hljs-keyword">return</span> va == vb;<br>    }<br>};<br></code></pre></td></tr></tbody></table></figure><h3 id="【模拟】同位字符串连接的最小长度">【模拟】同位字符串连接的最小长度</h3><p><a href="https://leetcode.cn/problems/minimum-length-of-anagram-concatenation/">https://leetcode.cn/problems/minimum-length-of-anagram-concatenation/</a></p><blockquote><p>题意：给定一个由若干个同位字符串 t 组成的字符串 s，问这个同位字符串最短是什么？定义同位字符串为字符数量相等且每种字符的数量相等的字符串</p><p>错误思路：很显然的一个分配问题。我们预统计每一种字符的数量，假设答案同位字符串长度为 len，则显然就需要将所有的字符等分为 <code>s.size() / len</code> 份，每一份的每一种字符数量均相等。也就是每一种字符都需要分为 <code>s.size() / len</code> 份。为了最小化 len，就需要分出尽可能多的同位字符串。每一种字符可以分出的份数是其因数，取所有字符可分出份数的最大值就是求解每一种字符数量的最大公因数！于是问题就转化为了求解每一种字符数量的最大公因数</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p><p>正确思路：上述思路忽略了同位字符串必须存在于原始字符串中这个约束。正确做法就更加无脑了，直接枚举字符串 s 的每一个因数，然后检查所有长度的子串是否可以作为同位字符串即可</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>128</mn><mo>×</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(128 \times n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">128</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><p>WA code</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span> {<br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">minAnagramLength</span><span class="hljs-params">(string s)</span> </span>{<br>        unordered_map&lt;<span class="hljs-type">char</span>, <span class="hljs-type">int</span>&gt; a;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span> c: s) a[c]++;<br>        <br>        vector&lt;<span class="hljs-type">int</span>&gt; v;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; it: a) v.<span class="hljs-built_in">push_back</span>(it.second);<br>        <br>        <span class="hljs-type">int</span> gcd;<br>        <span class="hljs-keyword">if</span> (v.<span class="hljs-built_in">size</span>() == <span class="hljs-number">1</span>) gcd = v[<span class="hljs-number">0</span>];<br>        <span class="hljs-keyword">else</span> {<br>            gcd = __gcd(v[<span class="hljs-number">0</span>], v[<span class="hljs-number">1</span>]);<br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">2</span>; i &lt; v.<span class="hljs-built_in">size</span>(); i++) {<br>                gcd = __gcd(gcd, v[i]);<br>            }<br>        }<br>        <br>        <span class="hljs-keyword">return</span> s.<span class="hljs-built_in">size</span>() / gcd;<br>    }<br>};<br></code></pre></td></tr></tbody></table></figure><p>AC code</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span> {<br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">minAnagramLength</span><span class="hljs-params">(string s)</span> </span>{<br>        <span class="hljs-type">int</span> n = s.<span class="hljs-built_in">size</span>();<br>        <br>        vector&lt;<span class="hljs-type">int</span>&gt; v;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n / i; i++) {<br>            <span class="hljs-keyword">if</span> (n % i == <span class="hljs-number">0</span>) {<br>                v.<span class="hljs-built_in">push_back</span>(i);<br>                v.<span class="hljs-built_in">push_back</span>(n / i);<br>            }<br>        }<br><br>        <span class="hljs-type">int</span> res = n + <span class="hljs-number">1</span>;<br><br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> len: v) {<br>            <span class="hljs-type">bool</span> ok = <span class="hljs-literal">true</span>;<br><br>            string pre = s.<span class="hljs-built_in">substr</span>(<span class="hljs-number">0</span>, len);<br>            <span class="hljs-built_in">sort</span>(pre.<span class="hljs-built_in">begin</span>(), pre.<span class="hljs-built_in">end</span>());<br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = len; i &lt;= n - len; i += len) {<br>                string now = s.<span class="hljs-built_in">substr</span>(i, len);<br>                <span class="hljs-built_in">sort</span>(now.<span class="hljs-built_in">begin</span>(), now.<span class="hljs-built_in">end</span>());<br>                <span class="hljs-keyword">if</span> (now != pre) {<br>                    ok = <span class="hljs-literal">false</span>;<br>                    <span class="hljs-keyword">break</span>;<br>                }<br>            }<br><br>            <span class="hljs-keyword">if</span> (ok) {<br>                res = <span class="hljs-built_in">min</span>(res, len);<br>            }<br>        }<br><br>        <span class="hljs-keyword">return</span> res;<br>    }<br>};<br></code></pre></td></tr></tbody></table></figure>]]></content>
     
     
     <categories>
       
-      <category>BackEnd</category>
+      <category>Algorithm</category>
       
     </categories>
     
@@ -1403,16 +1403,16 @@
   
   
   <entry>
-    <title>z_logic</title>
-    <link href="/Algorithm/z_logic/"/>
-    <url>/Algorithm/z_logic/</url>
+    <title>back-end-guide</title>
+    <link href="/BackEnd/back-end-guide/"/>
+    <url>/BackEnd/back-end-guide/</url>
     
-    <content type="html"><![CDATA[<h3 id="思维题">思维题</h3><p>多角度切入。</p><h3 id="【思维-哈希-排序】最长严格递增子序列">【思维/哈希/排序】最长严格递增子序列</h3><p><a href="https://www.acwing.com/problem/content/5273/">https://www.acwing.com/problem/content/5273/</a></p><blockquote><p>题意：给定长度为 n 的序列，问将这个序列拼接 n 次后，最长严格递增子序列的长度为多少？</p><p>思路：其实最终的思路很简单，将题目转化为在每个序列中选一个数，一共可以选出多少个不同的数。但是在产生这样的想法之前，先讲一下我的思考过程。我将序列脑补出一幅散点折线图，然后将这些点投影到y轴上，最终投影点的个数就是答案的数量，但是投影会有重合，因此答案最多就是 n 个数，最少 1 个数，从而想到就是在 n 个序列中选数，选的数依次增大即可，相应的就是一个求序列不重复数的个数的过程。去重即可。</p><p>注意点：由于 C++ 的 STL 的 unique 函数的前提是一个有序的序列，因此在unique之前需要将序列进行排序。</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span></span><br><span class="hljs-function"></span>{<br>    <span class="hljs-type">int</span> T;<br>    cin &gt;&gt; T;<br><br>    <span class="hljs-keyword">while</span> (T--)<br>    {<br>        <span class="hljs-type">int</span> n;<br>        cin &gt;&gt; n;<br><br>        vector&lt;<span class="hljs-type">int</span>&gt; a;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++)<br>        {<br>            <span class="hljs-type">int</span> x;<br>            cin &gt;&gt; x;<br>            a.<span class="hljs-built_in">emplace_back</span>(x);<br>        }<br><br>        <span class="hljs-built_in">sort</span>(a.<span class="hljs-built_in">begin</span>(), a.<span class="hljs-built_in">end</span>());<br>        a.<span class="hljs-built_in">erase</span>(<span class="hljs-built_in">unique</span>(a.<span class="hljs-built_in">begin</span>(), a.<span class="hljs-built_in">end</span>()), a.<span class="hljs-built_in">end</span>());<br><br>        cout &lt;&lt; a.<span class="hljs-built_in">size</span>() &lt;&lt; endl;<br>    }<br><br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【分讨-组合数学】三元组">【分讨/组合数学】三元组</h3><p><a href="https://www.acwing.com/problem/content/5280/">https://www.acwing.com/problem/content/5280/</a></p><blockquote><p>题意：在一个序列中如何选择一个三元组，使得三个数之积最小，给出情况数</p><p>思路：首先很容易得知，这三个数一定是序列排序后的前三个数，那么就是对这三个数可能的情况进行讨论</p><ul><li>情况1：<strong>前三个数都相等</strong> <code>（2 2 2 2 2 4 5）</code>，则就是在所有的 <code>a[0]</code> 中选3个，情况数就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>C</mi><mrow><mi>c</mi><mi>n</mi><mi>t</mi><mi mathvariant="normal">_</mi><mi>a</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo></mrow><mn>3</mn></msubsup></mrow><annotation encoding="application/x-tex">C_{cnt\_a[0]}^{3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3531em;vertical-align:-0.539em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.378em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">c</span><span class="mord mathnormal mtight">n</span><span class="mord mathnormal mtight">t</span><span class="mord mtight" style="margin-right:0.02778em;">_</span><span class="mord mathnormal mtight">a</span><span class="mopen mtight">[</span><span class="mord mtight">0</span><span class="mclose mtight">]</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.539em;"><span></span></span></span></span></span></span></span></span></span></li><li>情况2：<strong>前三个数中有两个数相等</strong>，由于数组经过了排序，因此情况2分为两种情况<ul><li>前两个数相等 <code>（1 1 3 3 3 5）</code>，则就是在所有的 <code>a[2]</code> 中选1个，情况数就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>C</mi><mrow><mi>c</mi><mi>n</mi><mi>t</mi><mi mathvariant="normal">_</mi><mi>a</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow><mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">C_{cnt\_a[2]}^{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3531em;vertical-align:-0.539em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.378em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">c</span><span class="mord mathnormal mtight">n</span><span class="mord mathnormal mtight">t</span><span class="mord mtight" style="margin-right:0.02778em;">_</span><span class="mord mathnormal mtight">a</span><span class="mopen mtight">[</span><span class="mord mtight">2</span><span class="mclose mtight">]</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.539em;"><span></span></span></span></span></span></span></span></span></span></li><li>后两个数相等 <code>（1 2 2 2 4 6）</code>，则就是在所有的 <code>a[1]</code> 中选2个，情况数就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>C</mi><mrow><mi>c</mi><mi>n</mi><mi>t</mi><mi mathvariant="normal">_</mi><mi>a</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">C_{cnt\_a[1]}^{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3531em;vertical-align:-0.539em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.378em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">c</span><span class="mord mathnormal mtight">n</span><span class="mord mathnormal mtight">t</span><span class="mord mtight" style="margin-right:0.02778em;">_</span><span class="mord mathnormal mtight">a</span><span class="mopen mtight">[</span><span class="mord mtight">1</span><span class="mclose mtight">]</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.539em;"><span></span></span></span></span></span></span></span></span></span></li></ul></li><li>情况3：<strong>前三个数中全都不相等</strong> <code>（1 2 4 4 4 5 7）</code>，这就是在所有的 <code>a[2]</code> 中选1个，情况数就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>C</mi><mrow><mi>c</mi><mi>n</mi><mi>t</mi><mi mathvariant="normal">_</mi><mi>a</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow><mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">C_{cnt\_a[2]}^{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3531em;vertical-align:-0.539em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.378em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">c</span><span class="mord mathnormal mtight">n</span><span class="mord mathnormal mtight">t</span><span class="mord mtight" style="margin-right:0.02778em;">_</span><span class="mord mathnormal mtight">a</span><span class="mopen mtight">[</span><span class="mord mtight">2</span><span class="mclose mtight">]</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.539em;"><span></span></span></span></span></span></span></span></span></span></li></ul><p>可以发现上述情况2的第一种和情况3的答案相等，故分为三种情况即可。</p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-keyword">typedef</span> <span class="hljs-type">long</span> <span class="hljs-type">long</span> ll;<br><br><span class="hljs-comment">// 计算组合数 C_{a}^{b}</span><br><span class="hljs-function">ll <span class="hljs-title">calc</span><span class="hljs-params">(ll a, ll b)</span> </span>{<br>    ll fz = <span class="hljs-number">1</span>, fm = <span class="hljs-number">1</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>, num = a; i &lt; b; i++, num--) {<br>        fz *= num;<br>    }<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= b; i++) {<br>        fm *= i;<br>    }<br>    <span class="hljs-keyword">return</span> fz / fm;<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-type">int</span> n; cin &gt;&gt; n;<br>    <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">a</span><span class="hljs-params">(n)</span></span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        cin &gt;&gt; a[i];<br>    }<br>    <br>    <span class="hljs-built_in">sort</span>(a.<span class="hljs-built_in">begin</span>(), a.<span class="hljs-built_in">begin</span>() + n);<br>    <br>    ll res = <span class="hljs-number">0</span>;<br>    <br>    <span class="hljs-keyword">if</span> (a[<span class="hljs-number">0</span>] == a[<span class="hljs-number">1</span>] &amp;&amp; a[<span class="hljs-number">1</span>] == a[<span class="hljs-number">2</span>]) {<br>        <span class="hljs-comment">// 情况1：前三个数都相等（2 2 2 2 2 4 5），则就是在所有的a[0]中选3个</span><br>        ll cnt = <span class="hljs-built_in">count</span>(a.<span class="hljs-built_in">begin</span>(), a.<span class="hljs-built_in">begin</span>() + n, a[<span class="hljs-number">0</span>]);<br>        res = <span class="hljs-built_in">calc</span>(cnt, <span class="hljs-number">3</span>);<br>    } <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (a[<span class="hljs-number">0</span>] == a[<span class="hljs-number">1</span>] || (a[<span class="hljs-number">0</span>] != a[<span class="hljs-number">1</span>] &amp;&amp; a[<span class="hljs-number">1</span>] != a[<span class="hljs-number">2</span>])) {<br>        <span class="hljs-comment">// 情况2：前两个数相等（1 1 3 3 3 4 5） or 三个数都不相等（1 2 4 4 4 5 7），则就是在所有的a[2]中选2个</span><br>        res = <span class="hljs-built_in">count</span>(a.<span class="hljs-built_in">begin</span>(), a.<span class="hljs-built_in">begin</span>() + n, a[<span class="hljs-number">2</span>]);<br>    } <span class="hljs-keyword">else</span> {<br>        <span class="hljs-comment">// 情况3：后两个数相等（1 2 2 2 4 6），则就是在所有的a[1]中选2个</span><br>        ll cnt = <span class="hljs-built_in">count</span>(a.<span class="hljs-built_in">begin</span>(), a.<span class="hljs-built_in">begin</span>() + n, a[<span class="hljs-number">1</span>]);<br>        res = <span class="hljs-built_in">calc</span>(cnt, <span class="hljs-number">2</span>);<br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>    <br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【分讨】删除元素">【分讨】删除元素</h3><p><a href="https://www.acwing.com/problem/content/5281/">https://www.acwing.com/problem/content/5281/</a></p><blockquote><p>题意：给定一个排列为1~n，定义一个数“有价值”为当前数的前面没有数比当前的数大。现在需要删除一个数，使得序列中增加尽可能多的“有价值”的数，如果这个数有多个，则删除最小的那个数</p><p>最开始想到的思路：枚举每一个数，如果删除，则序列中会增加多少个“有价值”的数，算法设计如下：</p><ol><li>首先判断每一个数是否是有价值的数<ul><li>创建一个变量来记录当前数的前面序列的最大值</li><li>比较判断当前数和前方最大值的关系，如果小于，则无价值，反之有价值</li></ul></li><li>接着枚举每一个数，如果删除该数，则序列会损失多少个有价值的数<ul><li>首先判断自己是不是有价值的数，如果是，则当前损失值 -1</li><li>接着判断删除当前数对后续的影响 <span class="github-emoji"><span>⭐</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/2b50.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span> ：我们在枚举后续数的时候，起始的newd（前驱除掉a[i]的最大值）应该就是当前的d，只不过需要使用新变量newd而非d是因为这一步与整体无关，不可以改变整体的d。否则会出现错误，比如对于<code>4 3 5 1 2</code>，如果我们在枚举后续数的时候直接对d进行迭代，那么在第一轮d就会被更新为5，就再也无法更新了</li></ul></li><li>最终根据维护的损失值数组cnt即可求解</li><li>时间复杂度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></li></ol><p>优化的思路：</p><ol><li><p>同样是枚举每一个数，如果当前数字是有价值的数，那么cnt[a[i]]就 -1</p><ul><li>取决于当前数字前面是否有比它大的数，如果没有，那么就是有价值的</li></ul></li><li><p>接下来我们不需要遍历j=i+1到j=n-1，而是讨论当前数a[i]不是有价值数时的所有情况。现在我们想要维护cnt数组。不难发现，对于某个位置上的数a[k]，能否成为有价值的数只取决于前排序列是否有比他大的数以及大的数的个数。那么理论上我们在枚举a[i]的时候维护cnt[1~i-1]就可以确保cnt数组的正确性了。</p><ul><li>假设a[k]的前排有一个比它大的数d：那么把d去掉之后，a[k]就会从无价值变为有价值</li><li>假设a[k]的前排有至少两个比它大的数d1,d2,…dj…：那么不管去掉哪一个 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">d_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9805em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>，我们都没法让a[k]变得有价值</li></ul><p>如此一来，cnt数组就可以正确维护了</p></li><li><p>最终根据维护的损失值数组cnt即可求解</p></li><li><p>时间复杂度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></li></ol></blockquote><p>暴力：</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-type">int</span> n;<br>    cin &gt;&gt; n;<br>    <br>    <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">a</span><span class="hljs-params">(n)</span></span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        cin &gt;&gt; a[i];<br>    }<br>    <br>    <span class="hljs-comment">// 判断每个数是否是有价值的数</span><br>    <span class="hljs-type">int</span> d = <span class="hljs-number">0</span>; <span class="hljs-comment">// 前方序列中的最大值</span><br>    <span class="hljs-function">vector&lt;<span class="hljs-type">bool</span>&gt; <span class="hljs-title">is_val</span><span class="hljs-params">(n + <span class="hljs-number">1</span>)</span></span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        <span class="hljs-keyword">if</span> (a[i] &gt; d) {<br>            is_val[a[i]] = <span class="hljs-literal">true</span>;<br>            d = a[i];<br>        }<br>    }<br>    <br>    <span class="hljs-comment">// 枚举每一个数，计算删除该数之后会增加的有价值的数的个数 </span><br>    d = <span class="hljs-number">0</span>; <span class="hljs-comment">// 同上</span><br>    <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">cnt</span><span class="hljs-params">(n + <span class="hljs-number">1</span>)</span></span>; <span class="hljs-comment">// cnt[x]表示删除数x之后可以增加的有价值的数的个数</span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        <span class="hljs-keyword">if</span> (is_val[a[i]]) {<br>            cnt[a[i]]--;<br>        }<br>        <br>        <span class="hljs-type">int</span> newd = d; <span class="hljs-comment">// 去掉a[i]后的前方序列的最大值 newd</span><br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = i + <span class="hljs-number">1</span>; j &lt; n; j++) {<br>            <span class="hljs-keyword">if</span> (is_val[a[j]]) {<br>                <span class="hljs-keyword">if</span> (a[j] &lt; newd) {<br>                    cnt[a[i]]--;<br>                }<br>            } <span class="hljs-keyword">else</span> {<br>                <span class="hljs-keyword">if</span> (a[j] &gt; newd) {<br>                    cnt[a[i]]++;<br>                }<br>            }<br>            <span class="hljs-keyword">if</span> (a[j] &gt; newd) {<br>                newd = a[j];<br>            }<br>        }<br>        <br>        <span class="hljs-comment">// 更新前方序列的最大值 d</span><br>        <span class="hljs-keyword">if</span> (a[i] &gt; d) {<br>            d = a[i];<br>        }<br>    }<br><br>    <span class="hljs-comment">// 找出可以增加的有价值的数的最大个数 ma</span><br>    <span class="hljs-type">int</span> ma = -n - <span class="hljs-number">1</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        <span class="hljs-keyword">if</span> (cnt[a[i]] &gt; ma) {<br>            ma = <span class="hljs-built_in">max</span>(ma, cnt[a[i]]);<br>        }<br>    }<br><br>    <span class="hljs-comment">// 答案是满足个数的最小的数字</span><br>    <span class="hljs-type">int</span> res = <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-keyword">if</span> (cnt[i] == ma) {<br>            res = i;<br>            <span class="hljs-keyword">break</span>;<br>        }<br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>    <br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><p>优化：</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-type">int</span> n; <br>    cin &gt;&gt; n;<br>    <br>    <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">a</span><span class="hljs-params">(n)</span></span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        cin &gt;&gt; a[i];<br>    }<br>     <br>    <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">cnt</span><span class="hljs-params">(n + <span class="hljs-number">1</span>)</span></span>;     <span class="hljs-comment">// cnt[x]表示删除数x之后可以增加的有价值的数的个数</span><br>    <span class="hljs-type">int</span> d1 = <span class="hljs-number">0</span>, d2 = <span class="hljs-number">0</span>;         <span class="hljs-comment">// d1: 最大值, d2: 次大值</span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        <span class="hljs-keyword">if</span> (a[i] &gt; d1) {<br>            <span class="hljs-comment">// 前方序列 没有数比当前的数大</span><br>            cnt[a[i]]--;<br>        } <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (a[i] &gt; d2) {<br>            <span class="hljs-comment">// 前方序列 只有一个数比当前大</span><br>            cnt[d1]++;<br>        }<br>        <br>        <span class="hljs-comment">// 更新最大值d1和次大值d2</span><br>        <span class="hljs-keyword">if</span> (a[i] &gt; d1) d2 = d1, d1 = a[i];<br>        <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (a[i] &gt; d2) d2 = a[i];<br>    }<br>    <br>    <span class="hljs-comment">// 找到可以增加的最大值</span><br>    <span class="hljs-type">int</span> ma = <span class="hljs-number">-1</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-keyword">if</span> (cnt[i] &gt; ma) {<br>            ma = cnt[i];<br>        }<br>    }<br>    <br>    <span class="hljs-comment">// 取元素的最小值</span><br>    <span class="hljs-type">int</span> res = n + <span class="hljs-number">1</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-keyword">if</span> (cnt[i] == ma) {<br>            res = i;<br>            <span class="hljs-keyword">break</span>;<br>        }<br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>    <br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【构造】Sorting-with-Twos">【构造】Sorting with Twos</h3><p><a href="https://codeforces.com/contest/1891/problem/A">https://codeforces.com/contest/1891/problem/A</a></p><blockquote><p>题意：给定一个序列，现在需要通过以下方法对序列进行升序排序</p><ul><li>可以选择前 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>m</mi></msup></mrow><annotation encoding="application/x-tex">2^m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6644em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">m</span></span></span></span></span></span></span></span></span></span></span> 个数执行 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span> 的操作（保证不会越界的情况下）</li></ul><p>现在需要确定给定的序列经过k次上述后能否变为升序序列</p><p>思路：我们可以将每 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>k</mi></msup></mrow><annotation encoding="application/x-tex">2^k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span></span></span> 个数看成一个数，可以不断的-1。那么可以发现执行一定的次数后，一定可以实现升序序列。但是现在是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>k</mi></msup></mrow><annotation encoding="application/x-tex">2^k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span></span></span> 个数，那么只需要这相邻区段的数是升序即可，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>→</mo><msup><mn>2</mn><mi>k</mi></msup></mrow><annotation encoding="application/x-tex">2^{k-1} \to 2^k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span></span></span> 之间的数是升序即可。按区间进行判断即可</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">judge</span><span class="hljs-params">(vector&lt;<span class="hljs-type">int</span>&gt;&amp; a, <span class="hljs-type">int</span> l, <span class="hljs-type">int</span> r, <span class="hljs-type">bool</span>&amp; ok, <span class="hljs-type">int</span> n)</span> </span>{<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = l + <span class="hljs-number">1</span>; i &lt;= r &amp;&amp; i &lt;= n; i++) {<br>        <span class="hljs-keyword">if</span> (a[i] &lt; a[i - <span class="hljs-number">1</span>]) {<br>            ok = <span class="hljs-literal">false</span>;<br>            <span class="hljs-keyword">return</span>;<br>        }<br>    }<br>}<br> <br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-type">int</span> n;<br>    cin &gt;&gt; n;<br> <br>    <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">a</span><span class="hljs-params">(n + <span class="hljs-number">1</span>)</span></span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        cin &gt;&gt; a[i];<br>    }<br> <br>    <span class="hljs-type">bool</span> ok = <span class="hljs-literal">true</span>;<br>    <span class="hljs-built_in">judge</span>(a, <span class="hljs-number">3</span>, <span class="hljs-number">4</span>, ok, n);<br>    <span class="hljs-built_in">judge</span>(a, <span class="hljs-number">5</span>, <span class="hljs-number">8</span>, ok, n);<br>    <span class="hljs-built_in">judge</span>(a, <span class="hljs-number">9</span>, <span class="hljs-number">16</span>, ok, n);<br>    <span class="hljs-built_in">judge</span>(a, <span class="hljs-number">17</span>, <span class="hljs-number">20</span>, ok, n);<br> <br>    <span class="hljs-keyword">if</span> (ok) {<br>        cout &lt;&lt; <span class="hljs-string">"YES\n"</span>;<br>    } <span class="hljs-keyword">else</span> {<br>        cout &lt;&lt; <span class="hljs-string">"NO\n"</span>;<br>    }<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【模拟】指针运动​-fire">【模拟】指针运动​ <span class="github-emoji"><span>🔥</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/1f525.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span></h3><p><a href="https://www.acwing.com/problem/content/description/5468/">https://www.acwing.com/problem/content/description/5468/</a></p><blockquote><ul><li>题意：给定一个序列，和一个从第一个数开始的指针，若当前指的数为 0 则输出当前数的下标结束程序，若不为 0 则全体正数减一并将指针后移，若指针越界则重新指向第一个数。问最终指向第几个数</li><li>思路：<ul><li>第一层思路：纯暴力。即按照题意进行模拟，时间复杂度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>×</mo><mi>max</mi><mo>⁡</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \times \max (a_i))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">max</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">))</span></span></span></span></li><li>第二层思路：枚举优化。假设当前枚举到了第 k 次，若第一次遇到当前的数小于 k，则该数就是终止指向的数，时间复杂度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>×</mo><mi>max</mi><mo>⁡</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \times \max(a_i))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">max</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">))</span></span></span></span></li><li>第三层思路：数列优化。</li></ul></li><li>时间复杂度：</li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><br></code></pre></td></tr></tbody></table></figure><h3 id="【哈希】套餐设计">【哈希】套餐设计</h3><p><a href="https://www.acwing.com/problem/content/5480/">https://www.acwing.com/problem/content/5480/</a></p><blockquote><p>题意：给定 n 个食物，其中可能有相同种类的，现在需要设计一个食物套餐包含 k 个食物，问如何设计套餐可以使得产生的套餐数最多，给出最多数量的套餐数</p><p>思路：如果不限定套餐中食物的种类数，那么我们可以从最多可生产的套餐数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">⌊</mo><mfrac><mi>n</mi><mi>k</mi></mfrac><mo fence="true">⌋</mo></mrow><annotation encoding="application/x-tex">\left \lfloor \frac{n}{k} \right \rfloor</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size1">⌊</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size1">⌋</span></span></span></span></span></span> 开始降序检查，直到一套都无法生产为止（即一套需要的食物数超过了总的食物数）。检查的逻辑很简单，我们知道对于当前的套餐数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>，每一种食物的数量如果超过了 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>，就至少可以对套餐贡献 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 个食物的数量。我们统计当前需要套餐数的局面下，所有食物可以贡献的食物数量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi><mi>n</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">cnt</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span></span></span></span>，如果超过了需要的套餐数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>，显然是可以满足需求的套餐数的。为了求得最大的套餐数，从降序开始检查直到可以满足即可。</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;unordered_map&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><span class="hljs-keyword">typedef</span> <span class="hljs-type">long</span> <span class="hljs-type">long</span> ll;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">110</span>;<br><br><span class="hljs-type">int</span> n, k, a[N];<br>unordered_map&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt; v;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; k &gt;&gt; n;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        cin &gt;&gt; a[i];<br>        v[a[i]]++;<br>    }<br>    <br>    <span class="hljs-type">int</span> res = <span class="hljs-number">0</span>;<br>    <br>    <span class="hljs-type">int</span> i; <span class="hljs-comment">// 当前需要生产的套餐数</span><br>    <span class="hljs-keyword">for</span> (i = n / k; i &gt;= <span class="hljs-number">1</span>; i--) {<br>        <span class="hljs-comment">// 所有食物对于当前套餐需要的食物数量可以贡献的最大食物数量</span><br>        <span class="hljs-type">int</span> cnt = <span class="hljs-number">0</span>; <br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; x: v) {<br>            <span class="hljs-comment">// 每种食物对于当前套餐需要的食物数量可以贡献的最大食物数量</span><br>            cnt += x.second / i;<br>        }<br>        <span class="hljs-keyword">if</span> (cnt &gt;= k) {<br>            res = i;<br>            <span class="hljs-keyword">break</span>;<br>        }<br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>), cin.<span class="hljs-built_in">tie</span>(<span class="hljs-number">0</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-number">0</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span>(T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【贪心-分讨】分班">【贪心/分讨】分班</h3><p><a href="https://www.acwing.com/problem/content/5481/">https://www.acwing.com/problem/content/5481/</a></p><blockquote><p>题意：给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>×</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">n \times k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个学生，需要将这些学生分成 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 组，每组 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个人，第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 个学生有一个属性值 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">a_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 。问如何分组可以使得在满足任意两组学生中最小值相差不超过 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>i</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">lim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">im</span></span></span></span> 的情况下，所有组最小值之和最大，给出这个最大值</p><p>思路：一开始觉得是二分答案，但是不知道在已知答案的情况下如何检查，因为我不知道应该如何分组最优，未遂。重读题目发现题中的 lim 约束不是相邻两组，而是任意两组，思路打开，那我直接根据这个条件将满足约束的所有学生枚举出来（很显然是相对于数值最小的学生而言），然后设计如何将这些学生安排到 n 组且保证他们是组里最小的即可。先看下面的学生编号 - 学生属性示意图：</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403030040949.png" alt="学生编号 - 学生属性示意图"></p><p>很显然我们需要将这 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>×</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">n \times k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个学生按数值升序排序，为了满足任意两组的最小值不超过 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>i</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">lim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">im</span></span></span></span>，我们找到前 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span> 个学生满足比最小的学生超过的数不超过 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>i</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">lim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">im</span></span></span></span> 即可。接下来就是从这 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span> 个学生中选择出 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个安排到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个组中，即可满足题中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>i</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">lim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">im</span></span></span></span> 的限制，现在就是考虑如何分组可以使得答案最大了。不难发现一共有三种情况</p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mo>&lt;</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">r&lt;n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> ：即可选择人数都没有组数多。那么分完组以后肯定无法满足任意两组的最小值不超过 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>i</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">lim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">im</span></span></span></span>，直接输出 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 即可</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mo>=</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">r=n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> ：即可选择人数和组数相等。这种情况最好理解，就是将这 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 个人安排到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 组，答案就是这 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 个人数值之和</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mo>&gt;</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">r&gt;n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> ：即可选择人数多于组数。如果纯粹的贪心，将第一个人放在第一组，然后从第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span> 个人开始将 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 个人安排在剩余的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 组。这样的确可以满足答案最大，但是忽略了一个点就是从第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span> 个人开始的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mo>−</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">r-n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个人何去何从？如果全部安排到了第一组那万事大吉，如果第一组塞不进去就不得不安排到别的组，这样答案就不是上面计算的结果了，应该更小才对！那么应该如何解决这个问题呢？我们可以从往第一组塞剩余的人得到启发：如果第一组可以容纳剩余的人活着还有空位就已经满足了第二种情况，就可以直接计算结果，如果第一组无法容纳从第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span> 个人开始的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mo>−</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">r-n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个人，就将第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>+</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">1+k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个人作为一组的最小值，继续贪心的容纳紧跟着的人，从而确保剩下的人数值尽可能的大。如此循环知道出现 <strong>剩余的人的数量=还未安排最小值的组的数量</strong> 为止，计算答案总和即可！</li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><span class="hljs-keyword">typedef</span> <span class="hljs-type">long</span> <span class="hljs-type">long</span> ll;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">1e5</span> + <span class="hljs-number">10</span>;<br><br>ll n, k, lim, a[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; k &gt;&gt; lim;<br><br>    <span class="hljs-type">int</span> num = n * k;<br><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= num; i++) cin &gt;&gt; a[i];<br><br>    <span class="hljs-built_in">sort</span>(a + <span class="hljs-number">1</span>, a + num + <span class="hljs-number">1</span>);<br><br>    <span class="hljs-comment">// 找到最大的满足 lim 限制的人的下标 r - 线性或二分均可</span><br>    <span class="hljs-type">int</span> l = <span class="hljs-number">1</span>, r = num;<br>    <span class="hljs-keyword">while</span> (l &lt; r) {<br>        <span class="hljs-type">int</span> mid = (l + r + <span class="hljs-number">1</span>) &gt;&gt; <span class="hljs-number">1</span>;<br>          <span class="hljs-keyword">if</span> (a[mid] &lt;= a[<span class="hljs-number">1</span>] + lim) l = mid;<br>          <span class="hljs-keyword">else</span> r = mid - <span class="hljs-number">1</span>;<br>    }<br><br>    ll res = <span class="hljs-number">0</span>;<br><br>    <span class="hljs-keyword">if</span> (r &lt; n) res = <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">else</span> {<br>        res += a[<span class="hljs-number">1</span>];<br>        <span class="hljs-type">int</span> idx = <span class="hljs-number">2</span>;   <span class="hljs-comment">// 指针</span><br>        <span class="hljs-type">int</span> in = <span class="hljs-number">1</span>;    <span class="hljs-comment">// 已安排作为组内最小值的人员数</span><br>        <span class="hljs-type">int</span> team = <span class="hljs-number">1</span>;  <span class="hljs-comment">// 当前组内的人员数</span><br>        <span class="hljs-keyword">while</span> (r - idx + <span class="hljs-number">1</span> &gt; n - in) { <span class="hljs-comment">// 当可选择人数多于组数时</span><br>            <span class="hljs-keyword">if</span> (team &lt; k) team++, idx++; <span class="hljs-comment">// 塞得下就继续塞</span><br>            <span class="hljs-keyword">else</span> team = <span class="hljs-number">1</span>, res += a[idx], idx++, in++; <span class="hljs-comment">// 塞不下就重开一组</span><br>        }<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = idx; i &lt;= r; i++) {<br>            res += a[i];<br>        }<br>    }<br><br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>), cin.<span class="hljs-built_in">tie</span>(<span class="hljs-number">0</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-number">0</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span>(T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【贪心-分讨】双端队列">【贪心/分讨】双端队列</h3><p><a href="https://www.acwing.com/problem/content/5484/">https://www.acwing.com/problem/content/5484/</a></p><blockquote><p>题意：给定一个序列，每次可以从序列头或尾取出一个数，在满足：按取出数的顺序是升序序列的条件下，给定的序列最多可以取出多少数？</p><p>思路：首先很容易想到的就是哪头小取哪头，因为取了较小者后较大者可以后续继续被选择，反之则不能。当然，还需要考虑的一个点就是如果较小者不符合题意，即较小者比上一次选出来的数更小从而无法组成严格升序序列时，还是得考虑较大值的那一头数。接下来考虑相等的情况，若两头数值相等，那一定只能选择一端且后续都只能从那端继续选择，我们直接两种方法都计算统计一遍可选择的数的数量，取较大者即可</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;queue&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;cstring&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;vector&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;unordered_map&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;algorithm&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;stack&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">200010</span>;<br><br><span class="hljs-type">int</span> n, a[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) cin &gt;&gt; a[i];<br>    <br>    <span class="hljs-type">int</span> res = <span class="hljs-number">0</span>, top = <span class="hljs-number">-1</span>, l = <span class="hljs-number">1</span>, r = n;<br>    <span class="hljs-keyword">while</span> (l &lt;= r) {<br>        <span class="hljs-comment">// 左端是较小值</span><br>        <span class="hljs-keyword">if</span> (a[l] &lt; a[r]) {<br>            <span class="hljs-keyword">if</span> (a[l] &gt; top) {<br>                res++;<br>                top = a[l++];<br>            } <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (a[r] &gt; top) {<br>                res++;<br>                top = a[r--];<br>            } <span class="hljs-keyword">else</span> {<br>                <span class="hljs-keyword">break</span>;<br>            }<br>        <span class="hljs-comment">// 右端是较小值</span><br>        } <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (a[l] &gt; a[r]) {<br>            <span class="hljs-keyword">if</span> (a[r] &gt; top) {<br>                res++;<br>                top = a[r--];<br>            } <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (a[l] &gt; top) {<br>                res++;<br>                top = a[l++];<br>            } <span class="hljs-keyword">else</span> {<br>                <span class="hljs-keyword">break</span>;<br>            }<br>        <span class="hljs-comment">// 两端数值相等</span><br>        } <span class="hljs-keyword">else</span> { <span class="hljs-comment">// a[l] == a[r]</span><br>            <span class="hljs-keyword">if</span> (a[l] &gt; top) {<br>                <span class="hljs-type">int</span> lcnt = <span class="hljs-number">1</span>, rcnt = <span class="hljs-number">1</span>;<br>                <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = l + <span class="hljs-number">1</span>; i &lt;= r; i++) {<br>                    <span class="hljs-keyword">if</span> (a[i] &gt; a[i - <span class="hljs-number">1</span>]) lcnt++;<br>                    <span class="hljs-keyword">else</span> <span class="hljs-keyword">break</span>;<br>                }<br>                <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = r - <span class="hljs-number">1</span>; i &gt;= l; i--) {<br>                    <span class="hljs-keyword">if</span> (a[i] &gt; a[i + <span class="hljs-number">1</span>]) rcnt++;<br>                    <span class="hljs-keyword">else</span> <span class="hljs-keyword">break</span>;<br>                }<br>                res += <span class="hljs-built_in">max</span>(lcnt, rcnt);<br>                <span class="hljs-keyword">break</span>;<br>            } <span class="hljs-keyword">else</span> {<br>                <span class="hljs-keyword">break</span>;<br>            }<br>        }<br>    }<br><br>    cout &lt;&lt; res;<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【思维】你说的对，但是这是签到-I">【思维】你说的对，但是这是签到 I</h3><p><a href="https://hydro.ac/d/nnu_contest/p/P1201">https://hydro.ac/d/nnu_contest/p/P1201</a></p><blockquote><p>题意：给定两个单调不减且只含有小写字母的字符串，求解其中最长公共子串长度</p><p>思路：</p><ul><li>一眼 dp？错误的，那时间复杂度就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>，显然不可能。可以发现与常规的最长公共子串的题目不同，本题中两个字符串是单调的，这个性质一定很有用。事实的确如此。</li><li>对于单调的两个字符串而言，如果<strong>连续的</strong>字符数量之和相等，即对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>c</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>c</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">c_i,c_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 之间的所有的字符对应的数量都相等 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi><mo stretchy="false">[</mo><msub><mi>c</mi><mi>k</mi></msub><mo stretchy="false">]</mo><mo>=</mo><mi>b</mi><mo stretchy="false">[</mo><msub><mi>c</mi><mi>k</mi></msub><mo stretchy="false">]</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mi>k</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo separator="true">,</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mi>j</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a[c_k]=b[c_k],(k=i+1,i+1, \cdots,j-1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">b</span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7429em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span>，则该段字符串就一定是公共子串。答案就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></munderover><mi>a</mi><mo stretchy="false">[</mo><msub><mi>c</mi><mi>k</mi></msub><mo stretchy="false">]</mo></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle  \sum_{k=i+1}^{j-1}a[c_k]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.2192em;vertical-align:-1.3604em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8588em;"><span style="top:-1.8479em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mathnormal mtight">i</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3471em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3604em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span> 或 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></munderover><mi>b</mi><mo stretchy="false">[</mo><msub><mi>c</mi><mi>k</mi></msub><mo stretchy="false">]</mo></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle  \sum_{k=i+1}^{j-1}b[c_k]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.2192em;vertical-align:-1.3604em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8588em;"><span style="top:-1.8479em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mathnormal mtight">i</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3471em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3604em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">b</span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span>。在枚举字符时，需要特判一下只有一种字符的情况。</li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Θ</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>2</mn><msup><mn>6</mn><mn>3</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta(n + 26^3)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Θ</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord">2</span><span class="mord"><span class="mord">6</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;cstring&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;vector&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;queue&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;stack&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;algorithm&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;unordered_map&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;set&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">26</span>;<br><br>string s, t;<br><span class="hljs-type">int</span> a[N], b[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; s &gt;&gt; t;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span> c: s) a[c - <span class="hljs-string">'a'</span>]++;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span> c: t) b[c - <span class="hljs-string">'a'</span>]++;<br>    <br>    <span class="hljs-type">int</span> res = <span class="hljs-number">-1</span>;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; N; i++) {<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = i; j &lt; N; j++) {<br>            <span class="hljs-keyword">if</span> (j == i) res = <span class="hljs-built_in">max</span>(res, <span class="hljs-built_in">min</span>(a[i], b[j]));<br>            <span class="hljs-keyword">else</span> {<br>                <span class="hljs-type">bool</span> ok = <span class="hljs-literal">true</span>;<br>                <span class="hljs-type">int</span> mid = <span class="hljs-number">0</span>;<br>                <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> k = i + <span class="hljs-number">1</span>; k &lt;= j - <span class="hljs-number">1</span>; k++) {<br>                    <span class="hljs-keyword">if</span> (a[k] != b[k]) ok = <span class="hljs-literal">false</span>;<br>                    mid += a[k];<br>                }<br>                <br>                <span class="hljs-keyword">if</span> (ok) {<br>                    res = <span class="hljs-built_in">max</span>(res, <span class="hljs-built_in">min</span>(a[i], b[i]) + mid + <span class="hljs-built_in">min</span>(a[j], b[j]));<br>                }<br>            }<br>        }<br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【贪心】从双倍数组中还原原数组">【贪心】从双倍数组中还原原数组</h3><p><a href="https://leetcode.cn/problems/find-original-array-from-doubled-array/description/">https://leetcode.cn/problems/find-original-array-from-doubled-array/description/</a></p><blockquote><p>题意：给定一个改变后数组，问其原始数组是什么？我们定义改变后数组为原始数组每一个元素乘以2以后加入原始数组，并随机打乱得到的数组</p><p>思路：答案与下标无关，我们考虑对原始数组升序排序。在哈希统计每一个元素出现的频率后，可以枚举每一个元素，如果当前元素与当前元素的二倍都存在，则说明当前元素一定存在于原始数组中，我们将其加入答案并对两个数的频率统计减一。</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span> {<br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">findOriginalArray</span><span class="hljs-params">(vector&lt;<span class="hljs-type">int</span>&gt;&amp; changed)</span> </span>{<br>        vector&lt;<span class="hljs-type">int</span>&gt; res;<br><br>        <span class="hljs-comment">// 剪枝：合法原始数组一定是偶数个</span><br>        <span class="hljs-type">int</span> n = changed.<span class="hljs-built_in">size</span>();<br>        <span class="hljs-keyword">if</span> (n &amp; <span class="hljs-number">1</span>) <span class="hljs-keyword">return</span> {};<br><br>        <span class="hljs-built_in">sort</span>(changed.<span class="hljs-built_in">begin</span>(), changed.<span class="hljs-built_in">end</span>());<br>        unordered_map&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt; cnt;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> x: changed) cnt[x]++;<br><br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> x: changed) {<br>            <span class="hljs-keyword">if</span> (cnt[x] == <span class="hljs-number">0</span>) <span class="hljs-keyword">continue</span>;<br><br>            cnt[x]--;<br>            <span class="hljs-keyword">if</span> (cnt[x * <span class="hljs-number">2</span>]) {<br>                cnt[x * <span class="hljs-number">2</span>]--;<br>                res.<span class="hljs-built_in">push_back</span>(x);<br>            }<br>            <span class="hljs-keyword">else</span> {<br>                <span class="hljs-keyword">return</span> {};<br>            }<br>        }<br><br>        <span class="hljs-keyword">return</span> res;<br>    }<br>};<br></code></pre></td></tr></tbody></table></figure><h3 id="【前缀和-双指针】可获得的最大点数">【前缀和/双指针】可获得的最大点数</h3><p><a href="https://leetcode.cn/problems/maximum-points-you-can-obtain-from-cards/description/">https://leetcode.cn/problems/maximum-points-you-can-obtain-from-cards/description/</a></p><blockquote><p>题意：给定一个序列，每次可以从头或尾取一个元素并弹出该元素，问如何取数可以使得取到的数总和最大</p><p>思路：<strong>前缀和、双指针</strong>。很显然，我们选到的数的组成一定是原数组的头部一部分和尾部一部分，从反面思考，其实就是让中间一部分的数值总和最小！因此我们用前缀和记录一下然后双指针枚举序列即可。之所以记录本地是因为双指针好久没写了，调了好久 : (</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span> {<br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">maxScore</span><span class="hljs-params">(vector&lt;<span class="hljs-type">int</span>&gt;&amp; cardPoints, <span class="hljs-type">int</span> k)</span> </span>{<br>        <span class="hljs-type">int</span> n = cardPoints.<span class="hljs-built_in">size</span>();<br>        <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">s</span><span class="hljs-params">(n + <span class="hljs-number">1</span>, <span class="hljs-number">0</span>)</span></span>;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>            s[i] = s[i - <span class="hljs-number">1</span>] + cardPoints[i - <span class="hljs-number">1</span>];<br>        }<br><br>        <span class="hljs-type">int</span> res = <span class="hljs-number">-1</span>;<br>        <span class="hljs-type">int</span> l = <span class="hljs-number">1</span>, r = l + n - k - <span class="hljs-number">1</span>;<br>        <span class="hljs-keyword">while</span> (r &lt;= n) {<br>            res = <span class="hljs-built_in">max</span>(res, s[n] - (s[r] - s[l - <span class="hljs-number">1</span>]));<br>            l++, r++;<br>        }<br><br>        <span class="hljs-keyword">return</span> res;<br>    }<br>};<br></code></pre></td></tr></tbody></table></figure><h3 id="【思维】确定两个字符串是否接近">【思维】确定两个字符串是否接近</h3><p><a href="https://leetcode.cn/problems/determine-if-two-strings-are-close/description/">https://leetcode.cn/problems/determine-if-two-strings-are-close/description/</a></p><blockquote><p>题意：给定两个字符串，可以对两个字符串进行任意次下面的操作：问在操作后使得两个字符串完全相等</p><ol><li>交换一个字符串中的任意两个字符</li><li>交换一个字符串中的任意两个相同字符的集合。比如 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><munder accentunder="true"><mrow><mi>a</mi><mi>a</mi></mrow><mo stretchy="true">‾</mo></munder><mi>c</mi><munder accentunder="true"><mrow><mi>a</mi><mi>b</mi><mi>b</mi></mrow><mo stretchy="true">‾</mo></munder><mo>→</mo><munder accentunder="true"><mrow><mi>b</mi><mi>b</mi></mrow><mo stretchy="true">‾</mo></munder><mi>c</mi><munder accentunder="true"><mrow><mi>b</mi><mi>a</mi><mi>a</mi></mrow><mo stretchy="true">‾</mo></munder></mrow><annotation encoding="application/x-tex">\underline {aa} c \underline{abb} \to \underline{bb}c\underline{baa}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8944em;vertical-align:-0.2em;"></span><span class="mord underline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.4306em;"><span style="top:-2.84em;"><span class="pstrut" style="height:3em;"></span><span class="underline-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">aa</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2em;"><span></span></span></span></span></span><span class="mord mathnormal">c</span><span class="mord underline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.84em;"><span class="pstrut" style="height:3em;"></span><span class="underline-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">abb</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8944em;vertical-align:-0.2em;"></span><span class="mord underline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.84em;"><span class="pstrut" style="height:3em;"></span><span class="underline-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">bb</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2em;"><span></span></span></span></span></span><span class="mord mathnormal">c</span><span class="mord underline"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6944em;"><span style="top:-2.84em;"><span class="pstrut" style="height:3em;"></span><span class="underline-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">baa</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2em;"><span></span></span></span></span></span></span></span></span></li></ol><p>思路：一道纯纯思维题。首先对于第一个操作，我们可以直接对两个字符串排序即可，至于第二个操作，我们需要确保两个字符串哈希值中，所有的键全部相等，即两个字符串排序去重后应该完全相等，同时还需要确保两个字符串哈希的结果中值排序的结果完全相等，这样就可以通过操作二使得两个字符串完全相等</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span> {<br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-function"><span class="hljs-type">bool</span> <span class="hljs-title">closeStrings</span><span class="hljs-params">(string word1, string word2)</span> </span>{<br>        <span class="hljs-comment">// 剪枝：两个字符串长度不相等，则操作后一定不可能相等</span><br>        <span class="hljs-keyword">if</span> (word<span class="hljs-number">1.</span><span class="hljs-built_in">size</span>() != word<span class="hljs-number">2.</span><span class="hljs-built_in">size</span>()) <span class="hljs-keyword">return</span> <span class="hljs-literal">false</span>;<br><br>        <span class="hljs-comment">// 执行操作1</span><br>        <span class="hljs-built_in">sort</span>(word<span class="hljs-number">1.</span><span class="hljs-built_in">begin</span>(), word<span class="hljs-number">1.</span><span class="hljs-built_in">end</span>());<br>        <span class="hljs-built_in">sort</span>(word<span class="hljs-number">2.</span><span class="hljs-built_in">begin</span>(), word<span class="hljs-number">2.</span><span class="hljs-built_in">end</span>());<br>        <br>        <span class="hljs-comment">// 剪枝：若两个字符串排序后相等就不需要执行操作2了，已经相等了</span><br>        <span class="hljs-keyword">if</span> (word1 == word2) <span class="hljs-keyword">return</span> <span class="hljs-literal">true</span>;<br><br>        <span class="hljs-comment">// 哈希两个字符串的字符与对应字符数量</span><br>        unordered_map&lt;<span class="hljs-type">char</span>, <span class="hljs-type">int</span>&gt; a, b;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; c: word1) a[c]++;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; c: word2) b[c]++;<br>        <br>        <span class="hljs-comment">// 剪枝：字符种类数不一样，则操作2后一定不可能相等</span><br>        <span class="hljs-keyword">if</span> (a.<span class="hljs-built_in">size</span>() != b.<span class="hljs-built_in">size</span>()) <span class="hljs-keyword">return</span> <span class="hljs-literal">false</span>;<br>        <br>        <span class="hljs-comment">// 剪枝：字符种类数一样但是种类不一样，则操作2后一定不可能相等</span><br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span> it: a) {<br>            <span class="hljs-keyword">if</span> (b[it.first] == <span class="hljs-number">0</span>) {<br>                <span class="hljs-keyword">return</span> <span class="hljs-literal">false</span>;<br>            }<br>        }<br>        <br>        <span class="hljs-comment">// 执行操作2：比较所有种类字符的数量</span><br>        vector&lt;<span class="hljs-type">int</span>&gt; va, vb;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; it: a) va.<span class="hljs-built_in">push_back</span>(it.second);<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; it: b) vb.<span class="hljs-built_in">push_back</span>(it.second);<br>        <span class="hljs-built_in">sort</span>(va.<span class="hljs-built_in">begin</span>(), va.<span class="hljs-built_in">end</span>());<br>        <span class="hljs-built_in">sort</span>(vb.<span class="hljs-built_in">begin</span>(), vb.<span class="hljs-built_in">end</span>());<br>        <span class="hljs-keyword">return</span> va == vb;<br>    }<br>};<br></code></pre></td></tr></tbody></table></figure><h3 id="【模拟】同位字符串连接的最小长度">【模拟】同位字符串连接的最小长度</h3><p><a href="https://leetcode.cn/problems/minimum-length-of-anagram-concatenation/">https://leetcode.cn/problems/minimum-length-of-anagram-concatenation/</a></p><blockquote><p>题意：给定一个由若干个同位字符串 t 组成的字符串 s，问这个同位字符串最短是什么？定义同位字符串为字符数量相等且每种字符的数量相等的字符串</p><p>错误思路：很显然的一个分配问题。我们预统计每一种字符的数量，假设答案同位字符串长度为 len，则显然就需要将所有的字符等分为 <code>s.size() / len</code> 份，每一份的每一种字符数量均相等。也就是每一种字符都需要分为 <code>s.size() / len</code> 份。为了最小化 len，就需要分出尽可能多的同位字符串。每一种字符可以分出的份数是其因数，取所有字符可分出份数的最大值就是求解每一种字符数量的最大公因数！于是问题就转化为了求解每一种字符数量的最大公因数</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p><p>正确思路：上述思路忽略了同位字符串必须存在于原始字符串中这个约束。正确做法就更加无脑了，直接枚举字符串 s 的每一个因数，然后检查所有长度的子串是否可以作为同位字符串即可</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>128</mn><mo>×</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(128 \times n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">128</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><p>WA code</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span> {<br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">minAnagramLength</span><span class="hljs-params">(string s)</span> </span>{<br>        unordered_map&lt;<span class="hljs-type">char</span>, <span class="hljs-type">int</span>&gt; a;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span> c: s) a[c]++;<br>        <br>        vector&lt;<span class="hljs-type">int</span>&gt; v;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; it: a) v.<span class="hljs-built_in">push_back</span>(it.second);<br>        <br>        <span class="hljs-type">int</span> gcd;<br>        <span class="hljs-keyword">if</span> (v.<span class="hljs-built_in">size</span>() == <span class="hljs-number">1</span>) gcd = v[<span class="hljs-number">0</span>];<br>        <span class="hljs-keyword">else</span> {<br>            gcd = __gcd(v[<span class="hljs-number">0</span>], v[<span class="hljs-number">1</span>]);<br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">2</span>; i &lt; v.<span class="hljs-built_in">size</span>(); i++) {<br>                gcd = __gcd(gcd, v[i]);<br>            }<br>        }<br>        <br>        <span class="hljs-keyword">return</span> s.<span class="hljs-built_in">size</span>() / gcd;<br>    }<br>};<br></code></pre></td></tr></tbody></table></figure><p>AC code</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span> {<br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">minAnagramLength</span><span class="hljs-params">(string s)</span> </span>{<br>        <span class="hljs-type">int</span> n = s.<span class="hljs-built_in">size</span>();<br>        <br>        vector&lt;<span class="hljs-type">int</span>&gt; v;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n / i; i++) {<br>            <span class="hljs-keyword">if</span> (n % i == <span class="hljs-number">0</span>) {<br>                v.<span class="hljs-built_in">push_back</span>(i);<br>                v.<span class="hljs-built_in">push_back</span>(n / i);<br>            }<br>        }<br><br>        <span class="hljs-type">int</span> res = n + <span class="hljs-number">1</span>;<br><br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> len: v) {<br>            <span class="hljs-type">bool</span> ok = <span class="hljs-literal">true</span>;<br><br>            string pre = s.<span class="hljs-built_in">substr</span>(<span class="hljs-number">0</span>, len);<br>            <span class="hljs-built_in">sort</span>(pre.<span class="hljs-built_in">begin</span>(), pre.<span class="hljs-built_in">end</span>());<br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = len; i &lt;= n - len; i += len) {<br>                string now = s.<span class="hljs-built_in">substr</span>(i, len);<br>                <span class="hljs-built_in">sort</span>(now.<span class="hljs-built_in">begin</span>(), now.<span class="hljs-built_in">end</span>());<br>                <span class="hljs-keyword">if</span> (now != pre) {<br>                    ok = <span class="hljs-literal">false</span>;<br>                    <span class="hljs-keyword">break</span>;<br>                }<br>            }<br><br>            <span class="hljs-keyword">if</span> (ok) {<br>                res = <span class="hljs-built_in">min</span>(res, len);<br>            }<br>        }<br><br>        <span class="hljs-keyword">return</span> res;<br>    }<br>};<br></code></pre></td></tr></tbody></table></figure>]]></content>
+    <content type="html"><![CDATA[<h2 id="后端开发指南">后端开发指南</h2><h2 id="前言">前言</h2><p>本文旨在介绍后端开发的基本理念和路径规划。正在不断完善中。</p><p>对于后端开发来说，语言只是一个工具和基础。除了<strong>语言本身</strong>和对应的<strong>开发框架</strong>外，其他要学的技术在后端开发中往往是通用的，比如：数据库、缓存、消息队列、搜索引擎、Linux、分布式、高并发、设计模式、架构设计等等。</p><h2 id="参考">参考</h2><p><a href="https://roadmap.sh/backend?r=backend-beginner">Backend Beginner Roadmap</a></p><p><a href="https://roadmap.sh/backend">Backend Roadmap</a></p>]]></content>
     
     
     <categories>
       
-      <category>Algorithm</category>
+      <category>BackEnd</category>
       
     </categories>
     
@@ -1422,11 +1422,11 @@
   
   
   <entry>
-    <title>geometry</title>
-    <link href="/Algorithm/geometry/"/>
-    <url>/Algorithm/geometry/</url>
+    <title>games</title>
+    <link href="/Algorithm/games/"/>
+    <url>/Algorithm/games/</url>
     
-    <content type="html"><![CDATA[<h3 id="计算几何">计算几何</h3><h3 id="【二维-数学】Minimum-Manhattan-Distance">【二维/数学】Minimum Manhattan Distance</h3><p><a href="https://codeforces.com/gym/104639/problem/J">https://codeforces.com/gym/104639/problem/J</a></p><blockquote><p>题意：给定两个圆的直径的两个点坐标，其中约束条件是两个圆一定是处在相离的两个角上。问如何在C2圆上或圆内找到一点p，使得点p到C1圆的所有点的曼哈顿距离的期望值最小</p><p>思路：</p><ul><li><p>看似需要积分，其实我们可以发现，对于点p到C1中某个点q1的曼哈顿距离，我们一定可以找到q1关于C1对称的点q2，那么点p到q1和q2的曼哈顿距离之和就是点p到C1的曼哈顿距离的两倍（证明就是中线定理）那么期望的最小值就是点p到C1的曼哈顿距离的最小值。目标转化后，我们开始思考如何计算此目标的最小值，思路如下图</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202402252321057.png" alt="image-20240116175917260"></p></li></ul><p>注意点：</p><ul><li>double的读取速度很慢，可以用 <code>int</code> or <code>long long</code> 读入，后续强制类型转换（显示 or 和浮点数计算）</li><li>注意输出答案的精度控制  <code>cout &lt;&lt; fixed &lt;&lt; setprecision(10) &lt;&lt; res &lt;&lt; "\n";</code></li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br><span class="hljs-type">double</span> x1, y1, x2, y2;<br><span class="hljs-type">long</span> <span class="hljs-type">long</span> a, b, c, d;<br><br>cin &gt;&gt; a &gt;&gt; b &gt;&gt; c &gt;&gt; d;<br>x1 = (a + c) / <span class="hljs-number">2.0</span>;<br>y1 = (b + d) / <span class="hljs-number">2.0</span>;<br><br>cin &gt;&gt; a &gt;&gt; b &gt;&gt; c &gt;&gt; d;<br>x2 = (a + c) / <span class="hljs-number">2.0</span>;<br>y2 = (b + d) / <span class="hljs-number">2.0</span>;<br><br><span class="hljs-type">double</span> r2 = <span class="hljs-built_in">sqrt</span>((a - c) * (a - c) + (b - d) * (b - d)) / <span class="hljs-number">2</span>;<br><br>cout &lt;&lt; fixed &lt;&lt; <span class="hljs-built_in">setprecision</span>(<span class="hljs-number">10</span>) &lt;&lt;  <span class="hljs-built_in">abs</span>(x1 - x2) + <span class="hljs-built_in">abs</span>(y1 - y2) - <span class="hljs-built_in">sqrt</span>(<span class="hljs-number">2</span>) * r2 &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【枚举】三角形">【枚举】三角形</h3><p><a href="https://www.acwing.com/problem/content/5383/">https://www.acwing.com/problem/content/5383/</a></p><blockquote><p>题意：给定两个直角三角形的两条直角边的长度 a, b，问能不能在坐标轴上找到三个整数点使得三点满足该直角三角形且三遍均不与坐标轴垂直</p><p>思路：首先确定两个直角边的顶点为原点 (0, 0)，接着根据对称性直接在第一象限中按照边长枚举其中一个顶点 A，对于每一个枚举到的顶点 A，按照斜率枚举最后一个顶点 B，如果满足长度以及不平行于坐标轴的条件就是合法的点。如果全部枚举完都没有找到就是没有合法组合，直接输出 NO 即可。</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>a</mi><mn>2</mn></msup><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(a^2b)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal">b</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><br><span class="hljs-type">int</span> a, b;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>cin &gt;&gt; a &gt;&gt; b;<br><br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt;= a; i++) {<br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">0</span>; j &lt;= a; j++) {<br><span class="hljs-keyword">if</span> (i * i + j * j == a * a &amp;&amp; i &amp;&amp; j) {<br><span class="hljs-type">int</span> gcd = __gcd(i, j);<br><span class="hljs-type">int</span> p = -i / gcd, q = j / gcd;<br><span class="hljs-type">int</span> y = p, x = q;<br><span class="hljs-keyword">while</span> (x * x + y * y &lt; b * b) {<br>x += q, y += p;<br>}<br><span class="hljs-keyword">if</span> (x * x + y * y == b * b &amp;&amp; x != i &amp;&amp; y != j) {<br>cout &lt;&lt; <span class="hljs-string">"YES\n"</span>;<br>cout &lt;&lt; <span class="hljs-number">0</span> &lt;&lt; <span class="hljs-string">' '</span> &lt;&lt; <span class="hljs-number">0</span> &lt;&lt; <span class="hljs-string">"\n"</span>;<br>cout &lt;&lt; i &lt;&lt; <span class="hljs-string">' '</span> &lt;&lt; j &lt;&lt; <span class="hljs-string">"\n"</span>;<br>cout &lt;&lt; x &lt;&lt; <span class="hljs-string">' '</span> &lt;&lt; y &lt;&lt; <span class="hljs-string">"\n"</span>;<br><span class="hljs-keyword">return</span>;<br>}<br>}<br>}<br>}<br><br>cout &lt;&lt; <span class="hljs-string">"NO\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br><span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//cin &gt;&gt; T;</span><br><span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br><span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【凸包】奶牛过马路">【凸包】奶牛过马路</h3><p><a href="https://www.acwing.com/problem/content/5572/">https://www.acwing.com/problem/content/5572/</a></p><blockquote><p>题意：给定一个二维坐标系，现在有两个东西可以进行平移。</p><ul><li>一个是起始位于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span> 的奶牛，可以沿着 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> 轴以 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mi>u</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0,u]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">u</span><span class="mclose">]</span></span></span></span> 的速度正向或负向移动</li><li>一个是起始位于第一象限的凸包形状的车辆，可以向 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 轴负半轴以恒定的速度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span></span> 移动</li></ul><p>在给定凸包的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个顶点的情况下，以及不允许奶牛被车辆撞到的情况下，奶牛前进到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mi>w</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,w)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mclose">)</span></span></span></span> 的最短时间</p><p>思路：由于是凸包，因此每一个顶点同时满足的性质，就可以代表整个凸包满足的性质。那么对于奶牛和凸包，就一共有三个局面：</p><ol><li>奶牛速度足够快。可以使得对于任意一个顶点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> 在到达 y 轴时，都小于此时奶牛的 y 值，用时间来表示就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mi>y</mi><mi>u</mi></mfrac><mo>≤</mo><mfrac><mi>x</mi><mi>v</mi></mfrac></mrow><annotation encoding="application/x-tex">\frac{y}{u} \le \frac{x}{v}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0925em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7475em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>≤</mo><mfrac><mi>u</mi><mi>v</mi></mfrac><mi>x</mi></mrow><annotation encoding="application/x-tex">y \le \frac{u}{v}x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">x</span></span></span></span>，表示凸包上所有的顶点都在直线 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mfrac><mi>u</mi><mi>v</mi></mfrac></mrow><annotation encoding="application/x-tex">y=\frac{u}{v}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 的下方</li><li>凸包速度足够快。可以使得对于任意一个顶点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> 在到达 y 轴时，都大于此时奶牛的 y 值，用时间来表示就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mi>y</mi><mi>u</mi></mfrac><mo>≥</mo><mfrac><mi>x</mi><mi>v</mi></mfrac></mrow><annotation encoding="application/x-tex">\frac{y}{u} \ge \frac{x}{v}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0925em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7475em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>≥</mo><mfrac><mi>u</mi><mi>v</mi></mfrac><mi>x</mi></mrow><annotation encoding="application/x-tex">y \ge \frac{u}{v}x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">x</span></span></span></span>，表示凸包上所有的顶点都在直线 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mfrac><mi>u</mi><mi>v</mi></mfrac></mrow><annotation encoding="application/x-tex">y=\frac{u}{v}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 的上方</li><li>上述两种都不满足，即直线 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mfrac><mi>u</mi><mi>v</mi></mfrac></mrow><annotation encoding="application/x-tex">y=\frac{u}{v}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 与凸包相交。此时奶牛就不能一直全速（u）前进到终点了，需要进行一定的减速 or 返回操作。有一个显然的结论就是，对于减速 or 返回的操作，都可以等价于后退一段距离后再全速（u）前进，因此对于需要减速 or 返回的操作，我们只需要计算出当前状态下应该后退的距离即可。再简洁的，我们就是需要平移直线 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mfrac><mi>u</mi><mi>v</mi></mfrac></mrow><annotation encoding="application/x-tex">y=\frac{u}{v}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>，显然只能向下平移而非向上平移，因此其实就是计算出直线 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mfrac><mi>u</mi><mi>v</mi></mfrac><mo>+</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">y=\frac{u}{v}+b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span> 的截距 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span> 使得最终的直线 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mfrac><mi>u</mi><mi>v</mi></mfrac><mo>+</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">y=\frac{u}{v}+b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span> 满足第二种条件即可。那么如何计算这个 b 呢？很显然我们可以遍历每一个顶点取满足所有顶点都在直线 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mfrac><mi>u</mi><mi>v</mi></mfrac><mo>+</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">y=\frac{u}{v}+b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span>​ 上方的最大截距 b 即可。</li></ol><p>注意点：</p><ul><li>浮点数比较注意精度误差，尽可能将除法转化为乘法</li><li>在比较相等时可以引入一个无穷小量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϵ</mi><mo>=</mo><mn>1</mn><msup><mn>0</mn><mrow><mo>−</mo><mn>7</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\epsilon=10^{-7}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ϵ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span></span></li><li>注意答案输出精度要求，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mrow><mo>−</mo><mn>6</mn></mrow></msup></mrow><annotation encoding="application/x-tex">10^{-6}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">6</span></span></span></span></span></span></span></span></span></span></span></span> 就需要我们出至少 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>7</mn></mrow><annotation encoding="application/x-tex">7</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">7</span></span></span></span> 位小数</li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><p>C++</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;cstring&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;algorithm&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iomanip&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><span class="hljs-keyword">using</span> ll = <span class="hljs-type">long</span> <span class="hljs-type">long</span>;<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br><span class="hljs-type">double</span> n, w, v, u;<br>cin &gt;&gt; n &gt;&gt; w &gt;&gt; v &gt;&gt; u;<br><br><span class="hljs-type">bool</span> up = <span class="hljs-literal">false</span>, down = <span class="hljs-literal">false</span>;<br><span class="hljs-type">double</span> b = <span class="hljs-number">0</span>;<br><br><span class="hljs-keyword">while</span> (n--) {<br><span class="hljs-type">double</span> x, y;<br>cin &gt;&gt; x &gt;&gt; y;<br>        <br>        <span class="hljs-comment">// 除法无法通过</span><br><span class="hljs-comment">// if (y / x &lt; u / v) down = true;</span><br><span class="hljs-comment">// else if (y / x &gt; u / v) up = true;</span><br>        <br>        <span class="hljs-comment">// 转化为乘法以后才能通过</span><br><span class="hljs-keyword">if</span> (y &lt; u / v * x) down = <span class="hljs-literal">true</span>;<br><span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (y &gt; u / v * x) up = <span class="hljs-literal">true</span>;<br>        <br>b = <span class="hljs-built_in">min</span>(b, y - u / v * x);<br>}<br><br><span class="hljs-keyword">if</span> (up &amp;&amp; down) cout &lt;&lt; <span class="hljs-built_in">setprecision</span>(<span class="hljs-number">7</span>) &lt;&lt; (w - b) / u;<br><span class="hljs-keyword">else</span> cout &lt;&lt; <span class="hljs-built_in">setprecision</span>(<span class="hljs-number">7</span>) &lt;&lt; w / u;<br><br><span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><p>Py</p><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br></pre></td><td class="code"><pre><code class="hljs python">read = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">map</span>(<span class="hljs-built_in">int</span>, <span class="hljs-built_in">input</span>().split())<br><br>n, w, v, u = read()<br>k = u / v<br>b = <span class="hljs-number">0</span><br>up = <span class="hljs-literal">False</span><br>down = <span class="hljs-literal">False</span><br><br><span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(n):<br>    x, y = read()<br>    <span class="hljs-keyword">if</span> (y &lt; k * x): down = <span class="hljs-literal">True</span><br>    <span class="hljs-keyword">if</span> (y &gt; k * x): up = <span class="hljs-literal">True</span><br>    b = <span class="hljs-built_in">min</span>(b, y - k * x)<br><br><span class="hljs-keyword">if</span> (up <span class="hljs-keyword">and</span> down): <span class="hljs-built_in">print</span>((w - b) / u)<br><span class="hljs-keyword">else</span>: <span class="hljs-built_in">print</span>(w / u)<br></code></pre></td></tr></tbody></table></figure>]]></content>
+    <content type="html"><![CDATA[<h3 id="博弈论">博弈论</h3><p>思考如何必胜态和必败态是什么以及如何构造这样的局面。</p><h3 id="【博弈-贪心-交互】Salyg1n-and-the-MEX-Game">【博弈/贪心/交互】Salyg1n and the MEX Game</h3><p><a href="https://codeforces.com/contest/1867/problem/C">https://codeforces.com/contest/1867/problem/C</a></p><blockquote><p>标签：博弈、贪心、交互</p><p>题面：对于给定n个数的数列，先手可以放入一个数列中不存在的数（0-1e9），后手可以从数列中拿掉一个数，但是这个数必须严格小于刚才先手放入的数。终止条件：后手没法拿数或者操作次数达到了 2n+1 次。问：当你是先手时，如何放数可以使得最终数列的 MEX 值最大。</p><p>思路：先手每次放入的数一定是当前数列的 MEX 值，此后不管后手拿掉什么数，先手都将刚刚被拿掉的数放进去即可。那么最多操作次数就刚好是 2n+1次，因为加入当前数列就是一个从 0 开始的连续整数数列，那么先手放入的数就是最大数 +1，即 n，那么假如后手从 n-1 开始拿，后手最多拿 n 次，先手再放 n 次，那么就是 2n+1 次。</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;vector&gt;</span></span><br><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span></span><br><span class="hljs-function"></span>{<br><span class="hljs-type">int</span> T; cin &gt;&gt; T;<br><br><span class="hljs-keyword">while</span> (T--)<br>{<br><span class="hljs-type">int</span> n; cin &gt;&gt; n;<br><span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">a</span><span class="hljs-params">(n)</span></span>;<br><br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i ++)<br>cin &gt;&gt; a[i];<br><br><span class="hljs-type">int</span> mex = n;<br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i ++)<br><span class="hljs-keyword">if</span> (a[i] != i)<br>{<br>mex = i;<br><span class="hljs-keyword">break</span>;<br>}<br><br>cout &lt;&lt; mex &lt;&lt; endl;<br><br><span class="hljs-type">int</span> remove;<br>cin &gt;&gt; remove;<br><br><span class="hljs-keyword">while</span> (remove != <span class="hljs-number">-1</span>)<br>{<br>cout &lt;&lt; remove &lt;&lt; endl;<br>cin &gt;&gt; remove;<br>}<br>}<br><br><span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>} <br></code></pre></td></tr></tbody></table></figure>]]></content>
     
     
     <categories>
@@ -1441,11 +1441,11 @@
   
   
   <entry>
-    <title>hashing</title>
-    <link href="/Algorithm/hashing/"/>
-    <url>/Algorithm/hashing/</url>
+    <title>geometry</title>
+    <link href="/Algorithm/geometry/"/>
+    <url>/Algorithm/geometry/</url>
     
-    <content type="html"><![CDATA[<h3 id="哈希">哈希</h3><h3 id="【哈希】分组">【哈希】分组</h3><p><a href="https://www.acwing.com/problem/content/5182/">https://www.acwing.com/problem/content/5182/</a></p><blockquote><p>存储不想同组和想同组的人员信息：存入数组，数据类型为一对字符串</p><p>存储所有的组队信息：存入哈希表，数据类型为“键:字符串”“值:一对字符串”</p><p>想要知道最终的分组情况，只需要查询数组中的队员情况与想同组 or 不想同组的成员名字是否一致即可</p><p>时间复杂度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>，空间复杂度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mtext>&nbsp;</mtext><mi>l</mi><mi>e</mi><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mi>a</mi><mi>m</mi><mi>e</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></mrow><annotation encoding="application/x-tex">O(n\ len(name))_{max}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">e</span><span class="mord mathnormal">n</span><span class="mopen">(</span><span class="mord mathnormal">nam</span><span class="mord mathnormal">e</span><span class="mclose">)</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">ma</span><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;vector&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;unordered_map&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span></span><br><span class="hljs-function"></span>{<br>    <span class="hljs-type">int</span> x;<br>    cin &gt;&gt; x;<br>    <br>    vector&lt;pair&lt;string, string&gt;&gt; <span class="hljs-built_in">X</span>(x);<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; x; i ++)<br>        cin &gt;&gt; X[i].first &gt;&gt; X[i].second;<br>    <br>    <span class="hljs-type">int</span> y;<br>    cin &gt;&gt; y;<br>    <br>    vector&lt;pair&lt;string, string&gt;&gt; <span class="hljs-built_in">Y</span>(y);<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; y; i ++)<br>        cin &gt;&gt; Y[i].first &gt;&gt; Y[i].second;<br>        <br>    <span class="hljs-type">int</span> sum;<br>    cin &gt;&gt; sum;<br>    <br>    unordered_map&lt;string, pair&lt;string, string&gt;&gt; a;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; sum; i ++)<br>    {<br>        string s, t, p;<br>        cin &gt;&gt; s &gt;&gt; t &gt;&gt; p;<br>        a[s] = {t, p};<br>        a[t] = {s, p};<br>        a[p] = {s, t};<br>    }<br>    <br>    <span class="hljs-type">int</span> res = <span class="hljs-number">0</span>;<br>    <br>    <span class="hljs-comment">// 想同组 </span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; x; i ++)<br>    {<br>        string s = X[i].first, t = X[i].second;<br>        <span class="hljs-keyword">if</span> (a[s].first != t &amp;&amp; a[s].second != t)<br>            res ++;<br>    }<br>    <br>    <span class="hljs-comment">// 不想同组 </span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; y; i ++)<br>    {<br>        string s = Y[i].first, t = Y[i].second;<br>        <span class="hljs-keyword">if</span> (a[s].first == t || a[s].second == t)<br>            res ++; <br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; endl; <br>    <br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【哈希】海港">【哈希】海港</h3><p><a href="https://www.luogu.com.cn/problem/P2058">https://www.luogu.com.cn/problem/P2058</a></p><blockquote><ul><li>题意：给定 n 艘船只的到达时间、载客信息（载客人数和每一个客人的国籍），现在需要知道对于每一艘抵达的船只，前 24 小时中抵达的客人的国籍总数</li><li>思路：本题思路很简单，就是一个队列的应用以及哈希客人国籍的过程。由于船只抵达的时间是顺序增加的，故每抵达一艘船只，就对新来的客人国籍进行哈希，为了计算前 24 小时的情况，需要对船只抵达队列进行删减，即只保留 24 小时以内的船只抵达信息。对于删除的船只信息，需要将这些船只上的客人国籍信息从哈希表中删除，故每一艘船只的访问次数为 2。</li><li><code>unordered_map </code>补充：在进行哈希统计时。为了判断当前 24 小时内客人国籍数，在删除哈希记录时，为了判断是否将当前国籍的游客全部删除时，需要统计哈希表中某个国籍是否减为了 0，我用了 <code>.count(x)</code> 内置方法，但这是不正确的，因为我想要统计的是 值 是否为 0，而 <code>.count(x)</code> 统计的是哈希表中 x 这个 键 的个数，而 <code>unordered_map </code>中是没有重复的键的，故 <code>.count(x)</code> 方法只会返回 0 或 1，返回 0 就表示当前哈希表中没有 x 这个键，返回 1 就表示哈希表中有 x 这个键，但是有这个键不代表对应的值就存在，可能是 <code>x: 0</code> 的情况，即键存在，但是值记录为 0</li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>2</mn><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(2 \sum x_i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">2</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> - 即两倍的所有游客数</li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-comment">// #include &lt;bits/stdc++.h&gt;</span><br><span class="hljs-comment">// #define int long long</span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;unordered_map&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;stack&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;queue&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">1e5</span> + <span class="hljs-number">10</span>;<br><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">Ship</span> { <span class="hljs-type">int</span> idx, t; };<br><br><span class="hljs-type">int</span> n;<br>queue&lt;Ship&gt; q;<br>vector&lt;<span class="hljs-type">int</span>&gt; G[N];<br>unordered_map&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt; cnt;<br><span class="hljs-type">int</span> kind;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n;<br><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-type">int</span> t, num;<br>        cin &gt;&gt; t &gt;&gt; num;<br>        <br>        q.<span class="hljs-built_in">push</span>({i, t});<br>        <br>        <span class="hljs-comment">// 哈希</span><br>        <span class="hljs-keyword">while</span> (num--) {<br>            <span class="hljs-type">int</span> id;<br>            cin &gt;&gt; id;<br>            <br>            <span class="hljs-keyword">if</span> (!cnt[id]) kind++;<br>            cnt[id]++;<br>            G[i].<span class="hljs-built_in">push_back</span>(id);<br>        }<br>        <br>        <span class="hljs-comment">// 去哈希</span><br>        Ship h = q.<span class="hljs-built_in">front</span>();<br>        <span class="hljs-keyword">while</span> (t - h.t &gt;= <span class="hljs-number">86400</span>) {<br>            <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; id: G[h.idx]) {<br>                cnt[id]--;<br>                <span class="hljs-keyword">if</span> (!cnt[id]) kind--;<br>            }<br>            q.<span class="hljs-built_in">pop</span>();<br>            h = q.<span class="hljs-built_in">front</span>();<br>        }<br>        <br>        cout &lt;&lt; kind &lt;&lt; <span class="hljs-string">"\n"</span>;<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【哈希】Cities-and-States-S">【哈希】Cities and States S</h3><p><a href="https://www.luogu.com.cn/problem/P3405">https://www.luogu.com.cn/problem/P3405</a></p><blockquote><p>题意：给定 n 个字符串，每一个字符串归属一个集合，现在需要统计字符串与集合名相反相等的对数</p><p>思路：很显然的哈希计数。难点有两个，如何哈希？如何计数？哈希可以采用扩展字符的方法进行，即第一个字符乘某一个较大的质数再加上第二个字符。此处采用一种较为巧妙的方法，直接将两个字符串与集合名加起来进行唯一性哈希，降低编码难度。计数有两种方式，第一种就是全部哈希结束之后，再遍历哈希表进行统计，最后将结果除二即可。第二种就是边哈希边计数，遇到相反相等的就直接计数，这样就不会重复计数了，也很巧妙</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;algorithm&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;unordered_map&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;stack&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;queue&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;set&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">int</span> n;<br>unordered_map&lt;string, <span class="hljs-type">int</span>&gt; a;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n;<br>    <br>    <span class="hljs-type">int</span> res = <span class="hljs-number">0</span>;<br>    <br>    <span class="hljs-keyword">while</span> (n--) {<br>        string s, t;<br>        cin &gt;&gt; s &gt;&gt; t;<br>        s = s.<span class="hljs-built_in">substr</span>(<span class="hljs-number">0</span>, <span class="hljs-number">2</span>);<br>        res += a[t + <span class="hljs-string">" "</span> + s] * (s != t);<br>        a[s + <span class="hljs-string">" "</span> + t]++;<br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【哈希-枚举-思维】Torn-Lucky-Ticket">【哈希/枚举/思维】Torn Lucky Ticket</h3><p><a href="https://codeforces.com/contest/1895/problem/C">https://codeforces.com/contest/1895/problem/C</a></p><blockquote><p>题意: 给定一个长度为 n 的字符串数组 nums, 数组的每一个元素长度不超过 5 且仅由数字组成. 问能找到多少对 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo separator="true">,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i,j)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">)</span></span></span></span> 可以使得拼接后的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi><mi>s</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo>+</mo><mi>n</mi><mi>u</mi><mi>m</mi><mi>s</mi><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">nums[i]+nums[j]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">]</span></span></span></span> 长度为偶数且左半部分的数字之和与右半部分的数字之和相等.</p><p>思路:</p><ul><li><p>首先最暴力的做法就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 枚举,所有的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo separator="true">,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i,j)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">)</span></span></span></span> 然后 check 合法性.</p></li><li><p>尝试优化掉第二层的枚举循环. 对于第二层循环, 我们就是要寻找合适的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi><mi>s</mi><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">nums[j]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">]</span></span></span></span> 并和当前的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi><mi>s</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">nums[i]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span></span></span></span> 拼接. 显然我们可以通过扫描当前的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi><mi>s</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">nums[i]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span></span></span></span> 并 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 的计算出所有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>e</mi><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mi>u</mi><mi>m</mi><mi>s</mi><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>≤</mo><mi>l</mi><mi>e</mi><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mi>u</mi><mi>m</mi><mi>s</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">len(nums[j])\le len(nums[i])</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">e</span><span class="mord mathnormal">n</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">])</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">e</span><span class="mord mathnormal">n</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">])</span></span></span></span> 且可以和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi><mi>s</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">nums[i]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span></span></span></span> 匹配的字符串的 <code>[长度][数字和]</code> 信息, 只需要一个二维数组预存储每一个字符串的 <strong>长度</strong> <strong>数字和</strong> 信息即可.</p></li><li><p>那么对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>e</mi><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mi>u</mi><mi>m</mi><mi>s</mi><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>&gt;</mo><mi>l</mi><mi>e</mi><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mi>u</mi><mi>m</mi><mi>s</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">len(nums[j])&gt; len(nums[i])</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">e</span><span class="mord mathnormal">n</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">])</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">e</span><span class="mord mathnormal">n</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">])</span></span></span></span> 的情况如何统计呢. 显然此时我们没法 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 的检查 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi><mi>s</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo>+</mo><mi>n</mi><mi>u</mi><mi>m</mi><mi>s</mi><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">nums[i]+nums[j]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">]</span></span></span></span> 的合法性. 不妨换一个角度, 当我们枚举 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi><mi>s</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">nums[i]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span></span></span></span> 时:</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>统计右侧拼接长度更大的&nbsp;nums[j]&nbsp;的合法情况数</mtext><mtext>  </mtext><mo>⟺</mo><mtext>  </mtext><mtext>统计左侧拼接长度更小的&nbsp;nums[j]&nbsp;的合法情况数</mtext></mrow><annotation encoding="application/x-tex">\text{统计右侧拼接长度更大的 nums[j] 的合法情况数} \iff \text{统计左侧拼接长度更小的 nums[j] 的合法情况数}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord cjk_fallback">统计右侧拼接长度更大的</span><span class="mord">&nbsp;nums[j]&nbsp;</span><span class="mord cjk_fallback">的合法情况数</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟺</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord cjk_fallback">统计左侧拼接长度更小的</span><span class="mord">&nbsp;nums[j]&nbsp;</span><span class="mord cjk_fallback">的合法情况数</span></span></span></span></span></span></p><p>于是合法情况数就可以表示为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><mo fence="false" stretchy="true" minsize="1.2em" maxsize="1.2em">[</mo><msub><mtext>cond</mtext><mn>1</mn></msub><mo stretchy="false">(</mo><mi>n</mi><mi>u</mi><mi>m</mi><mi>s</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>cond</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>n</mi><mi>u</mi><mi>m</mi><mi>s</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo fence="false" stretchy="true" minsize="1.2em" maxsize="1.2em">]</mo></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \sum_{i=0}^{n-1}\big[\text{cond}_1(nums[i])+\text{cond}_2(nums[i])\big]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0788em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8011em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="delimsizing size1">[</span></span><span class="mord"><span class="mord text"><span class="mord">cond</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">])</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord text"><span class="mord">cond</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">])</span><span class="mord"><span class="delimsizing size1">]</span></span></span></span></span>, 其中第一种情况 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>cond</mtext><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\text{cond}_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">cond</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 就是统计右侧拼接长度更小的字符串数量, 第二种情况 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>cond</mtext><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\text{cond}_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">cond</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 就是统计左侧拼接长度更小的字符串数量. 这两步可以同时计算.</p></li></ul><p>时间复杂度: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">from</span> typing <span class="hljs-keyword">import</span> <span class="hljs-type">List</span>, <span class="hljs-type">Tuple</span>, <span class="hljs-type">Dict</span>, <span class="hljs-type">Optional</span><br><span class="hljs-keyword">from</span> collections <span class="hljs-keyword">import</span> defaultdict, deque<br><span class="hljs-keyword">from</span> itertools <span class="hljs-keyword">import</span> combinations, permutations<br><span class="hljs-keyword">import</span> math, heapq, queue<br><br>II = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">int</span>(<span class="hljs-built_in">input</span>())<br>FI = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">float</span>(<span class="hljs-built_in">input</span>())<br>MII = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">tuple</span>(<span class="hljs-built_in">map</span>(<span class="hljs-built_in">int</span>, <span class="hljs-built_in">input</span>().split()))<br>LII = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">list</span>(<span class="hljs-built_in">map</span>(<span class="hljs-built_in">int</span>, <span class="hljs-built_in">input</span>().split()))<br>LSI = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">list</span>(<span class="hljs-built_in">map</span>(<span class="hljs-built_in">str</span>, <span class="hljs-built_in">input</span>().split()))<br><br><span class="hljs-keyword">def</span> <span class="hljs-title function_">solve</span>() -&gt; <span class="hljs-type">Optional</span>:<br>    n, nums = II(), LSI()<br>    <br>    f = [[<span class="hljs-number">0</span> <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">46</span>)] <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">6</span>)]<br>    <span class="hljs-keyword">for</span> num <span class="hljs-keyword">in</span> nums:<br>        m = <span class="hljs-built_in">len</span>(num)<br>        s = <span class="hljs-built_in">sum</span>([<span class="hljs-built_in">int</span>(c) <span class="hljs-keyword">for</span> c <span class="hljs-keyword">in</span> num])<br>        f[m][s] += <span class="hljs-number">1</span><br>    <br>    res = <span class="hljs-number">0</span><br>    <span class="hljs-keyword">for</span> num <span class="hljs-keyword">in</span> nums:<br>        m = <span class="hljs-built_in">len</span>(num)<br>        s = [<span class="hljs-number">0</span>] * (m + <span class="hljs-number">1</span>)<br>        <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(m - <span class="hljs-number">1</span>, -<span class="hljs-number">1</span>, -<span class="hljs-number">1</span>):<br>            s[i] = s[i + <span class="hljs-number">1</span>] + <span class="hljs-built_in">int</span>(num[i])<br>        <br>        <span class="hljs-comment"># cond1: now + right -&gt; len(now) &gt;= len(right)</span><br>        <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(m - <span class="hljs-number">1</span>, -<span class="hljs-number">1</span>, -<span class="hljs-number">1</span>):<br>            now_len, now_sum = i + <span class="hljs-number">1</span>, s[<span class="hljs-number">0</span>] - s[i + <span class="hljs-number">1</span>]<br>            r_len, r_sum = now_len - (m - <span class="hljs-number">1</span> - i), now_sum - s[i + <span class="hljs-number">1</span>]<br>            <span class="hljs-keyword">if</span> <span class="hljs-number">1</span> &lt;= r_len &lt;= now_len <span class="hljs-keyword">and</span> r_sum &gt;= <span class="hljs-number">0</span>:<br>                res += f[r_len][r_sum]<br>        <br>        <span class="hljs-comment"># cond2: left + now -&gt; len(left) &lt; len(now)</span><br>        <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(m):<br>            now_len, now_sum = m - i, s[i]<br>            l_len, l_sum = now_len - i, now_sum - (s[<span class="hljs-number">0</span>] - s[i])<br>            <span class="hljs-keyword">if</span> <span class="hljs-number">1</span> &lt;= l_len &lt; now_len <span class="hljs-keyword">and</span> l_sum &gt;= <span class="hljs-number">0</span>:<br>                res += f[l_len][l_sum]<br>    <br>    <span class="hljs-keyword">return</span> res<br><br><span class="hljs-keyword">if</span> __name__ == <span class="hljs-string">'__main__'</span>:<br>    OUTs = []<br>    N = <span class="hljs-number">1</span><br>    <span class="hljs-comment"># N = II()</span><br>    <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(N):<br>        OUTs.append(solve())<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">'\n'</span>.join(<span class="hljs-built_in">map</span>(<span class="hljs-built_in">str</span>, OUTs)))<br></code></pre></td></tr></tbody></table></figure>]]></content>
+    <content type="html"><![CDATA[<h3 id="计算几何">计算几何</h3><h3 id="【二维-数学】Minimum-Manhattan-Distance">【二维/数学】Minimum Manhattan Distance</h3><p><a href="https://codeforces.com/gym/104639/problem/J">https://codeforces.com/gym/104639/problem/J</a></p><blockquote><p>题意：给定两个圆的直径的两个点坐标，其中约束条件是两个圆一定是处在相离的两个角上。问如何在C2圆上或圆内找到一点p，使得点p到C1圆的所有点的曼哈顿距离的期望值最小</p><p>思路：</p><ul><li><p>看似需要积分，其实我们可以发现，对于点p到C1中某个点q1的曼哈顿距离，我们一定可以找到q1关于C1对称的点q2，那么点p到q1和q2的曼哈顿距离之和就是点p到C1的曼哈顿距离的两倍（证明就是中线定理）那么期望的最小值就是点p到C1的曼哈顿距离的最小值。目标转化后，我们开始思考如何计算此目标的最小值，思路如下图</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202402252321057.png" alt="image-20240116175917260"></p></li></ul><p>注意点：</p><ul><li>double的读取速度很慢，可以用 <code>int</code> or <code>long long</code> 读入，后续强制类型转换（显示 or 和浮点数计算）</li><li>注意输出答案的精度控制  <code>cout &lt;&lt; fixed &lt;&lt; setprecision(10) &lt;&lt; res &lt;&lt; "\n";</code></li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br><span class="hljs-type">double</span> x1, y1, x2, y2;<br><span class="hljs-type">long</span> <span class="hljs-type">long</span> a, b, c, d;<br><br>cin &gt;&gt; a &gt;&gt; b &gt;&gt; c &gt;&gt; d;<br>x1 = (a + c) / <span class="hljs-number">2.0</span>;<br>y1 = (b + d) / <span class="hljs-number">2.0</span>;<br><br>cin &gt;&gt; a &gt;&gt; b &gt;&gt; c &gt;&gt; d;<br>x2 = (a + c) / <span class="hljs-number">2.0</span>;<br>y2 = (b + d) / <span class="hljs-number">2.0</span>;<br><br><span class="hljs-type">double</span> r2 = <span class="hljs-built_in">sqrt</span>((a - c) * (a - c) + (b - d) * (b - d)) / <span class="hljs-number">2</span>;<br><br>cout &lt;&lt; fixed &lt;&lt; <span class="hljs-built_in">setprecision</span>(<span class="hljs-number">10</span>) &lt;&lt;  <span class="hljs-built_in">abs</span>(x1 - x2) + <span class="hljs-built_in">abs</span>(y1 - y2) - <span class="hljs-built_in">sqrt</span>(<span class="hljs-number">2</span>) * r2 &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【枚举】三角形">【枚举】三角形</h3><p><a href="https://www.acwing.com/problem/content/5383/">https://www.acwing.com/problem/content/5383/</a></p><blockquote><p>题意：给定两个直角三角形的两条直角边的长度 a, b，问能不能在坐标轴上找到三个整数点使得三点满足该直角三角形且三遍均不与坐标轴垂直</p><p>思路：首先确定两个直角边的顶点为原点 (0, 0)，接着根据对称性直接在第一象限中按照边长枚举其中一个顶点 A，对于每一个枚举到的顶点 A，按照斜率枚举最后一个顶点 B，如果满足长度以及不平行于坐标轴的条件就是合法的点。如果全部枚举完都没有找到就是没有合法组合，直接输出 NO 即可。</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>a</mi><mn>2</mn></msup><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(a^2b)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathnormal">b</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><br><span class="hljs-type">int</span> a, b;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>cin &gt;&gt; a &gt;&gt; b;<br><br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt;= a; i++) {<br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">0</span>; j &lt;= a; j++) {<br><span class="hljs-keyword">if</span> (i * i + j * j == a * a &amp;&amp; i &amp;&amp; j) {<br><span class="hljs-type">int</span> gcd = __gcd(i, j);<br><span class="hljs-type">int</span> p = -i / gcd, q = j / gcd;<br><span class="hljs-type">int</span> y = p, x = q;<br><span class="hljs-keyword">while</span> (x * x + y * y &lt; b * b) {<br>x += q, y += p;<br>}<br><span class="hljs-keyword">if</span> (x * x + y * y == b * b &amp;&amp; x != i &amp;&amp; y != j) {<br>cout &lt;&lt; <span class="hljs-string">"YES\n"</span>;<br>cout &lt;&lt; <span class="hljs-number">0</span> &lt;&lt; <span class="hljs-string">' '</span> &lt;&lt; <span class="hljs-number">0</span> &lt;&lt; <span class="hljs-string">"\n"</span>;<br>cout &lt;&lt; i &lt;&lt; <span class="hljs-string">' '</span> &lt;&lt; j &lt;&lt; <span class="hljs-string">"\n"</span>;<br>cout &lt;&lt; x &lt;&lt; <span class="hljs-string">' '</span> &lt;&lt; y &lt;&lt; <span class="hljs-string">"\n"</span>;<br><span class="hljs-keyword">return</span>;<br>}<br>}<br>}<br>}<br><br>cout &lt;&lt; <span class="hljs-string">"NO\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br><span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//cin &gt;&gt; T;</span><br><span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br><span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【凸包】奶牛过马路">【凸包】奶牛过马路</h3><p><a href="https://www.acwing.com/problem/content/5572/">https://www.acwing.com/problem/content/5572/</a></p><blockquote><p>题意：给定一个二维坐标系，现在有两个东西可以进行平移。</p><ul><li>一个是起始位于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,0)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span> 的奶牛，可以沿着 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> 轴以 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mi>u</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0,u]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">u</span><span class="mclose">]</span></span></span></span> 的速度正向或负向移动</li><li>一个是起始位于第一象限的凸包形状的车辆，可以向 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 轴负半轴以恒定的速度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span></span> 移动</li></ul><p>在给定凸包的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个顶点的情况下，以及不允许奶牛被车辆撞到的情况下，奶牛前进到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo separator="true">,</mo><mi>w</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,w)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mclose">)</span></span></span></span> 的最短时间</p><p>思路：由于是凸包，因此每一个顶点同时满足的性质，就可以代表整个凸包满足的性质。那么对于奶牛和凸包，就一共有三个局面：</p><ol><li>奶牛速度足够快。可以使得对于任意一个顶点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> 在到达 y 轴时，都小于此时奶牛的 y 值，用时间来表示就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mi>y</mi><mi>u</mi></mfrac><mo>≤</mo><mfrac><mi>x</mi><mi>v</mi></mfrac></mrow><annotation encoding="application/x-tex">\frac{y}{u} \le \frac{x}{v}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0925em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7475em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>≤</mo><mfrac><mi>u</mi><mi>v</mi></mfrac><mi>x</mi></mrow><annotation encoding="application/x-tex">y \le \frac{u}{v}x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">x</span></span></span></span>，表示凸包上所有的顶点都在直线 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mfrac><mi>u</mi><mi>v</mi></mfrac></mrow><annotation encoding="application/x-tex">y=\frac{u}{v}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 的下方</li><li>凸包速度足够快。可以使得对于任意一个顶点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> 在到达 y 轴时，都大于此时奶牛的 y 值，用时间来表示就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mi>y</mi><mi>u</mi></mfrac><mo>≥</mo><mfrac><mi>x</mi><mi>v</mi></mfrac></mrow><annotation encoding="application/x-tex">\frac{y}{u} \ge \frac{x}{v}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0925em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7475em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4461em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>≥</mo><mfrac><mi>u</mi><mi>v</mi></mfrac><mi>x</mi></mrow><annotation encoding="application/x-tex">y \ge \frac{u}{v}x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">x</span></span></span></span>，表示凸包上所有的顶点都在直线 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mfrac><mi>u</mi><mi>v</mi></mfrac></mrow><annotation encoding="application/x-tex">y=\frac{u}{v}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 的上方</li><li>上述两种都不满足，即直线 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mfrac><mi>u</mi><mi>v</mi></mfrac></mrow><annotation encoding="application/x-tex">y=\frac{u}{v}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 与凸包相交。此时奶牛就不能一直全速（u）前进到终点了，需要进行一定的减速 or 返回操作。有一个显然的结论就是，对于减速 or 返回的操作，都可以等价于后退一段距离后再全速（u）前进，因此对于需要减速 or 返回的操作，我们只需要计算出当前状态下应该后退的距离即可。再简洁的，我们就是需要平移直线 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mfrac><mi>u</mi><mi>v</mi></mfrac></mrow><annotation encoding="application/x-tex">y=\frac{u}{v}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>，显然只能向下平移而非向上平移，因此其实就是计算出直线 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mfrac><mi>u</mi><mi>v</mi></mfrac><mo>+</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">y=\frac{u}{v}+b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span> 的截距 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span> 使得最终的直线 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mfrac><mi>u</mi><mi>v</mi></mfrac><mo>+</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">y=\frac{u}{v}+b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span> 满足第二种条件即可。那么如何计算这个 b 呢？很显然我们可以遍历每一个顶点取满足所有顶点都在直线 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>=</mo><mfrac><mi>u</mi><mi>v</mi></mfrac><mo>+</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">y=\frac{u}{v}+b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0404em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6954em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">v</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span>​ 上方的最大截距 b 即可。</li></ol><p>注意点：</p><ul><li>浮点数比较注意精度误差，尽可能将除法转化为乘法</li><li>在比较相等时可以引入一个无穷小量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϵ</mi><mo>=</mo><mn>1</mn><msup><mn>0</mn><mrow><mo>−</mo><mn>7</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\epsilon=10^{-7}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">ϵ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">7</span></span></span></span></span></span></span></span></span></span></span></span></li><li>注意答案输出精度要求，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mrow><mo>−</mo><mn>6</mn></mrow></msup></mrow><annotation encoding="application/x-tex">10^{-6}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">6</span></span></span></span></span></span></span></span></span></span></span></span> 就需要我们出至少 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>7</mn></mrow><annotation encoding="application/x-tex">7</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">7</span></span></span></span> 位小数</li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><p>C++</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;cstring&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;algorithm&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iomanip&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><span class="hljs-keyword">using</span> ll = <span class="hljs-type">long</span> <span class="hljs-type">long</span>;<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br><span class="hljs-type">double</span> n, w, v, u;<br>cin &gt;&gt; n &gt;&gt; w &gt;&gt; v &gt;&gt; u;<br><br><span class="hljs-type">bool</span> up = <span class="hljs-literal">false</span>, down = <span class="hljs-literal">false</span>;<br><span class="hljs-type">double</span> b = <span class="hljs-number">0</span>;<br><br><span class="hljs-keyword">while</span> (n--) {<br><span class="hljs-type">double</span> x, y;<br>cin &gt;&gt; x &gt;&gt; y;<br>        <br>        <span class="hljs-comment">// 除法无法通过</span><br><span class="hljs-comment">// if (y / x &lt; u / v) down = true;</span><br><span class="hljs-comment">// else if (y / x &gt; u / v) up = true;</span><br>        <br>        <span class="hljs-comment">// 转化为乘法以后才能通过</span><br><span class="hljs-keyword">if</span> (y &lt; u / v * x) down = <span class="hljs-literal">true</span>;<br><span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (y &gt; u / v * x) up = <span class="hljs-literal">true</span>;<br>        <br>b = <span class="hljs-built_in">min</span>(b, y - u / v * x);<br>}<br><br><span class="hljs-keyword">if</span> (up &amp;&amp; down) cout &lt;&lt; <span class="hljs-built_in">setprecision</span>(<span class="hljs-number">7</span>) &lt;&lt; (w - b) / u;<br><span class="hljs-keyword">else</span> cout &lt;&lt; <span class="hljs-built_in">setprecision</span>(<span class="hljs-number">7</span>) &lt;&lt; w / u;<br><br><span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><p>Py</p><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br></pre></td><td class="code"><pre><code class="hljs python">read = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">map</span>(<span class="hljs-built_in">int</span>, <span class="hljs-built_in">input</span>().split())<br><br>n, w, v, u = read()<br>k = u / v<br>b = <span class="hljs-number">0</span><br>up = <span class="hljs-literal">False</span><br>down = <span class="hljs-literal">False</span><br><br><span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(n):<br>    x, y = read()<br>    <span class="hljs-keyword">if</span> (y &lt; k * x): down = <span class="hljs-literal">True</span><br>    <span class="hljs-keyword">if</span> (y &gt; k * x): up = <span class="hljs-literal">True</span><br>    b = <span class="hljs-built_in">min</span>(b, y - k * x)<br><br><span class="hljs-keyword">if</span> (up <span class="hljs-keyword">and</span> down): <span class="hljs-built_in">print</span>((w - b) / u)<br><span class="hljs-keyword">else</span>: <span class="hljs-built_in">print</span>(w / u)<br></code></pre></td></tr></tbody></table></figure>]]></content>
     
     
     <categories>
@@ -1460,11 +1460,11 @@
   
   
   <entry>
-    <title>games</title>
-    <link href="/Algorithm/games/"/>
-    <url>/Algorithm/games/</url>
+    <title>graphs</title>
+    <link href="/Algorithm/graphs/"/>
+    <url>/Algorithm/graphs/</url>
     
-    <content type="html"><![CDATA[<h3 id="博弈论">博弈论</h3><p>思考如何必胜态和必败态是什么以及如何构造这样的局面。</p><h3 id="【博弈-贪心-交互】Salyg1n-and-the-MEX-Game">【博弈/贪心/交互】Salyg1n and the MEX Game</h3><p><a href="https://codeforces.com/contest/1867/problem/C">https://codeforces.com/contest/1867/problem/C</a></p><blockquote><p>标签：博弈、贪心、交互</p><p>题面：对于给定n个数的数列，先手可以放入一个数列中不存在的数（0-1e9），后手可以从数列中拿掉一个数，但是这个数必须严格小于刚才先手放入的数。终止条件：后手没法拿数或者操作次数达到了 2n+1 次。问：当你是先手时，如何放数可以使得最终数列的 MEX 值最大。</p><p>思路：先手每次放入的数一定是当前数列的 MEX 值，此后不管后手拿掉什么数，先手都将刚刚被拿掉的数放进去即可。那么最多操作次数就刚好是 2n+1次，因为加入当前数列就是一个从 0 开始的连续整数数列，那么先手放入的数就是最大数 +1，即 n，那么假如后手从 n-1 开始拿，后手最多拿 n 次，先手再放 n 次，那么就是 2n+1 次。</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;vector&gt;</span></span><br><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span></span><br><span class="hljs-function"></span>{<br><span class="hljs-type">int</span> T; cin &gt;&gt; T;<br><br><span class="hljs-keyword">while</span> (T--)<br>{<br><span class="hljs-type">int</span> n; cin &gt;&gt; n;<br><span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">a</span><span class="hljs-params">(n)</span></span>;<br><br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i ++)<br>cin &gt;&gt; a[i];<br><br><span class="hljs-type">int</span> mex = n;<br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i ++)<br><span class="hljs-keyword">if</span> (a[i] != i)<br>{<br>mex = i;<br><span class="hljs-keyword">break</span>;<br>}<br><br>cout &lt;&lt; mex &lt;&lt; endl;<br><br><span class="hljs-type">int</span> remove;<br>cin &gt;&gt; remove;<br><br><span class="hljs-keyword">while</span> (remove != <span class="hljs-number">-1</span>)<br>{<br>cout &lt;&lt; remove &lt;&lt; endl;<br>cin &gt;&gt; remove;<br>}<br>}<br><br><span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>} <br></code></pre></td></tr></tbody></table></figure>]]></content>
+    <content type="html"><![CDATA[<h3 id="图论">图论</h3><h3 id="【拓扑】有向图的拓扑序列">【拓扑】有向图的拓扑序列</h3><p><a href="https://www.acwing.com/problem/content/850/">https://www.acwing.com/problem/content/850/</a></p><blockquote><p>题意：输出一个图的拓扑序，不存在则输出-1</p><p>思路：</p><ul><li>首先我们要知道拓扑图的概念，感官上就是一张图可以从一个方向拓展到全图，用数学语言就是：若一个由图中所有点构成的序列 A 满足：对于图中的每条边 (x,y)，x 在 A 中都出现在 y 之前，则称 A 是该图的一个拓扑序列</li><li>接着我们就想要寻找这样的序列 A 了，可以发现对于每一个可扩展的点，入度一定为0，那么我们就从这些点开始宽搜，将搜到的点的入度-1，即删除这条边，直到最后。如果全图的点的入度都变为了0，则此图可拓扑</li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n+m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">100010</span>;<br><br><span class="hljs-type">int</span> n, m;<br>vector&lt;<span class="hljs-type">int</span>&gt; G[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-comment">// 建图 </span><br>    cin &gt;&gt; n &gt;&gt; m;<br>    <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">d</span><span class="hljs-params">(n + <span class="hljs-number">1</span>, <span class="hljs-number">0</span>)</span></span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= m; i++) {<br>        <span class="hljs-type">int</span> a, b;<br>        cin &gt;&gt; a &gt;&gt; b;<br>        d[b]++;<br>        G[a].<span class="hljs-built_in">push_back</span>(b);<br>    }<br>    <br>    <span class="hljs-comment">// 预处理宽搜起始点集</span><br>    queue&lt;<span class="hljs-type">int</span>&gt; q;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++)<br>        <span class="hljs-keyword">if</span> (!d[i])<br>            q.<span class="hljs-built_in">push</span>(i);<br>    <br>    <span class="hljs-comment">// 宽搜处理</span><br>    vector&lt;<span class="hljs-type">int</span>&gt; res;<br>    <span class="hljs-keyword">while</span> (q.<span class="hljs-built_in">size</span>()) {<br>        <span class="hljs-keyword">auto</span> h = q.<span class="hljs-built_in">front</span>();<br>        q.<span class="hljs-built_in">pop</span>();<br>        res.<span class="hljs-built_in">push_back</span>(h);<br>        <br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; ch: G[h]) {<br>            d[ch]--;<br>            <span class="hljs-keyword">if</span> (!d[ch]) q.<span class="hljs-built_in">push</span>(ch);<br>        }<br>    }<br>    <br>    <span class="hljs-comment">// 输出合法拓扑序</span><br>    <span class="hljs-keyword">if</span> (res.<span class="hljs-built_in">size</span>() == n) {<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; x: res) {<br>            cout &lt;&lt; x &lt;&lt; <span class="hljs-string">" "</span>;<br>        }<br>    } <span class="hljs-keyword">else</span> {<br>        cout &lt;&lt; <span class="hljs-number">-1</span> &lt;&lt; <span class="hljs-string">"\n"</span>;<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【基环树-拓扑】Mad-City-fire">【基环树/拓扑】Mad City<span class="github-emoji"><span>🔥</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/1f525.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span></h3><p><a href="https://codeforces.com/contest/1873/problem/H">https://codeforces.com/contest/1873/problem/H</a></p><blockquote><p>tag：基环树、拓扑排序</p><p>题意：给定一个基环树，现在图上有两个点，分别叫做A，B。现在B想要逃脱A的抓捕，问对于给定的局面，B能否永远逃离A的抓捕</p><p>思路：思路很简单，我们只需要分B所在位置的两种情况讨论即可</p><ol><li>B不在环上：此时我们记距离B最近的环上的那个点叫 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mi>a</mi><mi>g</mi></mrow><annotation encoding="application/x-tex">tag</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8095em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">t</span><span class="mord mathnormal">a</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span></span></span></span>，我们需要比较的是A到tag点的距离 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">d_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 和B到tag的距离 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mi>B</mi></msub></mrow><annotation encoding="application/x-tex">d_B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，如果 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mi>B</mi></msub><mo>&lt;</mo><msub><mi>d</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">d_B &lt; d_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，则一定可以逃脱，否则一定不可以逃脱</li><li>B在环上：此时我们只需要判断当前的A点是否与B的位置重合即可，如果重合那就无法逃脱，反之B一定可以逃脱。</li></ol><p>代码实现：</p><ol><li>对于第一种情况，我们需要找到tag点以及计算A和B到tag点的距离，</li></ol><p>时间复杂度：</p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br><span class="line">82</span><br><span class="line">83</span><br><span class="line">84</span><br><span class="line">85</span><br><span class="line">86</span><br><span class="line">87</span><br><span class="line">88</span><br><span class="line">89</span><br><span class="line">90</span><br><span class="line">91</span><br><span class="line">92</span><br><span class="line">93</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">200010</span>;<br><br><span class="hljs-type">int</span> n, a, b;<br>vector&lt;<span class="hljs-type">int</span>&gt; G[N];<br><span class="hljs-type">int</span> rd[N], tag, d[N];<br><span class="hljs-type">bool</span> del[N], vis[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">init</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        G[i].<span class="hljs-built_in">clear</span>();        <span class="hljs-comment">// 存无向图 </span><br>        rd[i] = <span class="hljs-number">0</span>;            <span class="hljs-comment">// 统计每一个结点的入度 </span><br>        del[i] = <span class="hljs-literal">false</span>;        <span class="hljs-comment">// 拓扑删点删边时使用 </span><br>        d[i] = <span class="hljs-number">0</span>;            <span class="hljs-comment">// 图上所有点到 tag 点的距离 </span><br>        vis[i] = <span class="hljs-literal">false</span>;        <span class="hljs-comment">// bfs计算距离时使用 </span><br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">topu</span><span class="hljs-params">(<span class="hljs-type">int</span> now)</span> </span>{<br>    <span class="hljs-keyword">if</span> (rd[now] == <span class="hljs-number">1</span>) {<br>        rd[now]--;<br>        del[now] = <span class="hljs-literal">true</span>;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; ch: G[now]) {<br>            <span class="hljs-keyword">if</span> (del[ch]) <span class="hljs-keyword">continue</span>;<br>            rd[ch]--;<br>            <span class="hljs-keyword">if</span> (now == tag) {<br>                tag = ch;<br>            }<br>            <span class="hljs-built_in">topu</span>(ch);<br>        }<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">bfs</span><span class="hljs-params">()</span> </span>{<br>    queue&lt;<span class="hljs-type">int</span>&gt; q;<br>    q.<span class="hljs-built_in">push</span>(tag);<br>    d[tag] = <span class="hljs-number">0</span>;<br>    <br>    <span class="hljs-keyword">while</span> (q.<span class="hljs-built_in">size</span>()) {<br>        <span class="hljs-keyword">auto</span> now = q.<span class="hljs-built_in">front</span>();<br>        vis[now] = <span class="hljs-literal">true</span>;<br>        q.<span class="hljs-built_in">pop</span>();<br>        <br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; ch: G[now]) {<br>            <span class="hljs-keyword">if</span> (!vis[ch]) {<br>                d[ch] = d[now] + <span class="hljs-number">1</span>;<br>                q.<span class="hljs-built_in">push</span>(ch);<br>                vis[ch] = <span class="hljs-literal">true</span>;<br>            }<br>        }<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-comment">// 初始化</span><br>    cin &gt;&gt; n &gt;&gt; a &gt;&gt; b; <br>    <span class="hljs-built_in">init</span>();<br>    <br>    <span class="hljs-comment">// 建图 </span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-type">int</span> u, v;<br>        cin &gt;&gt; u &gt;&gt; v;<br>        G[u].<span class="hljs-built_in">push_back</span>(v), rd[v]++;<br>        G[v].<span class="hljs-built_in">push_back</span>(u), rd[u]++;<br>    }<br>    <br>    <span class="hljs-comment">// 拓扑删边 &amp; 缩b点</span><br>    tag = b;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-built_in">topu</span>(i);<br>    }<br><br>    <span class="hljs-comment">// 判断结果 &amp; 计算距离 </span><br>    <span class="hljs-keyword">if</span> (rd[b] == <span class="hljs-number">2</span> &amp;&amp; a != b) {<br>        <span class="hljs-comment">// b点在环上</span><br>        cout &lt;&lt; <span class="hljs-string">"Yes\n"</span>;<br>    } <span class="hljs-keyword">else</span> {<br>        <span class="hljs-comment">// b不在环上</span><br>        <span class="hljs-built_in">bfs</span>();<br>        cout &lt;&lt; (d[a] &gt; d[b] ? <span class="hljs-string">"Yes\n"</span> : <span class="hljs-string">"No\n"</span>);<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-number">0</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br>    cin &gt;&gt; T;<br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【二分图】染色法判定二分图">【二分图】染色法判定二分图</h3><p><a href="https://www.acwing.com/problem/content/862/">https://www.acwing.com/problem/content/862/</a></p><blockquote><p>题意：给定一个无向图，可能有重边和自环。问是否可以构成二分图。</p><p>二分图的定义：一个图可以被分成两个点集，每个点集内部没有边相连（可以不是连通图）</p><p>思路：利用<strong>染色法</strong>，遍历每一个连通分量，选择连通分量中的任意一点进行染色扩展</p><ul><li>如果扩展到的点没有染过色，则染成与当前点相对的颜色</li><li>如果扩展到的点已经被染过色了且染的颜色和当前点的颜色相同，则无法构成二分图（奇数环）</li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n+e)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">e</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">100010</span>;<br><br><span class="hljs-type">int</span> n, m;<br>vector&lt;<span class="hljs-type">int</span>&gt; G[N], <span class="hljs-built_in">col</span>(N);<br><br><span class="hljs-function"><span class="hljs-type">bool</span> <span class="hljs-title">bfs</span><span class="hljs-params">(<span class="hljs-type">int</span> u)</span> </span>{<br>    queue&lt;<span class="hljs-type">int</span>&gt; q;<br>    q.<span class="hljs-built_in">push</span>(u);<br>    col[u] = <span class="hljs-number">1</span>;<br><br>    <span class="hljs-keyword">while</span> (q.<span class="hljs-built_in">size</span>()) {<br>        <span class="hljs-type">int</span> now = q.<span class="hljs-built_in">front</span>();<br>        q.<span class="hljs-built_in">pop</span>();<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; ch: G[now]) {<br>            <span class="hljs-keyword">if</span> (!col[ch]) {<br>                col[ch] = -col[now];<br>                q.<span class="hljs-built_in">push</span>(ch);<br>            }<br>            <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (col[ch] == col[now]) {<br>                <span class="hljs-keyword">return</span> <span class="hljs-literal">false</span>;<br>            }<br>        }<br>    }<br><br>    <span class="hljs-keyword">return</span> <span class="hljs-literal">true</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; m;<br>    <span class="hljs-keyword">while</span> (m--) {<br>        <span class="hljs-type">int</span> u, v;<br>        cin &gt;&gt; u &gt;&gt; v;<br>        G[u].<span class="hljs-built_in">push_back</span>(v);<br>        G[v].<span class="hljs-built_in">push_back</span>(u);<br>    }<br><br>    <span class="hljs-comment">// 遍历每一个连通分量</span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-keyword">if</span> (!col[i]) {<br>            <span class="hljs-type">bool</span> ok = <span class="hljs-built_in">bfs</span>(i);<br>            <span class="hljs-keyword">if</span> (!ok) {<br>                cout &lt;&lt; <span class="hljs-string">"No\n"</span>;<br>                <span class="hljs-keyword">return</span>;<br>            }<br>        }<br>    }<br><br>    cout &lt;&lt; <span class="hljs-string">"Yes\n"</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【最小生成树】Kruskal算法求最小生成树">【最小生成树】Kruskal算法求最小生成树</h3><p><a href="https://www.acwing.com/problem/content/861/">https://www.acwing.com/problem/content/861/</a></p><blockquote><p>题意：给定一个无向图，可能含有重边和自环。试判断能否求解其中的最小生成树，如果可以给出最小生成树的权值</p><p>思路：根据数据量，可以发现顶点数很大，不适用 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mi>r</mi><mi>i</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">Prim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">im</span></span></span></span> 算法，只能用 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi><mi>r</mi><mi>u</mi><mi>s</mi><mi>k</mi><mi>a</mi><mi>l</mi></mrow><annotation encoding="application/x-tex">Kruskal</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">Kr</span><span class="mord mathnormal">u</span><span class="mord mathnormal">s</span><span class="mord mathnormal">ka</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span></span></span> 算法，下面简单介绍一下该算法的流程</p><ul><li>自环首先排除 - 显然这条边连接的“两个”顶点是不可能选进 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mi>S</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">MST</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">MST</span></span></span></span> 的</li><li>首先将每一个结点看成一个连通分量</li><li>接着按照权值将所有的边升序排序后，依次选择<ul><li>如果选出的这条边的两个顶点不在一个连通分量中，则选择这条边并将两个顶点所在的连通分量合并</li><li>如果选出的这条边的两个顶点在同一个连通分量中，则不能选择这条边（否则会使得构造的树形成环）</li></ul></li><li>最后统计选择的边的数量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">num</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span></span></span></span> 进行判断即可<ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi><mo>=</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">num=n-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，则可以生成最小生成树</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi><mo>&lt;</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">num&lt;n-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，则无法生成最小生成树</li></ul></li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>e</mi><mi>log</mi><mo>⁡</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(e\log e)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">e</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">e</span><span class="mclose">)</span></span></span></span>​ - 因为最大的时间开销在对所有的边的权值进行排序上</li></ul></blockquote><p>C++</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><span class="hljs-keyword">typedef</span> <span class="hljs-type">long</span> <span class="hljs-type">long</span> ll;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">100010</span>;<br><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">edge</span> {<br>    <span class="hljs-type">int</span> a, b;<br>    <span class="hljs-type">int</span> w;<br>};<br><br><span class="hljs-type">int</span> n, m;<br>vector&lt;edge&gt; edges;<br><span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">p</span><span class="hljs-params">(N)</span></span>;<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">Find</span><span class="hljs-params">(<span class="hljs-type">int</span> now)</span> </span>{<br>    <span class="hljs-keyword">if</span> (p[now] != now) {<br>        p[now] = <span class="hljs-built_in">Find</span>(p[now]);<br>    }<br>    <span class="hljs-keyword">return</span> p[now];<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; m;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= m; i++) {<br>        <span class="hljs-type">int</span> a, b, w;<br>        cin &gt;&gt; a &gt;&gt; b &gt;&gt; w;<br>        <span class="hljs-keyword">if</span> (a == b) {<br>            <span class="hljs-keyword">continue</span>;<br>        }<br>        edges.<span class="hljs-built_in">push_back</span>({a, b, w});<br>    }<br><br>    <span class="hljs-comment">// 按照边权升序排序</span><br>    <span class="hljs-built_in">sort</span>(edges.<span class="hljs-built_in">begin</span>(), edges.<span class="hljs-built_in">end</span>(), [&amp;](edge&amp; x, edge&amp; y) {<br>        <span class="hljs-keyword">return</span> x.w &lt; y.w;<br>    });<br><br>    <span class="hljs-comment">// 选边</span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        p[i] = i;<br>    }<br><br>    <span class="hljs-type">int</span> res = <span class="hljs-number">0</span>, num = <span class="hljs-number">0</span>;<br><br>    <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; e: edges) {<br>        <span class="hljs-type">int</span> pa = <span class="hljs-built_in">Find</span>(e.a), pb = <span class="hljs-built_in">Find</span>(e.b);<br>        <span class="hljs-keyword">if</span> (pa != pb) {<br>            num++;<br>            p[pa] = pb;<br>            res += e.w;<br>        }<br><br>        <span class="hljs-keyword">if</span> (num == n - <span class="hljs-number">1</span>) {<br>            <span class="hljs-keyword">break</span>;<br>        }<br>    }<br><br>    <span class="hljs-comment">// 特判：选出来的边数无法构成一棵树</span><br>    <span class="hljs-keyword">if</span> (num &lt; n - <span class="hljs-number">1</span>) {<br>        cout &lt;&lt; <span class="hljs-string">"impossible\n"</span>;<br>        <span class="hljs-keyword">return</span>;<br>    }<br><br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><p>Python</p><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">def</span> <span class="hljs-title function_">Find</span>(<span class="hljs-params">x: <span class="hljs-built_in">int</span>, p: <span class="hljs-built_in">list</span></span>) -&gt; <span class="hljs-built_in">int</span>:<br>    <span class="hljs-keyword">if</span> p[x] != x: p[x] = Find(p[x], p)<br>    <span class="hljs-keyword">return</span> p[x]<br><br><span class="hljs-keyword">def</span> <span class="hljs-title function_">kruskal</span>(<span class="hljs-params">n: <span class="hljs-built_in">int</span>, m: <span class="hljs-built_in">int</span>, edges: <span class="hljs-built_in">list</span></span>) -&gt; <span class="hljs-built_in">int</span>:<br>    <span class="hljs-comment"># 按边权对边进行降序排序</span><br>    edges.sort(key=<span class="hljs-keyword">lambda</span> edge: edge[-<span class="hljs-number">1</span>])<br><br>    <span class="hljs-comment"># dsu 初始化</span><br>    p = [<span class="hljs-literal">None</span>] + [i <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">1</span>, n + <span class="hljs-number">1</span>)]<br><br>    <span class="hljs-comment"># 选边</span><br>    cnt = <span class="hljs-built_in">sum</span> = <span class="hljs-number">0</span><br>    <span class="hljs-keyword">for</span> edge <span class="hljs-keyword">in</span> edges:<br>        <span class="hljs-keyword">if</span> cnt == n - <span class="hljs-number">1</span>: <span class="hljs-keyword">break</span><br><br>        pa, pb = Find(edge[<span class="hljs-number">0</span>], p), Find(edge[<span class="hljs-number">1</span>], p)<br>        <span class="hljs-keyword">if</span> pa != pb:<br>            p[pa] = pb<br>            cnt += <span class="hljs-number">1</span><br>            <span class="hljs-built_in">sum</span> += edge[<span class="hljs-number">2</span>]<br><br>    <span class="hljs-keyword">return</span> <span class="hljs-built_in">sum</span> <span class="hljs-keyword">if</span> cnt == n - <span class="hljs-number">1</span> <span class="hljs-keyword">else</span> <span class="hljs-number">0</span><br><br><br><span class="hljs-keyword">if</span> __name__ == <span class="hljs-string">"__main__"</span>:<br>    n, m = <span class="hljs-built_in">map</span>(<span class="hljs-built_in">int</span>, <span class="hljs-built_in">input</span>().split())<br><br>    edges = []<br>    <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(m):<br>        edge = <span class="hljs-built_in">tuple</span>(<span class="hljs-built_in">map</span>(<span class="hljs-built_in">int</span>, <span class="hljs-built_in">input</span>().split()))<br>        edges.append(edge)<br><br>    res = kruskal(n, m, edges)<br><br>    <span class="hljs-keyword">if</span> res: <span class="hljs-built_in">print</span>(res)<br>    <span class="hljs-keyword">else</span>: <span class="hljs-built_in">print</span>(<span class="hljs-string">"impossible"</span>)<br></code></pre></td></tr></tbody></table></figure><p>JavaScript</p><figure class="highlight javascript"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br></pre></td><td class="code"><pre><code class="hljs javascript"><span class="hljs-keyword">const</span> readline = <span class="hljs-built_in">require</span>(<span class="hljs-string">'readline'</span>);<br><span class="hljs-keyword">const</span> rl = readline.<span class="hljs-title function_">createInterface</span>({<br>    <span class="hljs-attr">input</span>: process.<span class="hljs-property">stdin</span>,<br>    <span class="hljs-attr">output</span>: process.<span class="hljs-property">stdout</span>,<br>});<br><br><span class="hljs-keyword">let</span> n = <span class="hljs-literal">null</span>;<br><span class="hljs-keyword">let</span> m = <span class="hljs-literal">null</span>;<br><span class="hljs-keyword">const</span> edges = [];<br><br>rl.<span class="hljs-title function_">on</span>(<span class="hljs-string">'line'</span>, <span class="hljs-function"><span class="hljs-params">line</span> =&gt;</span> {<br>    <span class="hljs-keyword">const</span> [a, b, c] = line.<span class="hljs-title function_">split</span>(<span class="hljs-string">' '</span>).<span class="hljs-title function_">map</span>(<span class="hljs-function"><span class="hljs-params">i</span> =&gt;</span> <span class="hljs-title class_">Number</span>(i));<br>    <span class="hljs-keyword">if</span> (n === <span class="hljs-literal">null</span>) {<br>        n = a;<br>        m = b;<br>    } <span class="hljs-keyword">else</span> {<br>        edges.<span class="hljs-title function_">push</span>([a, b, c]);<br>    }<br>});<br><br>rl.<span class="hljs-title function_">on</span>(<span class="hljs-string">'close'</span>, <span class="hljs-function">() =&gt;</span> {<br>    <span class="hljs-keyword">const</span> res = <span class="hljs-title function_">kurskal</span>(n, m, edges);<br>    <span class="hljs-variable language_">console</span>.<span class="hljs-title function_">log</span>(res === <span class="hljs-title class_">Infinity</span> ? <span class="hljs-string">'impossible'</span> : res);<br>});<br><br><span class="hljs-keyword">function</span> <span class="hljs-title function_">Find</span>(<span class="hljs-params">x, p</span>) {<br>    <span class="hljs-keyword">if</span> (p[x] != x) p[x] = <span class="hljs-title class_">Find</span>(p[x], p);<br>    <span class="hljs-keyword">return</span> p[x];<br>}<br><br><span class="hljs-keyword">function</span> <span class="hljs-title function_">kurskal</span>(<span class="hljs-params">n, m, edges</span>) {<br>    <span class="hljs-comment">// 对边进行升序排序</span><br>    edges.<span class="hljs-title function_">sort</span>(<span class="hljs-function">(<span class="hljs-params">a, b</span>) =&gt;</span> a[<span class="hljs-number">2</span>] - b[<span class="hljs-number">2</span>]);<br>    <br>    <span class="hljs-comment">// 初始化 dsu</span><br>    p = [];<br>    <span class="hljs-keyword">for</span> (<span class="hljs-keyword">let</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) p[i] = i;<br>    <br>    <span class="hljs-comment">// 选边</span><br>    <span class="hljs-keyword">let</span> cnt = <span class="hljs-number">0</span>, sum = <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-keyword">let</span> [a, b, w] <span class="hljs-keyword">of</span> edges) {<br>        <span class="hljs-keyword">if</span> (cnt == n - <span class="hljs-number">1</span>) {<br>            <span class="hljs-keyword">break</span>;<br>        }<br>        <br>        <span class="hljs-keyword">let</span> pa = <span class="hljs-title class_">Find</span>(a, p), pb = <span class="hljs-title class_">Find</span>(b, p);<br>        <span class="hljs-keyword">if</span> (pa !== pb) {<br>            cnt++;<br>            p[pa] = pb;<br>            sum += w;<br>        }<br>    }<br>    <br>    <span class="hljs-keyword">if</span> (cnt === n - <span class="hljs-number">1</span>) <span class="hljs-keyword">return</span> sum;<br>    <span class="hljs-keyword">else</span> <span class="hljs-keyword">return</span> <span class="hljs-title class_">Infinity</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【最小生成树】Prim算法求最小生成树">【最小生成树】Prim算法求最小生成树</h3><p><a href="https://www.acwing.com/problem/content/860/">https://www.acwing.com/problem/content/860/</a></p><blockquote><p>题意：给定一个稠密无向图，有重边和自环。求出最小生成树</p><p>思路：根据题目的数据量，可以使用邻接矩阵存储的方法配合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mi>r</mi><mi>i</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">Prim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">im</span></span></span></span> 算法求解最小生成树，下面给出该算法的流程</p><ul><li>首先明确一下变量的定义：<ul><li><code>g[i][j]</code> 为无向图的邻接矩阵存储结构</li><li><code>MST[i]</code> 表示 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 号点是否加入了 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mi>S</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">MST</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">MST</span></span></span></span> 集合</li><li><code>d[i]</code> 表示 <code>i</code> 号点到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mi>S</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">MST</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">MST</span></span></span></span> 集合的最短边长度</li></ul></li><li>自环不存储，重边只保留最短的一条</li><li>任选一个点到集合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mi>S</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">MST</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">MST</span></span></span></span> 中，并且更新 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span></span></span></span> 数组</li><li>选择剩余的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 个点，每次选择有以下流程<ul><li>找到最短边，记录最短边长度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span> 和相应的在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</mi><mo>−</mo><mi>M</mi><mi>S</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">U-MST</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">MST</span></span></span></span> 集合中对应的顶点序号 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span></span></li><li>将 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span></span> 号点加入 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mi>S</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">MST</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">MST</span></span></span></span> 集合，同时根据此时选出的最短边的长度来判断是否存在最小生成树</li><li>根据 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span></span> 号点，更新 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span></span></span></span> 数组，即更新在集合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</mi><mo>−</mo><mi>M</mi><mi>S</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">U-MST</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">MST</span></span></span></span> 中的点到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mi>S</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">MST</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">MST</span></span></span></span> 集合中的点的交叉边的最短长度</li></ul></li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><span class="hljs-keyword">typedef</span> <span class="hljs-type">long</span> <span class="hljs-type">long</span> ll;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">510</span>;<br><br><span class="hljs-type">int</span> n, m;<br>vector&lt;vector&lt;<span class="hljs-type">int</span>&gt;&gt; <span class="hljs-built_in">g</span>(N, <span class="hljs-built_in">vector</span>&lt;<span class="hljs-type">int</span>&gt;(N, INT_MAX));<br><span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">d</span><span class="hljs-params">(N, INT_MAX)</span></span>; <span class="hljs-comment">// d[i]表示i号点到MST集合中的最短边长度</span><br><span class="hljs-type">bool</span> MST[N];<br><span class="hljs-type">int</span> res;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">prim</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-comment">// 选任意一个点到MST中并更新d数组</span><br>    MST[<span class="hljs-number">1</span>] = <span class="hljs-literal">true</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++)<br>        <span class="hljs-keyword">if</span> (!MST[i])<br>            d[i] = <span class="hljs-built_in">min</span>(d[i], g[i][<span class="hljs-number">1</span>]);<br><br>    <span class="hljs-comment">// 选剩下的n-1个点到MST中</span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">2</span>; i &lt;= n; i++) {<br>        <span class="hljs-comment">// 1. 找到最短边</span><br>        <span class="hljs-type">int</span> e = INT_MAX, v = <span class="hljs-number">-1</span>; <span class="hljs-comment">// e: 最短边长度，v: 最短边不在MST集合中的顶点</span><br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">1</span>; j &lt;= n; j++)<br>            <span class="hljs-keyword">if</span> (!MST[j] &amp;&amp; d[j] &lt; e)<br>                e = d[j], v = j;<br><br>        <span class="hljs-comment">// 2. 加入MST集合</span><br>        MST[v] = <span class="hljs-literal">true</span>;<br>        <span class="hljs-keyword">if</span> (e == INT_MAX) {<br>            <span class="hljs-comment">// 特判无法构造MST的情况</span><br>            cout &lt;&lt; <span class="hljs-string">"impossible\n"</span>;<br>            <span class="hljs-keyword">return</span>;<br>        } <span class="hljs-keyword">else</span> {<br>            res += e;<br>        }<br><br>        <span class="hljs-comment">// 3. 更新交叉边 - 迭代（覆盖更新）</span><br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">1</span>; j &lt;= n; j++)<br>            <span class="hljs-keyword">if</span> (!MST[j])<br>                d[j] = <span class="hljs-built_in">min</span>(d[j], g[j][v]);<br>    }<br><br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; m;<br>    <span class="hljs-keyword">while</span> (m--) {<br>        <span class="hljs-type">int</span> a, b, w;<br>        cin &gt;&gt; a &gt;&gt; b &gt;&gt; w;<br><br>        <span class="hljs-keyword">if</span> (a == b) {<br>            <span class="hljs-keyword">continue</span>;<br>        }<br><br>        <span class="hljs-keyword">if</span> (g[a][b] == INT_MAX) {<br>            g[a][b] = w;<br>            g[b][a] = w;<br>        } <span class="hljs-keyword">else</span> {<br>            g[a][b] = <span class="hljs-built_in">min</span>(g[a][b], w);<br>            g[b][a] = <span class="hljs-built_in">min</span>(g[b][a], w);<br>        }<br>    }<br><br>    <span class="hljs-built_in">prim</span>();<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【最短路】Dijkstra求最短路-fire">【最短路】Dijkstra求最短路 <span class="github-emoji"><span>🔥</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/1f525.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span></h3><p>朴素版 - <a href="https://www.acwing.com/problem/content/851/">https://www.acwing.com/problem/content/851/</a></p><p>堆优化 - <a href="https://www.acwing.com/problem/content/852/">https://www.acwing.com/problem/content/852/</a></p><blockquote><p>题意：给定一个正边权的有向图，可能存在重边与自环，问 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 号点到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 号点的最短路径长度是多少，如果不可达就输出 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span>。</p><p>思路一：朴素版。点数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>500</mn></mrow><annotation encoding="application/x-tex">1\le n \le 500</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">500</span></span></span></span>，边数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>m</mi><mo>≤</mo><mn>1</mn><msup><mn>0</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">1 \le m\le 10^5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></p><ul><li><p>思路：根据数据量，我们采用邻接矩阵的方式存储「点少边多」的稠密图。我们定义 <code>d[i]</code> 数组表示起点到 <code>i</code> 号点的最短距离。先将起点放入 <code>SPT (Shortest Path Tree)</code> 集合，然后更新所有 <code>V-SPT</code> 中的点到 <code>SPT</code> 集合的最短路径长度。接着循环 <code>n-1</code> 次迭代更新剩余的 <code>n-1</code> 个点，每次迭代的过程中，首先选择距离起点最近的点 <code>vex</code>，然后将该点加入 <code>SPT</code> 集合，最后利用该点更新 <code>V-SPT</code> 集合中和该点有连边的点到起点的最短距离。最终的 <code>d[end]</code> 就是起点 <code>start</code> 到终点 <code>end</code> 的最短距离。</p></li><li><p>总结：算法整体采用贪心与动态规划的思路。与 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Prim</mtext></mrow><annotation encoding="application/x-tex">\text{Prim}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord">Prim</span></span></span></span></span> 算法仔细比对可知，其中的贪心过程几乎一致，即每次选择加入 SPT 集合的点均为当前局面 <code>V-SPT</code> 集合中距离起点最近的点。而动态规划的过程体现在，在求解出集合 <code>V-SPT</code> 中到集合 <code>STP</code> 最短距离的点 <code>vex</code> 之后，利用该点对「在 <code>V-SPT</code> 集合且和 vex 点有连边的点 <code>i</code>」更新 <code>d[i]</code> 的过程。更新前的状态都是在之前的子结构下的最优解。</p></li><li><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></p></li></ul><p>思路二：堆优化。点数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>1.5</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">1\le n \le 1.5 \times 10^5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1.5</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span>，边数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>m</mi><mo>≤</mo><mn>1.5</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">1 \le m \le 1.5 \times 10^5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1.5</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></p><ul><li><p>思路：根据数据量，我们采用邻接表的方式存储「点多边少」的稀疏图。如果采用上述朴素 Dijkstra 算法进行求解必然会因为点数过多而超时，因此我们利用数据结构「堆」进行时间开销上的优化。不难发现朴素 Dijkstra 算法在迭代过程中主要有三部分：</p><ol><li>选择距离起点最近的点 <code>vex</code>。因为需要枚举所有的顶点，因此总的时间复杂度为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></li><li>将该点加入 <code>SPT</code> 集合。因为只是简单的打个标记，因此总的时间复杂度为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li><li>利用该点更新 <code>V-SPT</code> 集合中和该点相连的点到起点的最短距离。因为此时枚举的是该点所有的连边，而邻接表的图存储方式无法进行重边的删除，因此最坏情况下会枚举所有的边，时间复杂度为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span></li></ol></li><li><p>时间复杂度：</p></li></ul></blockquote><p>朴素版C++：</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><br><span class="hljs-keyword">using</span> ll = <span class="hljs-type">long</span> <span class="hljs-type">long</span>;<br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">dijkstra_ori</span><span class="hljs-params">(std::vector&lt;std::vector&lt;<span class="hljs-type">int</span>&gt;&gt;&amp; g, <span class="hljs-type">int</span> start, <span class="hljs-type">int</span> end)</span> </span>{<br>    <span class="hljs-type">int</span> n = g.<span class="hljs-built_in">size</span>() - <span class="hljs-number">1</span>;<br>    <span class="hljs-function">std::vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">d</span><span class="hljs-params">(n + <span class="hljs-number">1</span>, INT_MAX &gt;&gt; <span class="hljs-number">1</span>)</span></span>;<br>    <span class="hljs-function">std::vector&lt;<span class="hljs-type">bool</span>&gt; <span class="hljs-title">SPT</span><span class="hljs-params">(n + <span class="hljs-number">1</span>, <span class="hljs-literal">false</span>)</span></span>;<br>    <br>    <span class="hljs-comment">// update start vex</span><br>    d[start] = <span class="hljs-number">0</span>;<br>    SPT[start] = <span class="hljs-literal">true</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-keyword">if</span> (!SPT[i] &amp;&amp; g[start][i] != INT_MAX &gt;&gt; <span class="hljs-number">1</span>) {<br>            d[i] = std::<span class="hljs-built_in">min</span>(d[i], d[start] + g[start][i]);<br>        }<br>    }<br>    <br>    <span class="hljs-comment">// update remain n-1 vex</span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> k = <span class="hljs-number">0</span>; k &lt; n - <span class="hljs-number">1</span>; k++) {<br>        <span class="hljs-type">int</span> vex = <span class="hljs-number">-1</span>;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>            <span class="hljs-keyword">if</span> (!SPT[i] &amp;&amp; (vex == <span class="hljs-number">-1</span> || d[i] &lt; d[vex])) {<br>                vex = i;<br>            }<br>        }<br>        SPT[vex] = <span class="hljs-literal">true</span>;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>            <span class="hljs-keyword">if</span> (!SPT[i] &amp;&amp; g[vex][i] != INT_MAX &gt;&gt; <span class="hljs-number">1</span>) {<br>                d[i] = std::<span class="hljs-built_in">min</span>(d[i], d[vex] + g[vex][i]);<br>            }<br>        }<br>    }<br>    <br>    <span class="hljs-keyword">return</span> d[end] == INT_MAX &gt;&gt; <span class="hljs-number">1</span> ? <span class="hljs-number">-1</span> : d[end];<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-type">int</span> n, m;<br>    cin &gt;&gt; n &gt;&gt; m;<br>    <br>    vector&lt;vector&lt;<span class="hljs-type">int</span>&gt;&gt; <span class="hljs-built_in">g</span>(n + <span class="hljs-number">1</span>, <span class="hljs-built_in">vector</span>&lt;<span class="hljs-type">int</span>&gt;(n + <span class="hljs-number">1</span>, INT_MAX &gt;&gt; <span class="hljs-number">1</span>));<br>    <br>    <span class="hljs-keyword">while</span> (m--) {<br>        <span class="hljs-type">int</span> u, v, w;<br>        cin &gt;&gt; u &gt;&gt; v &gt;&gt; w;<br>        g[u][v] = <span class="hljs-built_in">min</span>(g[u][v], w);<br>    }<br>    <br>    cout &lt;&lt; <span class="hljs-built_in">dijkstra_ori</span>(g, <span class="hljs-number">1</span>, n) &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    std::ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    std::cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    std::cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><p>朴素版Python：</p><figure class="highlight python"><figcaption><span>[]</span></figcaption><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">import</span> heapq<br><span class="hljs-keyword">from</span> collections <span class="hljs-keyword">import</span> defaultdict<br><span class="hljs-keyword">from</span> typing <span class="hljs-keyword">import</span> <span class="hljs-type">List</span>, <span class="hljs-type">Tuple</span><br><span class="hljs-keyword">import</span> math<br><span class="hljs-keyword">from</span> itertools <span class="hljs-keyword">import</span> combinations<br><br>II = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">int</span>(<span class="hljs-built_in">input</span>())<br>FI = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">float</span>(<span class="hljs-built_in">input</span>())<br>MII = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">tuple</span>(<span class="hljs-built_in">map</span>(<span class="hljs-built_in">int</span>, <span class="hljs-built_in">input</span>().split()))<br>LII = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">list</span>(<span class="hljs-built_in">map</span>(<span class="hljs-built_in">int</span>, <span class="hljs-built_in">input</span>().split()))<br><br><br><span class="hljs-keyword">def</span> <span class="hljs-title function_">dijkstra_ori</span>(<span class="hljs-params">g: <span class="hljs-type">List</span>[<span class="hljs-type">List</span>[<span class="hljs-built_in">int</span>]], start: <span class="hljs-built_in">int</span>, end: <span class="hljs-built_in">int</span></span>) -&gt; <span class="hljs-built_in">int</span>:<br>    n = <span class="hljs-built_in">len</span>(g) - <span class="hljs-number">1</span><br>    d = [<span class="hljs-number">10</span> ** <span class="hljs-number">5</span>] * (n + <span class="hljs-number">1</span>)<br>    SPT = [<span class="hljs-literal">False</span>] * (n + <span class="hljs-number">1</span>)<br>    <br>    d[start] = <span class="hljs-number">0</span><br>    SPT[start] = <span class="hljs-literal">True</span><br>    <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">1</span>, n + <span class="hljs-number">1</span>):<br>        <span class="hljs-keyword">if</span> <span class="hljs-keyword">not</span> SPT[i] <span class="hljs-keyword">and</span> g[start][i] != <span class="hljs-number">10</span> ** <span class="hljs-number">5</span>:<br>            d[i] = <span class="hljs-built_in">min</span>(d[i], d[start] + g[start][i])<br>    <br>    <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(n - <span class="hljs-number">1</span>):<br>        vex = -<span class="hljs-number">1</span><br>        <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">1</span>, n + <span class="hljs-number">1</span>):<br>            <span class="hljs-keyword">if</span> <span class="hljs-keyword">not</span> SPT[i] <span class="hljs-keyword">and</span> (vex == -<span class="hljs-number">1</span> <span class="hljs-keyword">or</span> d[i] &lt; d[vex]):<br>                vex = i<br>        SPT[vex] = <span class="hljs-literal">True</span><br>        <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">1</span>, n + <span class="hljs-number">1</span>):<br>            <span class="hljs-keyword">if</span> <span class="hljs-keyword">not</span> SPT[i] <span class="hljs-keyword">and</span> g[vex][i] != <span class="hljs-number">10</span> ** <span class="hljs-number">5</span>:<br>                d[i] = <span class="hljs-built_in">min</span>(d[i], d[vex] + g[vex][i])<br>    <br>    <span class="hljs-keyword">return</span> -<span class="hljs-number">1</span> <span class="hljs-keyword">if</span> d[end] == <span class="hljs-number">10</span> ** <span class="hljs-number">5</span> <span class="hljs-keyword">else</span> d[end]<br><br><br><span class="hljs-keyword">def</span> <span class="hljs-title function_">solve</span>() -&gt; <span class="hljs-literal">None</span>:<br>    n, m = MII()<br>    g = [[<span class="hljs-number">10</span> ** <span class="hljs-number">5</span>] * (n + <span class="hljs-number">1</span>) <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(n + <span class="hljs-number">1</span>)]<br>    <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(m):<br>        u, v, w = MII()<br>        g[u][v] = <span class="hljs-built_in">min</span>(g[u][v], w)<br>    <span class="hljs-built_in">print</span>(dijkstra_ori(g, <span class="hljs-number">1</span>, n))<br><br><br><span class="hljs-keyword">if</span> __name__ == <span class="hljs-string">'__main__'</span>:<br>    T = <span class="hljs-number">1</span><br>    <span class="hljs-comment"># T = II()</span><br>    <span class="hljs-keyword">while</span> T: solve(); T -= <span class="hljs-number">1</span><br></code></pre></td></tr></tbody></table></figure><p>堆优化版C++：</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br></pre></td><td class="code"><pre><code class="hljs cpp">```<br><br>堆优化版Python：<br><br>```python<br><br></code></pre></td></tr></tbody></table></figure><h3 id="【最短路】Floyd求最短路">【最短路】Floyd求最短路</h3><p><a href="https://www.acwing.com/problem/content/856/">https://www.acwing.com/problem/content/856/</a></p><blockquote><p>题意：给定一个稠密有向图，可能存在重边与自环，给出多个询问，需要给出每一个询问的两个点之前的最短路径长度</p><p>思路：我们采用动态规划的思路。在此使用多阶段决策的方法，即每一个路径状态为选择 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>→</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">1\to k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个点的情况下的最短路径长度</p><ul><li>状态表示：<code>f[k][i][j]</code> 表示在前 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个顶点中进行选择（中转），<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 号点到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> 号点的最短路径长度</li><li>状态转移：对于第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个顶点，我们可以选择中转，也可以不中转。<ul><li>对于不选择中转的情况：<code>f[k][i][j] = f[k-1][i][j]</code></li><li>对于可选择中转的情况：<code>f[k][i][j] = f[k-1][i][k] + f[k-1][k][j]</code></li><li>在其中取最小值即可，但是有一个注意点：对于第二种情况，选择是有一个约束的：即如果选择了 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 号点进行转移的话，那么 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 号点到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 号点以及 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 号点到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> 号点都是需要有路径可达的，从而可以选择最小距离</li></ul></li><li>初始化：即选择 0 个站点进行中转时，即 <code>f[0][i][j]</code> 的情况中，<ul><li>如果 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 号点与 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> 号点自环，则取 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></li><li>如果 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 号点与 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> 号点之间有边，则取重边的最小值</li><li>如果 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 号点与 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> 号点之间无边，则初始化为正无穷</li></ul></li><li>答案状态：对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> 号点到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span> 号点之间的最小路径长度，就是 <code>f[n][a][b]</code></li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>3</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^3)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></li><li>空间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>3</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^3)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></li></ul><p>空间优化推导：我们尝试优化掉记忆数组的第一维度</p><ul><li><p>对于不选择的情况：由于决策局面 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 是从前往后枚举，故当前状态 <code>f[k][i][j]</code> 可以<strong>直接依赖于已经更新出来且不会被当前状态之后的状态再次覆盖的最优子结构 <code>f[i][j]</code></strong>。即上一个局面的选择情况，就是不选择第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个顶点的情况</p></li><li><p>对于选择的情况：如果删除第一维度，我们担心的是当前状态 <code>f[k][i][j]</code> 依赖的两个状态 <code>f[i][k]</code> 与 <code>f[k][j]</code> 会不会被后续覆盖掉，即<strong>我们不确定 <code>f[i][k]</code> 与 <code>f[k][j]</code> 是否是当前第 k 个局面的最优子结构</strong>。尝试推导：</p><blockquote><p>为了确定 <code>f[i][k]</code> 与 <code>f[k][j]</code> 是否是当前第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个局面的最优子结构，其实就是确定对于当前第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个局面，这两个状态会不会在当前状态 <code>f[i][j]</code> 之后被更新覆盖，那么我们就看这两个状态是从哪里转移过来进行更新的。如果 <code>f[i][k]</code> 与 <code>f[k][j]</code> 这两个状态的转移会依赖于当前状态之后的状态，那么删除第一维度就是错误的，反之就是成立的。</p><p>尝试推导 <code>f[i][k]</code> 与 <code>f[k][j]</code> 从何转移更新：利用我们未删除维度时正确的状态转移方程进行推演</p><p>我们知道：<code>f[k][i][k] = min(f[k-1][i][k], f[k-1][i][k] + f[k-1][k][k])</code>，其中的 <code>f[k-1][k][k]</code> 就是一个自环的路径长度，由于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mi>l</mi><mi>o</mi><mi>y</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">floyd</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal" style="margin-right:0.03588em;">oy</span><span class="mord mathnormal">d</span></span></span></span> 算法的约束条件是没有负环，因此 <code>f[k-1][k][k]</code> 一定大于零，故 <code>f[k][i][k]</code> 一定取前者，即 <code>f[k][i][k] = f[k-1][i][k]</code></p><p>同理可知：</p><p><code>f[k][k][j] = f[k-1][k][j]</code></p></blockquote><p>基于上述推导我们可以知道，当前第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个决策局面中的 <code>f[k][i][k]</code> 与 <code>f[k][k][j]</code> 是依赖于上一个决策局面 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 的，也就是说这<strong>两个状态一定是早于当前状态 <code>f[i][j]</code> 被更新覆盖的</strong>，故 <code>f[i][k]</code> 与 <code>f[k][j]</code> 就是当前第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个局面的最优子结构，证毕，可以进行维度的删除</p></li><li><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>3</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^3)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></p></li><li><p>空间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></p></li></ul></blockquote><p>不优化空间</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">210</span>, INF = <span class="hljs-number">0x3f3f3f3f</span>;<br><br><span class="hljs-type">int</span> n, m, Q;<br><span class="hljs-type">int</span> f[N][N][N];<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; m &gt;&gt; Q;<br>    <br>    <span class="hljs-comment">// init</span><br>    <span class="hljs-built_in">memset</span>(f, INF, <span class="hljs-keyword">sizeof</span> f);<br>    <br>    <span class="hljs-comment">// add edges and generate base</span><br>    <span class="hljs-keyword">while</span> (m--) {<br>        <span class="hljs-type">int</span> a, b, w;<br>        cin &gt;&gt; a &gt;&gt; b &gt;&gt; w;<br>        <span class="hljs-keyword">if</span> (a == b) <span class="hljs-keyword">continue</span>;                           <span class="hljs-comment">// 重边就不赋值</span><br>        <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (f[<span class="hljs-number">0</span>][a][b] == INF) f[<span class="hljs-number">0</span>][a][b] = w;     <span class="hljs-comment">// 第一次加边则直接赋值</span><br>        <span class="hljs-keyword">else</span> f[<span class="hljs-number">0</span>][a][b] = <span class="hljs-built_in">min</span>(f[<span class="hljs-number">0</span>][a][b], w);           <span class="hljs-comment">// 再次赋边权就取最小值</span><br>    }<br>    <br>    <span class="hljs-comment">// generate base again</span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++)<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">1</span>; j &lt;= n; j++)<br>            <span class="hljs-keyword">if</span> (i == j) <br>                f[<span class="hljs-number">0</span>][i][j] = <span class="hljs-number">0</span>;                         <span class="hljs-comment">// 自环取边权为 0</span><br>    <br>    <span class="hljs-comment">// dp </span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> k = <span class="hljs-number">1</span>; k &lt;= n; k++)<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++)<br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">1</span>; j &lt;= n; j++) {<br>                <span class="hljs-comment">// 不选第k个顶点</span><br>                f[k][i][j] = f[k - <span class="hljs-number">1</span>][i][j];<br>                <br>                <span class="hljs-comment">// 选择第k个顶点</span><br>                <span class="hljs-keyword">if</span> (f[k - <span class="hljs-number">1</span>][i][k] != INF &amp;&amp; f[k - <span class="hljs-number">1</span>][k][j] != INF)<br>                    f[k][i][j] = <span class="hljs-built_in">min</span>(f[k][i][j], f[k - <span class="hljs-number">1</span>][i][k] + f[k - <span class="hljs-number">1</span>][k][j]);<br>            }<br><br>    <span class="hljs-comment">// query</span><br>    <span class="hljs-keyword">while</span> (Q--) {<br>        <span class="hljs-type">int</span> a, b;<br>        cin &gt;&gt; a &gt;&gt; b;<br>        <span class="hljs-keyword">if</span> (f[n][a][b] == INF) cout &lt;&lt; <span class="hljs-string">"impossible\n"</span>;<br>        <span class="hljs-keyword">else</span> cout &lt;&lt; f[n][a][b] &lt;&lt; <span class="hljs-string">"\n"</span>;<br>    }<br>    <br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><p>优化空间</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">210</span>, INF = <span class="hljs-number">0x3f3f3f3f</span>;<br><br><span class="hljs-type">int</span> n, m, Q;<br><span class="hljs-type">int</span> f[N][N];<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; m &gt;&gt; Q;<br>    <br>    <span class="hljs-comment">// init</span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++)<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">1</span>; j &lt;= n; j++)<br>            <span class="hljs-keyword">if</span> (i == j) f[i][j] = <span class="hljs-number">0</span>;<br>            <span class="hljs-keyword">else</span> f[i][j] = INF;<br>            <br>    <span class="hljs-comment">// base</span><br>    <span class="hljs-keyword">while</span> (m--) {<br>        <span class="hljs-type">int</span> a, b, w;<br>        cin &gt;&gt; a &gt;&gt; b &gt;&gt; w;<br>        <span class="hljs-keyword">if</span> (a == b) <span class="hljs-keyword">continue</span>;<br>        <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (f[a][b] == INF) f[a][b] = w;<br>        <span class="hljs-keyword">else</span> f[a][b] = <span class="hljs-built_in">min</span>(f[a][b], w);<br>    }<br>    <br>    <span class="hljs-comment">// dp</span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> k = <span class="hljs-number">1</span>; k &lt;= n; k++)<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++)<br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">1</span>; j &lt;= n; j++)<br>                <span class="hljs-keyword">if</span> (f[i][k] != INF &amp;&amp; f[k][j] != INF)<br>                    f[i][j] = <span class="hljs-built_in">min</span>(f[i][j], f[i][k] + f[k][j]);<br>    <br>    <span class="hljs-comment">// query</span><br>    <span class="hljs-keyword">while</span> (Q--) {<br>        <span class="hljs-type">int</span> a, b;<br>        cin &gt;&gt; a &gt;&gt; b;<br>        <span class="hljs-keyword">if</span> (f[a][b] == INF) cout &lt;&lt; <span class="hljs-string">"impossible\n"</span>;<br>        <span class="hljs-keyword">else</span> cout &lt;&lt; f[a][b] &lt;&lt; <span class="hljs-string">"\n"</span>;<br>    }<br>    <br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【最短路】关闭分部的可行集合数目">【最短路】关闭分部的可行集合数目</h3><p><a href="https://leetcode.cn/problems/number-of-possible-sets-of-closing-branches/">https://leetcode.cn/problems/number-of-possible-sets-of-closing-branches/</a></p><blockquote><p>标签：二进制枚举、最短路</p><p>题意：给定一个含有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个顶点的无向图，如何删点可以使得剩余的图中顶点两两可达且最大距离不超过 maxDistance？返回所有删点的方案数。</p><p>思路：由于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 的数据范围只有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>→</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">1 \to 10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span></span></span></span>，我们可以直接枚举所有的删点方案。那么如何检查一个方案的合法性呢？直接使用最短路算法检查「所有顶点到每一个顶点」的最远距离即可。这里我们采用朴素 dijkstra 算法。</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mn>2</mn><mi>n</mi></msup><mo>×</mo><msup><mi>n</mi><mn>3</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(2^n \times n^3)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> - 其中枚举需要 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mn>2</mn><mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(2^n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>、计算所有顶点到某个顶点的最远距离需要 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>、检查所有顶点需要 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><figcaption><span>[]</span></figcaption><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span> {<br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">numberOfSets</span><span class="hljs-params">(<span class="hljs-type">int</span> n, <span class="hljs-type">int</span> maxDistance, vector&lt;vector&lt;<span class="hljs-type">int</span>&gt;&gt;&amp; roads)</span> </span>{<br>        vector&lt;vector&lt;<span class="hljs-type">int</span>&gt;&gt; <span class="hljs-built_in">g</span>(n, <span class="hljs-built_in">vector</span>&lt;<span class="hljs-type">int</span>&gt;(n, INT_MAX &gt;&gt; <span class="hljs-number">1</span>));<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; r: roads) {<br>            <span class="hljs-type">int</span> u = r[<span class="hljs-number">0</span>], v = r[<span class="hljs-number">1</span>], w = r[<span class="hljs-number">2</span>];<br>            g[u][v] = g[v][u] = <span class="hljs-built_in">min</span>(g[u][v], w);<br>        }<br><br>        <span class="hljs-keyword">auto</span> get_max_dist = [&amp;](<span class="hljs-type">int</span> mask, <span class="hljs-type">int</span> v) {<br>            vector&lt;<span class="hljs-type">bool</span>&gt; <span class="hljs-built_in">SPT</span>(n);<br>            <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">d</span><span class="hljs-params">(n, INT_MAX)</span></span>;<br>            <br>            d[v] = <span class="hljs-number">0</span>;<br>            SPT[v] = <span class="hljs-literal">true</span>;<br>            <br>            <span class="hljs-type">int</span> cnt = <span class="hljs-number">0</span>;<br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>                <span class="hljs-keyword">if</span> (mask &amp; (<span class="hljs-number">1</span> &lt;&lt; i) &amp;&amp; !SPT[i]) {<br>                    cnt++;<br>                    d[i] = <span class="hljs-built_in">min</span>(d[i], d[v] + g[v][i]);<br>                }<br>            }<br><br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> k = <span class="hljs-number">1</span>; k &lt;= cnt - <span class="hljs-number">1</span>; k++) {<br>                <span class="hljs-type">int</span> vex = <span class="hljs-number">-1</span>;<br>                <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>                    <span class="hljs-keyword">if</span> (mask &amp; (<span class="hljs-number">1</span> &lt;&lt; i) &amp;&amp; !SPT[i] &amp;&amp; (vex == <span class="hljs-number">-1</span> || d[i] &lt; d[vex])) {<br>                        vex = i;<br>                    }<br>                }<br>                SPT[vex] = <span class="hljs-literal">true</span>;<br>                <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>                    <span class="hljs-keyword">if</span> (mask &amp; (<span class="hljs-number">1</span> &lt;&lt; i) &amp;&amp; !SPT[i]) {<br>                        d[i] = <span class="hljs-built_in">min</span>(d[i], d[vex] + g[vex][i]);<br>                    }<br>                }<br>            }<br>            <br>            <span class="hljs-type">int</span> max_dist = <span class="hljs-number">-1</span>;<br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>                <span class="hljs-keyword">if</span> (mask &amp; (<span class="hljs-number">1</span> &lt;&lt; i)) {<br>                    max_dist = <span class="hljs-built_in">max</span>(max_dist, d[i]);<br>                }<br>            }<br>            <br>            <span class="hljs-keyword">return</span> max_dist;<br>        };<br><br>        <span class="hljs-type">int</span> res = <span class="hljs-number">0</span>;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> mask = <span class="hljs-number">0</span>; mask &lt; <span class="hljs-number">1</span> &lt;&lt; n; mask++) {<br>            <span class="hljs-type">bool</span> ok = <span class="hljs-literal">true</span>;<br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>                <span class="hljs-keyword">if</span> (mask &amp; (<span class="hljs-number">1</span> &lt;&lt; i) &amp;&amp; <span class="hljs-built_in">get_max_dist</span>(mask, i) &gt; maxDistance) {<br>                    ok = <span class="hljs-literal">false</span>;<br>                    <span class="hljs-keyword">break</span>;<br>                }<br>            }<br>            res += ok;<br>        }<br><br>        <span class="hljs-keyword">return</span> res;<br>    }<br>};<br></code></pre></td></tr></tbody></table></figure><figure class="highlight python"><figcaption><span>[]</span></figcaption><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span>:<br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">numberOfSets</span>(<span class="hljs-params">self, n: <span class="hljs-built_in">int</span>, maxDistance: <span class="hljs-built_in">int</span>, roads: <span class="hljs-type">List</span>[<span class="hljs-type">List</span>[<span class="hljs-built_in">int</span>]]</span>) -&gt; <span class="hljs-built_in">int</span>:<br>        g = [[<span class="hljs-number">10</span> ** <span class="hljs-number">6</span> <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(n)] <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(n)]<br>        <span class="hljs-keyword">for</span> u, v, w <span class="hljs-keyword">in</span> roads:<br>            g[u][v] = g[v][u] = <span class="hljs-built_in">min</span>(g[u][v], w)<br>        <br>        <span class="hljs-keyword">def</span> <span class="hljs-title function_">get_max_dist</span>(<span class="hljs-params">mask: <span class="hljs-built_in">int</span>, v: <span class="hljs-built_in">int</span></span>):<br>            SPT = [<span class="hljs-literal">False</span> <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(n)]<br>            d = [<span class="hljs-number">10</span> ** <span class="hljs-number">6</span> <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(n)]<br><br>            SPT[v] = <span class="hljs-literal">True</span><br>            d[v] = <span class="hljs-number">0</span><br><br>            cnt = <span class="hljs-number">0</span><br>            <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(n):<br>                <span class="hljs-keyword">if</span> mask &amp; (<span class="hljs-number">1</span> &lt;&lt; i) <span class="hljs-keyword">and</span> <span class="hljs-keyword">not</span> SPT[i]:<br>                    cnt += <span class="hljs-number">1</span><br>                    d[i] = <span class="hljs-built_in">min</span>(d[i], d[v] + g[v][i])<br>            <br>            <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(cnt - <span class="hljs-number">1</span>):<br>                vex = -<span class="hljs-number">1</span><br>                <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(n):<br>                    <span class="hljs-keyword">if</span> mask &amp; (<span class="hljs-number">1</span> &lt;&lt; i) <span class="hljs-keyword">and</span> <span class="hljs-keyword">not</span> SPT[i] <span class="hljs-keyword">and</span> (vex == -<span class="hljs-number">1</span> <span class="hljs-keyword">or</span> d[i] &lt; d[vex]):<br>                        vex = i<br>                SPT[vex] = <span class="hljs-literal">True</span><br>                <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(n):<br>                    <span class="hljs-keyword">if</span> mask &amp; (<span class="hljs-number">1</span> &lt;&lt; i) <span class="hljs-keyword">and</span> <span class="hljs-keyword">not</span> SPT[i]:<br>                        d[i] = <span class="hljs-built_in">min</span>(d[i], d[vex] + g[vex][i])<br>                <br>            max_dist = -<span class="hljs-number">1</span><br>            <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(n):<br>                <span class="hljs-keyword">if</span> mask &amp; (<span class="hljs-number">1</span> &lt;&lt; i):<br>                    max_dist = <span class="hljs-built_in">max</span>(max_dist, d[i])<br>            <br>            <span class="hljs-keyword">return</span> max_dist<br><br>        res = <span class="hljs-number">0</span><br>        <span class="hljs-keyword">for</span> mask <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">1</span> &lt;&lt; n):<br>            ok = <span class="hljs-literal">True</span><br>            <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(n):<br>                <span class="hljs-keyword">if</span> mask &amp; (<span class="hljs-number">1</span> &lt;&lt; i) <span class="hljs-keyword">and</span> get_max_dist(mask, i) &gt; maxDistance:<br>                    ok = <span class="hljs-literal">False</span><br>                    <span class="hljs-keyword">break</span><br>            res += ok<br><br>        <span class="hljs-keyword">return</span> res<br></code></pre></td></tr></tbody></table></figure><h3 id="【思维-遍历】图的遍历">【思维/遍历】图的遍历</h3><p><a href="https://www.luogu.com.cn/problem/P3916">https://www.luogu.com.cn/problem/P3916</a></p><blockquote><p>题意：给定一个有向图，求解每一个点可以到达的编号最大的点</p><p>思路：如果从正向考虑，很显然的一个暴力方法就是对于每一个点都跑一遍 dfs 或者 bfs 获取可达的最大点编号，时间复杂度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>，如果想要在遍历的过程中同时更新其余的点，那只有起点到最大点之间的点可以被更新，可以通过递归时记录路径点进行，时间复杂度几乎不变。我们尝试反向考虑：<strong>反向建边</strong>。既然正向考虑时需要标记的点为最大点与起点的路径，那不如直接从最大值点开始遍历搜索，在将所有的边全部反向以后，从最大值点开始遍历图，这样就可以在线性时间复杂度内解决问题</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n+m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span></p></blockquote><p>bfs 代码</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;queue&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;cstring&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;vector&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">100010</span>;<br><br><span class="hljs-type">int</span> n, m;<br>vector&lt;<span class="hljs-type">int</span>&gt; g[N], <span class="hljs-built_in">res</span>(N);<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">bfs</span><span class="hljs-params">(<span class="hljs-type">int</span> now)</span> </span>{<br>    queue&lt;<span class="hljs-type">int</span>&gt; q;<br><br>    res[now] = now;<br>    q.<span class="hljs-built_in">push</span>(now);<br>    <br>    <span class="hljs-keyword">while</span> (q.<span class="hljs-built_in">size</span>()) {<br>        <span class="hljs-type">int</span> h = q.<span class="hljs-built_in">front</span>();<br>        q.<span class="hljs-built_in">pop</span>();<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; ch: g[h]) {<br>            <span class="hljs-keyword">if</span> (!res[ch]) {<br>                res[ch] = now;<br>                q.<span class="hljs-built_in">push</span>(ch);<br>            }<br>        }<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; m;<br>    <span class="hljs-keyword">while</span> (m--) {<br>        <span class="hljs-type">int</span> a, b;<br>        cin &gt;&gt; a &gt;&gt; b;<br>        g[b].<span class="hljs-built_in">push_back</span>(a);<br>    }<br><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = n; i &gt;= <span class="hljs-number">1</span>; i--) {<br>        <span class="hljs-keyword">if</span> (!res[i]) {<br>            <span class="hljs-built_in">bfs</span>(i);<br>        }<br>    }<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        cout &lt;&lt; res[i] &lt;&lt; <span class="hljs-string">' '</span>;<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><p>dfs 代码</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;queue&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;cstring&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;vector&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">100010</span>;<br><br><span class="hljs-type">int</span> n, m, val;<br>vector&lt;<span class="hljs-type">int</span>&gt; g[N], <span class="hljs-built_in">res</span>(N);<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">dfs</span><span class="hljs-params">(<span class="hljs-type">int</span> now)</span> </span>{<br>    res[now] = val;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; ch: g[now]) {<br>        <span class="hljs-keyword">if</span> (!res[ch]) {<br>            <span class="hljs-built_in">dfs</span>(ch);<br>        }<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; m;<br>    <span class="hljs-keyword">while</span> (m--) {<br>        <span class="hljs-type">int</span> a, b;<br>        cin &gt;&gt; a &gt;&gt; b;<br>        g[b].<span class="hljs-built_in">push_back</span>(a);<br>    }<br><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = n; i &gt;= <span class="hljs-number">1</span>; i--) {<br>        <span class="hljs-keyword">if</span> (!res[i]) {<br>            val = i;<br>            <span class="hljs-built_in">dfs</span>(i);<br>        }<br>    }<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        cout &lt;&lt; res[i] &lt;&lt; <span class="hljs-string">' '</span>;<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【遍历-并查集】孤立点数量">【遍历/并查集】孤立点数量</h3><p><a href="https://www.acwing.com/problem/content/5560/">https://www.acwing.com/problem/content/5560/</a></p><blockquote><p>题意：给定一个无向图，可能不连通，没有重边和自环。现在需要给图中的每一条无向边定向，要求所有的边定完向以后 0 入度的点尽可能的少，给出最少的 0 入度点的数量</p><p>思路：我们知道对于一棵树而言，n 个结点一共有 n-1 条边，也就可以贡献 n-1 个入度，因此至少有一个点的入度为 0。而如果不是一棵树，就会有至少 n 条边，也就至少可以贡献 n 个入度，那么 n 个结点就至少全都有入度了。显然的，一个图至少含有 n 条边时就一定有环。有了上述思路以后就可以发现，这道题本质上就是在判断连通分量是否含有环，如果有环，那么该连通分量定向边以后就不会产生 0 入度的顶点，反之，如果没有环，那么定向边以后最少产生 1 个 0 入度的点。</p><ul><li><p>算法一：遍历图。我们采用 dfs 遍历的方式即可解决问题。一边遍历一边打标记，遇到已经打过标记的非父结点就表明当前连通分量有环。我们使用 C++ 实现</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n+m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span></p></li><li><p>算法二：并查集。由于没有重边，因此在判断每一个连通分量是否含有环时，可以直接通过该连通分量中点和边的数量关系得到结果。我们使用 Python 和 JavaScript 实现</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></li></ul></blockquote><p>C++</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-comment">// 实现 dfs 算法</span><br><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;cstring&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;vector&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;queue&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;stack&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;algorithm&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;unordered_map&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;set&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">100010</span>;<br><br><span class="hljs-type">int</span> n, m;<br>vector&lt;<span class="hljs-type">int</span>&gt; G[N];<br><span class="hljs-type">bool</span> vis[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">dfs</span><span class="hljs-params">(<span class="hljs-type">int</span> fa, <span class="hljs-type">int</span> now, <span class="hljs-type">bool</span>&amp; hasLoop)</span> </span>{<br>    vis[now] = <span class="hljs-literal">true</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; ch: G[now]) {<br>        <span class="hljs-keyword">if</span> (ch != fa) {<br>            <span class="hljs-keyword">if</span> (vis[ch]) hasLoop = <span class="hljs-literal">true</span>;<br>            <span class="hljs-keyword">else</span> <span class="hljs-built_in">dfs</span>(now, ch, hasLoop);<br>        }<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; m;<br>    <span class="hljs-keyword">while</span> (m--) {<br>        <span class="hljs-type">int</span> a, b;<br>        cin &gt;&gt; a &gt;&gt; b;<br>        G[a].<span class="hljs-built_in">push_back</span>(b);<br>        G[b].<span class="hljs-built_in">push_back</span>(a);<br>    }<br>    <br>    <span class="hljs-type">int</span> res = <span class="hljs-number">0</span>;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-keyword">if</span> (!vis[i]) {<br>            <span class="hljs-type">bool</span> hasLoop = <span class="hljs-literal">false</span>;<br>            <span class="hljs-built_in">dfs</span>(<span class="hljs-number">-1</span>, i, hasLoop);<br>            <span class="hljs-keyword">if</span> (!hasLoop) res++;<br>        }<br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><p>Python</p><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-comment"># 实现 dsu 算法</span><br><br>p = [_ <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">100010</span>)]<br><br><br><span class="hljs-keyword">def</span> <span class="hljs-title function_">Find</span>(<span class="hljs-params">x: <span class="hljs-built_in">int</span></span>) -&gt; <span class="hljs-built_in">int</span>:<br>    <span class="hljs-keyword">if</span> x != p[x]: p[x] = Find(p[x])<br>    <span class="hljs-keyword">return</span> p[x]<br><br><br><span class="hljs-keyword">def</span> <span class="hljs-title function_">solve</span>() -&gt; <span class="hljs-literal">None</span>:<br>    n, m = <span class="hljs-built_in">map</span>(<span class="hljs-built_in">int</span>, <span class="hljs-built_in">input</span>().split())<br><br>    edgeNum = [<span class="hljs-number">0</span>] * (n + <span class="hljs-number">1</span>) <span class="hljs-comment"># 每个点的连边数</span><br><br>    <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(m):<br>        u, v = <span class="hljs-built_in">map</span>(<span class="hljs-built_in">int</span>, <span class="hljs-built_in">input</span>().split())<br>        edgeNum[u] += <span class="hljs-number">1</span><br>        edgeNum[v] += <span class="hljs-number">1</span><br>        p[Find(u)] = Find(v)<br><br>    union = {}<br><br>    <span class="hljs-keyword">class</span> <span class="hljs-title class_">node</span>:<br>        <span class="hljs-keyword">def</span> <span class="hljs-title function_">__init__</span>(<span class="hljs-params">self</span>):<br>            <span class="hljs-variable language_">self</span>.v = <span class="hljs-variable language_">self</span>.e = <span class="hljs-number">0</span><br><br>    <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">1</span>, n + <span class="hljs-number">1</span>):<br>        nowp = Find(i)<br>        <span class="hljs-keyword">if</span> nowp <span class="hljs-keyword">not</span> <span class="hljs-keyword">in</span> union: union[nowp] = node()<br><br>        union[nowp].v += <span class="hljs-number">1</span><br>        union[nowp].e += edgeNum[i]<br><br>    res = <span class="hljs-number">0</span><br><br>    <span class="hljs-keyword">for</span> comp <span class="hljs-keyword">in</span> union:<br>        <span class="hljs-keyword">if</span> union[comp].e &gt;&gt; <span class="hljs-number">1</span> == union[comp].v - <span class="hljs-number">1</span>:<br>            res += <span class="hljs-number">1</span><br><br>    <span class="hljs-built_in">print</span>(res)<br><br><br><span class="hljs-keyword">if</span> __name__ == <span class="hljs-string">"__main__"</span>:<br>    solve()<br></code></pre></td></tr></tbody></table></figure><p>JavaScript</p><figure class="highlight javascript"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br></pre></td><td class="code"><pre><code class="hljs javascript"><span class="hljs-comment">// 实现 dsu 算法</span><br><br><span class="hljs-keyword">const</span> readline = <span class="hljs-built_in">require</span>(<span class="hljs-string">'readline'</span>);<br><span class="hljs-keyword">const</span> rl = readline.<span class="hljs-title function_">createInterface</span>({<br>    <span class="hljs-attr">input</span>: process.<span class="hljs-property">stdin</span>,<br>    <span class="hljs-attr">output</span>: process.<span class="hljs-property">stdout</span>,<br>});<br><br><span class="hljs-keyword">let</span> n = <span class="hljs-literal">null</span>;<br><span class="hljs-keyword">let</span> m = <span class="hljs-literal">null</span>;<br><span class="hljs-keyword">let</span> p = [], edgeNum = [];<br><br>rl.<span class="hljs-title function_">on</span>(<span class="hljs-string">'line'</span>, <span class="hljs-function"><span class="hljs-params">line</span> =&gt;</span> {<br>    <span class="hljs-keyword">const</span> [a, b] = line.<span class="hljs-title function_">split</span>(<span class="hljs-string">' '</span>).<span class="hljs-title function_">map</span>(<span class="hljs-function"><span class="hljs-params">i</span> =&gt;</span> <span class="hljs-title class_">Number</span>(i));<br>    <span class="hljs-keyword">if</span> (n === <span class="hljs-literal">null</span>) {<br>        n = a;<br>        m = b;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">let</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>            p[i] = i;<br>            edgeNum[i] = <span class="hljs-number">0</span>;<br>        }<br>    } <span class="hljs-keyword">else</span> {<br>        edgeNum[a]++;<br>        edgeNum[b]++;<br>        p[<span class="hljs-title class_">Find</span>(a)] = <span class="hljs-title class_">Find</span>(b);<br>    }<br>});<br><br>rl.<span class="hljs-title function_">on</span>(<span class="hljs-string">'close'</span>, <span class="hljs-function">() =&gt;</span> {<br>    <span class="hljs-keyword">const</span> res = <span class="hljs-title function_">solve</span>();<br>    <span class="hljs-variable language_">console</span>.<span class="hljs-title function_">log</span>(res);<br>});<br><br><span class="hljs-keyword">function</span> <span class="hljs-title function_">Find</span>(<span class="hljs-params">x</span>) {<br>    <span class="hljs-keyword">if</span> (x != p[x]) p[x] = <span class="hljs-title class_">Find</span>(p[x]);<br>    <span class="hljs-keyword">return</span> p[x];<br>}<br><br><span class="hljs-keyword">function</span> <span class="hljs-title function_">solve</span>(<span class="hljs-params"></span>) {<br>    <span class="hljs-keyword">let</span> res = <span class="hljs-number">0</span>;<br><br>    <span class="hljs-comment">// 自定义结构体</span><br>    <span class="hljs-keyword">class</span> <span class="hljs-title class_">Node</span> {<br>        <span class="hljs-title function_">constructor</span>(<span class="hljs-params"></span>) {<br>            <span class="hljs-variable language_">this</span>.<span class="hljs-property">v</span> = <span class="hljs-number">0</span>;<br>            <span class="hljs-variable language_">this</span>.<span class="hljs-property">e</span> = <span class="hljs-number">0</span>;<br>        }<br>    }<br><br>    <span class="hljs-comment">/*</span><br><span class="hljs-comment">        另一种结构体定义方法</span><br><span class="hljs-comment">        function Node() {</span><br><span class="hljs-comment">            this.v = 0;</span><br><span class="hljs-comment">            this.e = 0;</span><br><span class="hljs-comment">        }</span><br><span class="hljs-comment">    */</span><br><br>    <span class="hljs-comment">// 哈希</span><br>    <span class="hljs-keyword">let</span> union = <span class="hljs-keyword">new</span> <span class="hljs-title class_">Map</span>();<br><br>    <span class="hljs-keyword">for</span> (<span class="hljs-keyword">let</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-keyword">let</span> nowp = <span class="hljs-title class_">Find</span>(i); <span class="hljs-comment">// 当前结点的祖宗结点 nowp</span><br>        <span class="hljs-keyword">if</span> (!union.<span class="hljs-title function_">has</span>(nowp)) union.<span class="hljs-title function_">set</span>(nowp, <span class="hljs-keyword">new</span> <span class="hljs-title class_">Node</span>());<br><br>        union.<span class="hljs-title function_">get</span>(nowp).<span class="hljs-property">v</span> += <span class="hljs-number">1</span>;<br>        union.<span class="hljs-title function_">get</span>(nowp).<span class="hljs-property">e</span> += edgeNum[i];<br>    }<br><br>    <span class="hljs-comment">// 判断</span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-keyword">let</span> i <span class="hljs-keyword">of</span> union.<span class="hljs-title function_">keys</span>()) {<br>        <span class="hljs-keyword">if</span> (union.<span class="hljs-title function_">get</span>(i).<span class="hljs-property">e</span> &gt;&gt; <span class="hljs-number">1</span> === union.<span class="hljs-title function_">get</span>(i).<span class="hljs-property">v</span> - <span class="hljs-number">1</span>) {<br>            res++;<br>        }<br>    }<br><br>    <span class="hljs-keyword">return</span> res;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【LCA】树的直径">【LCA】树的直径</h3><p><a href="https://www.acwing.com/problem/content/5563/">https://www.acwing.com/problem/content/5563/</a></p><blockquote><p>题意：给定一棵树，初始时含有 4 个结点分别为 1 到 4，其中 1 号为根结点，2 到 4 均为根结点的叶子结点。现在进行 Q 次操作，每次指定一个已经存在的结点向其插入两个新结点作为叶节点。现在需要在每次操作以后输出这棵树的直径。我们定义<strong>树的直径</strong>为：树中距离最远的两个点之间的距离。</p><p>思路一：暴力搜索。</p><ul><li><p>我们将树重构为无向图，对于每一个状态的无向图，首先从任意一个已存在的结点 A 开始搜索到距离他最远的点 B，然后从 B 点出发搜索到离他最远的点 C，则 B 与 C 之间的距离就是当前状态的树的直径。由于每一个状态的树都要遍历两遍树，于是时间复杂度就是平方阶</p></li><li><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>q</mi><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(qn)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></li></ul><p>思路二：最近公共祖先 LCA。</p><ul><li><p><strong>从树的直径出发</strong>。我们知道，树的直径由直径的两个结点之间的距离决定，因此我们着眼于这两个结点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> 展开。不妨设当前局面直径的两个结点已知为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>，现在插入两个叶子结点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">L_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">L_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>。是否改变了树的直径大小取决于新插入的两个结点对于当前树的影响情况。如果 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">L_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 或 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">L_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 可以替代 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 或 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>，则树的直径就会改变。很显然新插入的两个叶子结点对于直径的两个端点影响是同效果的，因此我们统称新插入的叶子结点为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span>。</p></li><li><p><strong>什么时候树的直径会改变</strong>？对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> 来说，直径是否改变取决于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> 能否替代 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 或 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>，一共有六种情况。我们记 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>dist</mtext><mo stretchy="false">(</mo><mi>A</mi><mo separator="true">,</mo><mi>L</mi><mo stretchy="false">)</mo><mo>=</mo><mi>d</mi><mi>a</mi></mrow><annotation encoding="application/x-tex">\text{dist}(A,L)=da</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">dist</span></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">L</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">a</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>dist</mtext><mo stretchy="false">(</mo><mi>B</mi><mo separator="true">,</mo><mi>L</mi><mo stretchy="false">)</mo><mo>=</mo><mi>d</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">\text{dist}(B,L)=db</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">dist</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">L</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">b</span></span></span></span>，当前树的直径为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mi>e</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">res</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">res</span></span></span></span>，六种情况如下：</p><ol><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>max</mtext><mo stretchy="false">(</mo><mi>d</mi><mi>a</mi><mo separator="true">,</mo><mi>d</mi><mi>b</mi><mo stretchy="false">)</mo><mo>≤</mo><mtext>res</mtext></mrow><annotation encoding="application/x-tex">\text{max}(da, db) \le \text{res}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">max</span></span><span class="mopen">(</span><span class="mord mathnormal">d</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord text"><span class="mord">res</span></span></span></span></span>，交换 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> 得到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span> 种</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>min</mtext><mo stretchy="false">(</mo><mi>d</mi><mi>a</mi><mo separator="true">,</mo><mi>d</mi><mi>b</mi><mo stretchy="false">)</mo><mo>≥</mo><mtext>res</mtext></mrow><annotation encoding="application/x-tex">\text{min}(da,db) \ge \text{res}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">min</span></span><span class="mopen">(</span><span class="mord mathnormal">d</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord text"><span class="mord">res</span></span></span></span></span>，交换 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> 得到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span> 种</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>max</mtext><mo stretchy="false">(</mo><mi>d</mi><mi>a</mi><mo separator="true">,</mo><mi>d</mi><mi>b</mi><mo stretchy="false">)</mo><mo>&gt;</mo><mi>r</mi><mi>e</mi><mi>s</mi><mo separator="true">,</mo><mtext>min</mtext><mo stretchy="false">(</mo><mi>d</mi><mi>a</mi><mo separator="true">,</mo><mi>d</mi><mi>b</mi><mo stretchy="false">)</mo><mo>&lt;</mo><mtext>res</mtext></mrow><annotation encoding="application/x-tex">\text{max}(da,db) &gt;res,\text{min}(da,db) &lt; \text{res}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">max</span></span><span class="mopen">(</span><span class="mord mathnormal">d</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">res</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord text"><span class="mord">min</span></span><span class="mopen">(</span><span class="mord mathnormal">d</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord text"><span class="mord">res</span></span></span></span></span>，交换 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> 得到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span>​ 种</li></ol><p>如图：我们只需要在其中的最大值严格超过当前树的直径 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>res</mtext></mrow><annotation encoding="application/x-tex">\text{res}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord text"><span class="mord">res</span></span></span></span></span> 时更新<strong>直径对应的结点</strong>以及<strong>直径的长度</strong>即可</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403282344777.jpg" alt="六种情况"></p></li><li><p><strong>如何快速计算树上任意两个点之间的距离</strong>？我们可以使用最近公共祖先 LCA 算法。则树上任意两点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo separator="true">,</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x,y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> 之间的距离 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>dist</mtext><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{dist}(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">dist</span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> 为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>dist</mtext><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mtext>dist</mtext><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>r</mi><mi>o</mi><mi>o</mi><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mtext>dist</mtext><mo stretchy="false">(</mo><mi>y</mi><mo separator="true">,</mo><mi>r</mi><mi>o</mi><mi>o</mi><mi>t</mi><mo stretchy="false">)</mo><mo>−</mo><mn>2</mn><mo>×</mo><mtext>dist</mtext><mo stretchy="false">(</mo><mtext>lca</mtext><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>r</mi><mi>o</mi><mi>o</mi><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{dist}(x,y) = \text{dist}(x,root) + \text{dist}(y,root) - 2 \times \text{dist}(\text{lca}(x,y),root)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">dist</span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">dist</span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">roo</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">dist</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">roo</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">dist</span></span><span class="mopen">(</span><span class="mord text"><span class="mord">lca</span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">roo</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span></span></p></li><li><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>q</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(q \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></li></ul></blockquote><p>暴力搜索代码</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;cstring&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;vector&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;queue&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;stack&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;algorithm&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;unordered_map&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;set&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">500010</span>;<br><br>vector&lt;<span class="hljs-type">int</span>&gt; g[N];<br><span class="hljs-type">int</span> d[N];<br><span class="hljs-type">bool</span> vis[N];<br>pair&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt; res;     <span class="hljs-comment">// first 为最远距离；second 为对应结点编号</span><br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">dfs</span><span class="hljs-params">(<span class="hljs-type">int</span> pre, <span class="hljs-type">int</span> now)</span> </span>{<br>    <span class="hljs-keyword">if</span> (vis[now]) <span class="hljs-keyword">return</span>;<br>    <br>    vis[now] = <span class="hljs-literal">true</span>;<br>    <br>    <span class="hljs-keyword">if</span> (pre != <span class="hljs-number">-1</span>) {<br>        d[now] = d[pre] + <span class="hljs-number">1</span>;<br>        <span class="hljs-keyword">if</span> (d[now] &gt; res.first) {<br>            res = {d[now], now};<br>        }<br>    }<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; ch: g[now]) {<br>        <span class="hljs-built_in">dfs</span>(now, ch);<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-comment">// init</span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">2</span>; i &lt;= <span class="hljs-number">4</span>; i++) {<br>        g[<span class="hljs-number">1</span>].<span class="hljs-built_in">push_back</span>(i);<br>        g[i].<span class="hljs-built_in">push_back</span>(<span class="hljs-number">1</span>);<br>    }<br>    <br>    <span class="hljs-type">int</span> now = <span class="hljs-number">4</span>;<br>    <br>    <span class="hljs-type">int</span> Q;<br>    cin &gt;&gt; Q;<br>    <span class="hljs-keyword">while</span> (Q--) {<br>        <span class="hljs-type">int</span> id;<br>        cin &gt;&gt; id;<br>        <br>        g[id].<span class="hljs-built_in">push_back</span>(++now);<br>        g[now].<span class="hljs-built_in">push_back</span>(id);<br>        <br>        g[id].<span class="hljs-built_in">push_back</span>(++now);<br>        g[now].<span class="hljs-built_in">push_back</span>(id);<br>        <br>        res = {<span class="hljs-number">-1</span>, <span class="hljs-number">-1</span>};<br>        <br>        <span class="hljs-comment">// 第一趟</span><br>        <span class="hljs-built_in">memset</span>(vis, <span class="hljs-literal">false</span>, <span class="hljs-keyword">sizeof</span> vis);<br>        <span class="hljs-built_in">memset</span>(d, <span class="hljs-number">0</span>, <span class="hljs-keyword">sizeof</span> d);<br>        d[<span class="hljs-number">1</span>] = <span class="hljs-number">0</span>;<br>        <span class="hljs-built_in">dfs</span>(<span class="hljs-number">-1</span>, <span class="hljs-number">1</span>);<br>        <br>        <span class="hljs-comment">// 第二趟</span><br>        <span class="hljs-built_in">memset</span>(vis, <span class="hljs-literal">false</span>, <span class="hljs-keyword">sizeof</span> vis);<br>        <span class="hljs-built_in">memset</span>(d, <span class="hljs-number">0</span>, <span class="hljs-keyword">sizeof</span> d);<br>        d[res.second] = <span class="hljs-number">0</span>;<br>        <span class="hljs-built_in">dfs</span>(<span class="hljs-number">-1</span>, res.second);<br>        <br>        cout &lt;&lt; res.first &lt;&lt; <span class="hljs-string">"\n"</span>;<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><p>LCA 代码</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br><span class="line">82</span><br><span class="line">83</span><br><span class="line">84</span><br><span class="line">85</span><br><span class="line">86</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;cstring&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;vector&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;queue&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;stack&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;algorithm&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;unordered_map&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;set&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">1000010</span>, M = <span class="hljs-number">20</span>;<br><br><span class="hljs-type">int</span> d[N];        <span class="hljs-comment">// d[i] 表示 i 号点到根结点的距离</span><br><span class="hljs-type">int</span> to[N][M];    <span class="hljs-comment">// to[i][j] 表示 i 号点向上跳 2^j 步后到达的结点编号</span><br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">lca</span><span class="hljs-params">(<span class="hljs-type">int</span> a, <span class="hljs-type">int</span> b)</span> </span>{<br>    <span class="hljs-keyword">if</span> (d[a] &lt; d[b]) <span class="hljs-built_in">swap</span>(a, b);<br><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> k = M - <span class="hljs-number">1</span>; k &gt;= <span class="hljs-number">0</span>; k--)<br>        <span class="hljs-keyword">if</span> (d[to[a][k]] &gt;= d[b])<br>            a = to[a][k];<br><br>    <span class="hljs-keyword">if</span> (a == b) <span class="hljs-keyword">return</span> a;<br><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> k = M - <span class="hljs-number">1</span>; k &gt;= <span class="hljs-number">0</span>; k--)<br>        <span class="hljs-keyword">if</span> (to[a][k] != to[b][k])<br>            a = to[a][k], b = to[b][k];<br><br>    <span class="hljs-keyword">return</span> to[a][<span class="hljs-number">0</span>];<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">dist</span><span class="hljs-params">(<span class="hljs-type">int</span> a, <span class="hljs-type">int</span> b)</span> </span>{<br>    <span class="hljs-keyword">return</span> d[a] + d[b] - <span class="hljs-number">2</span> * d[<span class="hljs-built_in">lca</span>(a, b)];<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-type">int</span> Q;<br>    cin &gt;&gt; Q;<br><br>    <span class="hljs-comment">// init lca</span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">2</span>; i &lt;= <span class="hljs-number">4</span>; i++) {<br>        d[i] = <span class="hljs-number">1</span>;<br>        to[i][<span class="hljs-number">0</span>] = <span class="hljs-number">1</span>;<br>    }<br><br>    <span class="hljs-type">int</span> A = <span class="hljs-number">2</span>, B = <span class="hljs-number">4</span>, now = <span class="hljs-number">4</span>, res = <span class="hljs-number">2</span>;<br><br>    <span class="hljs-keyword">while</span> (Q--) {<br>        <span class="hljs-type">int</span> fa;<br>        cin &gt;&gt; fa;<br><br>        <span class="hljs-type">int</span> L1 = ++now, L2 = ++now;<br><br>        <span class="hljs-comment">// upd lca</span><br>        d[L1] = d[fa] + <span class="hljs-number">1</span>;<br>        d[L2] = d[fa] + <span class="hljs-number">1</span>;<br>        to[L1][<span class="hljs-number">0</span>] = fa;<br>        to[L2][<span class="hljs-number">0</span>] = fa;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> k = <span class="hljs-number">1</span>; k &lt;= M - <span class="hljs-number">1</span>; k++) {<br>            to[L1][k] = to[ to[L1][k<span class="hljs-number">-1</span>] ][ k<span class="hljs-number">-1</span> ];<br>            to[L2][k] = to[ to[L2][k<span class="hljs-number">-1</span>] ][ k<span class="hljs-number">-1</span> ];<br>        }<br><br>        <span class="hljs-type">int</span> da = <span class="hljs-built_in">dist</span>(A, L1), db = <span class="hljs-built_in">dist</span>(B, L1);<br><br>        <span class="hljs-keyword">if</span> (<span class="hljs-built_in">max</span>(da, db) &lt;= res) res = res;<br>        <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (<span class="hljs-built_in">min</span>(da, db) &gt;= res) {<br>            <span class="hljs-keyword">if</span> (da &gt; db) res = da, B = L1;<br>            <span class="hljs-keyword">else</span> res = db, A = L1;<br>        } <span class="hljs-keyword">else</span> {<br>            <span class="hljs-keyword">if</span> (da &gt; db) res = da, B = L1;<br>            <span class="hljs-keyword">else</span> res = db, A = L1;<br>        }<br><br>        cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure>]]></content>
     
     
     <categories>
@@ -1479,11 +1479,11 @@
   
   
   <entry>
-    <title>graphs</title>
-    <link href="/Algorithm/graphs/"/>
-    <url>/Algorithm/graphs/</url>
+    <title>greedy</title>
+    <link href="/Algorithm/greedy/"/>
+    <url>/Algorithm/greedy/</url>
     
-    <content type="html"><![CDATA[<h3 id="图论">图论</h3><h3 id="【拓扑】有向图的拓扑序列">【拓扑】有向图的拓扑序列</h3><p><a href="https://www.acwing.com/problem/content/850/">https://www.acwing.com/problem/content/850/</a></p><blockquote><p>题意：输出一个图的拓扑序，不存在则输出-1</p><p>思路：</p><ul><li>首先我们要知道拓扑图的概念，感官上就是一张图可以从一个方向拓展到全图，用数学语言就是：若一个由图中所有点构成的序列 A 满足：对于图中的每条边 (x,y)，x 在 A 中都出现在 y 之前，则称 A 是该图的一个拓扑序列</li><li>接着我们就想要寻找这样的序列 A 了，可以发现对于每一个可扩展的点，入度一定为0，那么我们就从这些点开始宽搜，将搜到的点的入度-1，即删除这条边，直到最后。如果全图的点的入度都变为了0，则此图可拓扑</li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n+m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">100010</span>;<br><br><span class="hljs-type">int</span> n, m;<br>vector&lt;<span class="hljs-type">int</span>&gt; G[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-comment">// 建图 </span><br>    cin &gt;&gt; n &gt;&gt; m;<br>    <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">d</span><span class="hljs-params">(n + <span class="hljs-number">1</span>, <span class="hljs-number">0</span>)</span></span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= m; i++) {<br>        <span class="hljs-type">int</span> a, b;<br>        cin &gt;&gt; a &gt;&gt; b;<br>        d[b]++;<br>        G[a].<span class="hljs-built_in">push_back</span>(b);<br>    }<br>    <br>    <span class="hljs-comment">// 预处理宽搜起始点集</span><br>    queue&lt;<span class="hljs-type">int</span>&gt; q;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++)<br>        <span class="hljs-keyword">if</span> (!d[i])<br>            q.<span class="hljs-built_in">push</span>(i);<br>    <br>    <span class="hljs-comment">// 宽搜处理</span><br>    vector&lt;<span class="hljs-type">int</span>&gt; res;<br>    <span class="hljs-keyword">while</span> (q.<span class="hljs-built_in">size</span>()) {<br>        <span class="hljs-keyword">auto</span> h = q.<span class="hljs-built_in">front</span>();<br>        q.<span class="hljs-built_in">pop</span>();<br>        res.<span class="hljs-built_in">push_back</span>(h);<br>        <br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; ch: G[h]) {<br>            d[ch]--;<br>            <span class="hljs-keyword">if</span> (!d[ch]) q.<span class="hljs-built_in">push</span>(ch);<br>        }<br>    }<br>    <br>    <span class="hljs-comment">// 输出合法拓扑序</span><br>    <span class="hljs-keyword">if</span> (res.<span class="hljs-built_in">size</span>() == n) {<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; x: res) {<br>            cout &lt;&lt; x &lt;&lt; <span class="hljs-string">" "</span>;<br>        }<br>    } <span class="hljs-keyword">else</span> {<br>        cout &lt;&lt; <span class="hljs-number">-1</span> &lt;&lt; <span class="hljs-string">"\n"</span>;<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【基环树-拓扑】Mad-City-fire">【基环树/拓扑】Mad City<span class="github-emoji"><span>🔥</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/1f525.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span></h3><p><a href="https://codeforces.com/contest/1873/problem/H">https://codeforces.com/contest/1873/problem/H</a></p><blockquote><p>tag：基环树、拓扑排序</p><p>题意：给定一个基环树，现在图上有两个点，分别叫做A，B。现在B想要逃脱A的抓捕，问对于给定的局面，B能否永远逃离A的抓捕</p><p>思路：思路很简单，我们只需要分B所在位置的两种情况讨论即可</p><ol><li>B不在环上：此时我们记距离B最近的环上的那个点叫 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mi>a</mi><mi>g</mi></mrow><annotation encoding="application/x-tex">tag</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8095em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">t</span><span class="mord mathnormal">a</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span></span></span></span>，我们需要比较的是A到tag点的距离 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">d_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 和B到tag的距离 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mi>B</mi></msub></mrow><annotation encoding="application/x-tex">d_B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，如果 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mi>B</mi></msub><mo>&lt;</mo><msub><mi>d</mi><mi>A</mi></msub></mrow><annotation encoding="application/x-tex">d_B &lt; d_A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，则一定可以逃脱，否则一定不可以逃脱</li><li>B在环上：此时我们只需要判断当前的A点是否与B的位置重合即可，如果重合那就无法逃脱，反之B一定可以逃脱。</li></ol><p>代码实现：</p><ol><li>对于第一种情况，我们需要找到tag点以及计算A和B到tag点的距离，</li></ol><p>时间复杂度：</p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br><span class="line">82</span><br><span class="line">83</span><br><span class="line">84</span><br><span class="line">85</span><br><span class="line">86</span><br><span class="line">87</span><br><span class="line">88</span><br><span class="line">89</span><br><span class="line">90</span><br><span class="line">91</span><br><span class="line">92</span><br><span class="line">93</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">200010</span>;<br><br><span class="hljs-type">int</span> n, a, b;<br>vector&lt;<span class="hljs-type">int</span>&gt; G[N];<br><span class="hljs-type">int</span> rd[N], tag, d[N];<br><span class="hljs-type">bool</span> del[N], vis[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">init</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        G[i].<span class="hljs-built_in">clear</span>();        <span class="hljs-comment">// 存无向图 </span><br>        rd[i] = <span class="hljs-number">0</span>;            <span class="hljs-comment">// 统计每一个结点的入度 </span><br>        del[i] = <span class="hljs-literal">false</span>;        <span class="hljs-comment">// 拓扑删点删边时使用 </span><br>        d[i] = <span class="hljs-number">0</span>;            <span class="hljs-comment">// 图上所有点到 tag 点的距离 </span><br>        vis[i] = <span class="hljs-literal">false</span>;        <span class="hljs-comment">// bfs计算距离时使用 </span><br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">topu</span><span class="hljs-params">(<span class="hljs-type">int</span> now)</span> </span>{<br>    <span class="hljs-keyword">if</span> (rd[now] == <span class="hljs-number">1</span>) {<br>        rd[now]--;<br>        del[now] = <span class="hljs-literal">true</span>;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; ch: G[now]) {<br>            <span class="hljs-keyword">if</span> (del[ch]) <span class="hljs-keyword">continue</span>;<br>            rd[ch]--;<br>            <span class="hljs-keyword">if</span> (now == tag) {<br>                tag = ch;<br>            }<br>            <span class="hljs-built_in">topu</span>(ch);<br>        }<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">bfs</span><span class="hljs-params">()</span> </span>{<br>    queue&lt;<span class="hljs-type">int</span>&gt; q;<br>    q.<span class="hljs-built_in">push</span>(tag);<br>    d[tag] = <span class="hljs-number">0</span>;<br>    <br>    <span class="hljs-keyword">while</span> (q.<span class="hljs-built_in">size</span>()) {<br>        <span class="hljs-keyword">auto</span> now = q.<span class="hljs-built_in">front</span>();<br>        vis[now] = <span class="hljs-literal">true</span>;<br>        q.<span class="hljs-built_in">pop</span>();<br>        <br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; ch: G[now]) {<br>            <span class="hljs-keyword">if</span> (!vis[ch]) {<br>                d[ch] = d[now] + <span class="hljs-number">1</span>;<br>                q.<span class="hljs-built_in">push</span>(ch);<br>                vis[ch] = <span class="hljs-literal">true</span>;<br>            }<br>        }<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-comment">// 初始化</span><br>    cin &gt;&gt; n &gt;&gt; a &gt;&gt; b; <br>    <span class="hljs-built_in">init</span>();<br>    <br>    <span class="hljs-comment">// 建图 </span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-type">int</span> u, v;<br>        cin &gt;&gt; u &gt;&gt; v;<br>        G[u].<span class="hljs-built_in">push_back</span>(v), rd[v]++;<br>        G[v].<span class="hljs-built_in">push_back</span>(u), rd[u]++;<br>    }<br>    <br>    <span class="hljs-comment">// 拓扑删边 &amp; 缩b点</span><br>    tag = b;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-built_in">topu</span>(i);<br>    }<br><br>    <span class="hljs-comment">// 判断结果 &amp; 计算距离 </span><br>    <span class="hljs-keyword">if</span> (rd[b] == <span class="hljs-number">2</span> &amp;&amp; a != b) {<br>        <span class="hljs-comment">// b点在环上</span><br>        cout &lt;&lt; <span class="hljs-string">"Yes\n"</span>;<br>    } <span class="hljs-keyword">else</span> {<br>        <span class="hljs-comment">// b不在环上</span><br>        <span class="hljs-built_in">bfs</span>();<br>        cout &lt;&lt; (d[a] &gt; d[b] ? <span class="hljs-string">"Yes\n"</span> : <span class="hljs-string">"No\n"</span>);<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-number">0</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br>    cin &gt;&gt; T;<br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【二分图】染色法判定二分图">【二分图】染色法判定二分图</h3><p><a href="https://www.acwing.com/problem/content/862/">https://www.acwing.com/problem/content/862/</a></p><blockquote><p>题意：给定一个无向图，可能有重边和自环。问是否可以构成二分图。</p><p>二分图的定义：一个图可以被分成两个点集，每个点集内部没有边相连（可以不是连通图）</p><p>思路：利用<strong>染色法</strong>，遍历每一个连通分量，选择连通分量中的任意一点进行染色扩展</p><ul><li>如果扩展到的点没有染过色，则染成与当前点相对的颜色</li><li>如果扩展到的点已经被染过色了且染的颜色和当前点的颜色相同，则无法构成二分图（奇数环）</li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n+e)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">e</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">100010</span>;<br><br><span class="hljs-type">int</span> n, m;<br>vector&lt;<span class="hljs-type">int</span>&gt; G[N], <span class="hljs-built_in">col</span>(N);<br><br><span class="hljs-function"><span class="hljs-type">bool</span> <span class="hljs-title">bfs</span><span class="hljs-params">(<span class="hljs-type">int</span> u)</span> </span>{<br>    queue&lt;<span class="hljs-type">int</span>&gt; q;<br>    q.<span class="hljs-built_in">push</span>(u);<br>    col[u] = <span class="hljs-number">1</span>;<br><br>    <span class="hljs-keyword">while</span> (q.<span class="hljs-built_in">size</span>()) {<br>        <span class="hljs-type">int</span> now = q.<span class="hljs-built_in">front</span>();<br>        q.<span class="hljs-built_in">pop</span>();<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; ch: G[now]) {<br>            <span class="hljs-keyword">if</span> (!col[ch]) {<br>                col[ch] = -col[now];<br>                q.<span class="hljs-built_in">push</span>(ch);<br>            }<br>            <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (col[ch] == col[now]) {<br>                <span class="hljs-keyword">return</span> <span class="hljs-literal">false</span>;<br>            }<br>        }<br>    }<br><br>    <span class="hljs-keyword">return</span> <span class="hljs-literal">true</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; m;<br>    <span class="hljs-keyword">while</span> (m--) {<br>        <span class="hljs-type">int</span> u, v;<br>        cin &gt;&gt; u &gt;&gt; v;<br>        G[u].<span class="hljs-built_in">push_back</span>(v);<br>        G[v].<span class="hljs-built_in">push_back</span>(u);<br>    }<br><br>    <span class="hljs-comment">// 遍历每一个连通分量</span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-keyword">if</span> (!col[i]) {<br>            <span class="hljs-type">bool</span> ok = <span class="hljs-built_in">bfs</span>(i);<br>            <span class="hljs-keyword">if</span> (!ok) {<br>                cout &lt;&lt; <span class="hljs-string">"No\n"</span>;<br>                <span class="hljs-keyword">return</span>;<br>            }<br>        }<br>    }<br><br>    cout &lt;&lt; <span class="hljs-string">"Yes\n"</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【最小生成树】Kruskal算法求最小生成树">【最小生成树】Kruskal算法求最小生成树</h3><p><a href="https://www.acwing.com/problem/content/861/">https://www.acwing.com/problem/content/861/</a></p><blockquote><p>题意：给定一个无向图，可能含有重边和自环。试判断能否求解其中的最小生成树，如果可以给出最小生成树的权值</p><p>思路：根据数据量，可以发现顶点数很大，不适用 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mi>r</mi><mi>i</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">Prim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">im</span></span></span></span> 算法，只能用 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi><mi>r</mi><mi>u</mi><mi>s</mi><mi>k</mi><mi>a</mi><mi>l</mi></mrow><annotation encoding="application/x-tex">Kruskal</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">Kr</span><span class="mord mathnormal">u</span><span class="mord mathnormal">s</span><span class="mord mathnormal">ka</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span></span></span> 算法，下面简单介绍一下该算法的流程</p><ul><li>自环首先排除 - 显然这条边连接的“两个”顶点是不可能选进 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mi>S</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">MST</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">MST</span></span></span></span> 的</li><li>首先将每一个结点看成一个连通分量</li><li>接着按照权值将所有的边升序排序后，依次选择<ul><li>如果选出的这条边的两个顶点不在一个连通分量中，则选择这条边并将两个顶点所在的连通分量合并</li><li>如果选出的这条边的两个顶点在同一个连通分量中，则不能选择这条边（否则会使得构造的树形成环）</li></ul></li><li>最后统计选择的边的数量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">num</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span></span></span></span> 进行判断即可<ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi><mo>=</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">num=n-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，则可以生成最小生成树</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi><mo>&lt;</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">num&lt;n-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，则无法生成最小生成树</li></ul></li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>e</mi><mi>log</mi><mo>⁡</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(e\log e)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">e</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">e</span><span class="mclose">)</span></span></span></span>​ - 因为最大的时间开销在对所有的边的权值进行排序上</li></ul></blockquote><p>C++</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><span class="hljs-keyword">typedef</span> <span class="hljs-type">long</span> <span class="hljs-type">long</span> ll;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">100010</span>;<br><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">edge</span> {<br>    <span class="hljs-type">int</span> a, b;<br>    <span class="hljs-type">int</span> w;<br>};<br><br><span class="hljs-type">int</span> n, m;<br>vector&lt;edge&gt; edges;<br><span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">p</span><span class="hljs-params">(N)</span></span>;<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">Find</span><span class="hljs-params">(<span class="hljs-type">int</span> now)</span> </span>{<br>    <span class="hljs-keyword">if</span> (p[now] != now) {<br>        p[now] = <span class="hljs-built_in">Find</span>(p[now]);<br>    }<br>    <span class="hljs-keyword">return</span> p[now];<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; m;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= m; i++) {<br>        <span class="hljs-type">int</span> a, b, w;<br>        cin &gt;&gt; a &gt;&gt; b &gt;&gt; w;<br>        <span class="hljs-keyword">if</span> (a == b) {<br>            <span class="hljs-keyword">continue</span>;<br>        }<br>        edges.<span class="hljs-built_in">push_back</span>({a, b, w});<br>    }<br><br>    <span class="hljs-comment">// 按照边权升序排序</span><br>    <span class="hljs-built_in">sort</span>(edges.<span class="hljs-built_in">begin</span>(), edges.<span class="hljs-built_in">end</span>(), [&amp;](edge&amp; x, edge&amp; y) {<br>        <span class="hljs-keyword">return</span> x.w &lt; y.w;<br>    });<br><br>    <span class="hljs-comment">// 选边</span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        p[i] = i;<br>    }<br><br>    <span class="hljs-type">int</span> res = <span class="hljs-number">0</span>, num = <span class="hljs-number">0</span>;<br><br>    <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; e: edges) {<br>        <span class="hljs-type">int</span> pa = <span class="hljs-built_in">Find</span>(e.a), pb = <span class="hljs-built_in">Find</span>(e.b);<br>        <span class="hljs-keyword">if</span> (pa != pb) {<br>            num++;<br>            p[pa] = pb;<br>            res += e.w;<br>        }<br><br>        <span class="hljs-keyword">if</span> (num == n - <span class="hljs-number">1</span>) {<br>            <span class="hljs-keyword">break</span>;<br>        }<br>    }<br><br>    <span class="hljs-comment">// 特判：选出来的边数无法构成一棵树</span><br>    <span class="hljs-keyword">if</span> (num &lt; n - <span class="hljs-number">1</span>) {<br>        cout &lt;&lt; <span class="hljs-string">"impossible\n"</span>;<br>        <span class="hljs-keyword">return</span>;<br>    }<br><br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><p>Python</p><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">def</span> <span class="hljs-title function_">Find</span>(<span class="hljs-params">x: <span class="hljs-built_in">int</span>, p: <span class="hljs-built_in">list</span></span>) -&gt; <span class="hljs-built_in">int</span>:<br>    <span class="hljs-keyword">if</span> p[x] != x: p[x] = Find(p[x], p)<br>    <span class="hljs-keyword">return</span> p[x]<br><br><span class="hljs-keyword">def</span> <span class="hljs-title function_">kruskal</span>(<span class="hljs-params">n: <span class="hljs-built_in">int</span>, m: <span class="hljs-built_in">int</span>, edges: <span class="hljs-built_in">list</span></span>) -&gt; <span class="hljs-built_in">int</span>:<br>    <span class="hljs-comment"># 按边权对边进行降序排序</span><br>    edges.sort(key=<span class="hljs-keyword">lambda</span> edge: edge[-<span class="hljs-number">1</span>])<br><br>    <span class="hljs-comment"># dsu 初始化</span><br>    p = [<span class="hljs-literal">None</span>] + [i <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">1</span>, n + <span class="hljs-number">1</span>)]<br><br>    <span class="hljs-comment"># 选边</span><br>    cnt = <span class="hljs-built_in">sum</span> = <span class="hljs-number">0</span><br>    <span class="hljs-keyword">for</span> edge <span class="hljs-keyword">in</span> edges:<br>        <span class="hljs-keyword">if</span> cnt == n - <span class="hljs-number">1</span>: <span class="hljs-keyword">break</span><br><br>        pa, pb = Find(edge[<span class="hljs-number">0</span>], p), Find(edge[<span class="hljs-number">1</span>], p)<br>        <span class="hljs-keyword">if</span> pa != pb:<br>            p[pa] = pb<br>            cnt += <span class="hljs-number">1</span><br>            <span class="hljs-built_in">sum</span> += edge[<span class="hljs-number">2</span>]<br><br>    <span class="hljs-keyword">return</span> <span class="hljs-built_in">sum</span> <span class="hljs-keyword">if</span> cnt == n - <span class="hljs-number">1</span> <span class="hljs-keyword">else</span> <span class="hljs-number">0</span><br><br><br><span class="hljs-keyword">if</span> __name__ == <span class="hljs-string">"__main__"</span>:<br>    n, m = <span class="hljs-built_in">map</span>(<span class="hljs-built_in">int</span>, <span class="hljs-built_in">input</span>().split())<br><br>    edges = []<br>    <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(m):<br>        edge = <span class="hljs-built_in">tuple</span>(<span class="hljs-built_in">map</span>(<span class="hljs-built_in">int</span>, <span class="hljs-built_in">input</span>().split()))<br>        edges.append(edge)<br><br>    res = kruskal(n, m, edges)<br><br>    <span class="hljs-keyword">if</span> res: <span class="hljs-built_in">print</span>(res)<br>    <span class="hljs-keyword">else</span>: <span class="hljs-built_in">print</span>(<span class="hljs-string">"impossible"</span>)<br></code></pre></td></tr></tbody></table></figure><p>JavaScript</p><figure class="highlight javascript"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br></pre></td><td class="code"><pre><code class="hljs javascript"><span class="hljs-keyword">const</span> readline = <span class="hljs-built_in">require</span>(<span class="hljs-string">'readline'</span>);<br><span class="hljs-keyword">const</span> rl = readline.<span class="hljs-title function_">createInterface</span>({<br>    <span class="hljs-attr">input</span>: process.<span class="hljs-property">stdin</span>,<br>    <span class="hljs-attr">output</span>: process.<span class="hljs-property">stdout</span>,<br>});<br><br><span class="hljs-keyword">let</span> n = <span class="hljs-literal">null</span>;<br><span class="hljs-keyword">let</span> m = <span class="hljs-literal">null</span>;<br><span class="hljs-keyword">const</span> edges = [];<br><br>rl.<span class="hljs-title function_">on</span>(<span class="hljs-string">'line'</span>, <span class="hljs-function"><span class="hljs-params">line</span> =&gt;</span> {<br>    <span class="hljs-keyword">const</span> [a, b, c] = line.<span class="hljs-title function_">split</span>(<span class="hljs-string">' '</span>).<span class="hljs-title function_">map</span>(<span class="hljs-function"><span class="hljs-params">i</span> =&gt;</span> <span class="hljs-title class_">Number</span>(i));<br>    <span class="hljs-keyword">if</span> (n === <span class="hljs-literal">null</span>) {<br>        n = a;<br>        m = b;<br>    } <span class="hljs-keyword">else</span> {<br>        edges.<span class="hljs-title function_">push</span>([a, b, c]);<br>    }<br>});<br><br>rl.<span class="hljs-title function_">on</span>(<span class="hljs-string">'close'</span>, <span class="hljs-function">() =&gt;</span> {<br>    <span class="hljs-keyword">const</span> res = <span class="hljs-title function_">kurskal</span>(n, m, edges);<br>    <span class="hljs-variable language_">console</span>.<span class="hljs-title function_">log</span>(res === <span class="hljs-title class_">Infinity</span> ? <span class="hljs-string">'impossible'</span> : res);<br>});<br><br><span class="hljs-keyword">function</span> <span class="hljs-title function_">Find</span>(<span class="hljs-params">x, p</span>) {<br>    <span class="hljs-keyword">if</span> (p[x] != x) p[x] = <span class="hljs-title class_">Find</span>(p[x], p);<br>    <span class="hljs-keyword">return</span> p[x];<br>}<br><br><span class="hljs-keyword">function</span> <span class="hljs-title function_">kurskal</span>(<span class="hljs-params">n, m, edges</span>) {<br>    <span class="hljs-comment">// 对边进行升序排序</span><br>    edges.<span class="hljs-title function_">sort</span>(<span class="hljs-function">(<span class="hljs-params">a, b</span>) =&gt;</span> a[<span class="hljs-number">2</span>] - b[<span class="hljs-number">2</span>]);<br>    <br>    <span class="hljs-comment">// 初始化 dsu</span><br>    p = [];<br>    <span class="hljs-keyword">for</span> (<span class="hljs-keyword">let</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) p[i] = i;<br>    <br>    <span class="hljs-comment">// 选边</span><br>    <span class="hljs-keyword">let</span> cnt = <span class="hljs-number">0</span>, sum = <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-keyword">let</span> [a, b, w] <span class="hljs-keyword">of</span> edges) {<br>        <span class="hljs-keyword">if</span> (cnt == n - <span class="hljs-number">1</span>) {<br>            <span class="hljs-keyword">break</span>;<br>        }<br>        <br>        <span class="hljs-keyword">let</span> pa = <span class="hljs-title class_">Find</span>(a, p), pb = <span class="hljs-title class_">Find</span>(b, p);<br>        <span class="hljs-keyword">if</span> (pa !== pb) {<br>            cnt++;<br>            p[pa] = pb;<br>            sum += w;<br>        }<br>    }<br>    <br>    <span class="hljs-keyword">if</span> (cnt === n - <span class="hljs-number">1</span>) <span class="hljs-keyword">return</span> sum;<br>    <span class="hljs-keyword">else</span> <span class="hljs-keyword">return</span> <span class="hljs-title class_">Infinity</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【最小生成树】Prim算法求最小生成树">【最小生成树】Prim算法求最小生成树</h3><p><a href="https://www.acwing.com/problem/content/860/">https://www.acwing.com/problem/content/860/</a></p><blockquote><p>题意：给定一个稠密无向图，有重边和自环。求出最小生成树</p><p>思路：根据题目的数据量，可以使用邻接矩阵存储的方法配合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mi>r</mi><mi>i</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">Prim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">im</span></span></span></span> 算法求解最小生成树，下面给出该算法的流程</p><ul><li>首先明确一下变量的定义：<ul><li><code>g[i][j]</code> 为无向图的邻接矩阵存储结构</li><li><code>MST[i]</code> 表示 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 号点是否加入了 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mi>S</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">MST</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">MST</span></span></span></span> 集合</li><li><code>d[i]</code> 表示 <code>i</code> 号点到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mi>S</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">MST</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">MST</span></span></span></span> 集合的最短边长度</li></ul></li><li>自环不存储，重边只保留最短的一条</li><li>任选一个点到集合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mi>S</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">MST</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">MST</span></span></span></span> 中，并且更新 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span></span></span></span> 数组</li><li>选择剩余的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 个点，每次选择有以下流程<ul><li>找到最短边，记录最短边长度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">e</span></span></span></span> 和相应的在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</mi><mo>−</mo><mi>M</mi><mi>S</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">U-MST</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">MST</span></span></span></span> 集合中对应的顶点序号 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span></span></li><li>将 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span></span> 号点加入 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mi>S</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">MST</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">MST</span></span></span></span> 集合，同时根据此时选出的最短边的长度来判断是否存在最小生成树</li><li>根据 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span></span></span></span> 号点，更新 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span></span></span></span> 数组，即更新在集合 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>U</mi><mo>−</mo><mi>M</mi><mi>S</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">U-MST</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">MST</span></span></span></span> 中的点到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi><mi>S</mi><mi>T</mi></mrow><annotation encoding="application/x-tex">MST</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">MST</span></span></span></span> 集合中的点的交叉边的最短长度</li></ul></li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><span class="hljs-keyword">typedef</span> <span class="hljs-type">long</span> <span class="hljs-type">long</span> ll;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">510</span>;<br><br><span class="hljs-type">int</span> n, m;<br>vector&lt;vector&lt;<span class="hljs-type">int</span>&gt;&gt; <span class="hljs-built_in">g</span>(N, <span class="hljs-built_in">vector</span>&lt;<span class="hljs-type">int</span>&gt;(N, INT_MAX));<br><span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">d</span><span class="hljs-params">(N, INT_MAX)</span></span>; <span class="hljs-comment">// d[i]表示i号点到MST集合中的最短边长度</span><br><span class="hljs-type">bool</span> MST[N];<br><span class="hljs-type">int</span> res;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">prim</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-comment">// 选任意一个点到MST中并更新d数组</span><br>    MST[<span class="hljs-number">1</span>] = <span class="hljs-literal">true</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++)<br>        <span class="hljs-keyword">if</span> (!MST[i])<br>            d[i] = <span class="hljs-built_in">min</span>(d[i], g[i][<span class="hljs-number">1</span>]);<br><br>    <span class="hljs-comment">// 选剩下的n-1个点到MST中</span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">2</span>; i &lt;= n; i++) {<br>        <span class="hljs-comment">// 1. 找到最短边</span><br>        <span class="hljs-type">int</span> e = INT_MAX, v = <span class="hljs-number">-1</span>; <span class="hljs-comment">// e: 最短边长度，v: 最短边不在MST集合中的顶点</span><br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">1</span>; j &lt;= n; j++)<br>            <span class="hljs-keyword">if</span> (!MST[j] &amp;&amp; d[j] &lt; e)<br>                e = d[j], v = j;<br><br>        <span class="hljs-comment">// 2. 加入MST集合</span><br>        MST[v] = <span class="hljs-literal">true</span>;<br>        <span class="hljs-keyword">if</span> (e == INT_MAX) {<br>            <span class="hljs-comment">// 特判无法构造MST的情况</span><br>            cout &lt;&lt; <span class="hljs-string">"impossible\n"</span>;<br>            <span class="hljs-keyword">return</span>;<br>        } <span class="hljs-keyword">else</span> {<br>            res += e;<br>        }<br><br>        <span class="hljs-comment">// 3. 更新交叉边 - 迭代（覆盖更新）</span><br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">1</span>; j &lt;= n; j++)<br>            <span class="hljs-keyword">if</span> (!MST[j])<br>                d[j] = <span class="hljs-built_in">min</span>(d[j], g[j][v]);<br>    }<br><br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; m;<br>    <span class="hljs-keyword">while</span> (m--) {<br>        <span class="hljs-type">int</span> a, b, w;<br>        cin &gt;&gt; a &gt;&gt; b &gt;&gt; w;<br><br>        <span class="hljs-keyword">if</span> (a == b) {<br>            <span class="hljs-keyword">continue</span>;<br>        }<br><br>        <span class="hljs-keyword">if</span> (g[a][b] == INT_MAX) {<br>            g[a][b] = w;<br>            g[b][a] = w;<br>        } <span class="hljs-keyword">else</span> {<br>            g[a][b] = <span class="hljs-built_in">min</span>(g[a][b], w);<br>            g[b][a] = <span class="hljs-built_in">min</span>(g[b][a], w);<br>        }<br>    }<br><br>    <span class="hljs-built_in">prim</span>();<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【最短路】Dijkstra求最短路-fire">【最短路】Dijkstra求最短路 <span class="github-emoji"><span>🔥</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/1f525.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span></h3><p>朴素版 - <a href="https://www.acwing.com/problem/content/851/">https://www.acwing.com/problem/content/851/</a></p><p>堆优化 - <a href="https://www.acwing.com/problem/content/852/">https://www.acwing.com/problem/content/852/</a></p><blockquote><p>题意：给定一个正边权的有向图，可能存在重边与自环，问 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 号点到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 号点的最短路径长度是多少，如果不可达就输出 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span>。</p><p>思路一：朴素版。点数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>500</mn></mrow><annotation encoding="application/x-tex">1\le n \le 500</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">500</span></span></span></span>，边数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>m</mi><mo>≤</mo><mn>1</mn><msup><mn>0</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">1 \le m\le 10^5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></p><ul><li><p>思路：根据数据量，我们采用邻接矩阵的方式存储「点少边多」的稠密图。我们定义 <code>d[i]</code> 数组表示起点到 <code>i</code> 号点的最短距离。先将起点放入 <code>SPT (Shortest Path Tree)</code> 集合，然后更新所有 <code>V-SPT</code> 中的点到 <code>SPT</code> 集合的最短路径长度。接着循环 <code>n-1</code> 次迭代更新剩余的 <code>n-1</code> 个点，每次迭代的过程中，首先选择距离起点最近的点 <code>vex</code>，然后将该点加入 <code>SPT</code> 集合，最后利用该点更新 <code>V-SPT</code> 集合中和该点有连边的点到起点的最短距离。最终的 <code>d[end]</code> 就是起点 <code>start</code> 到终点 <code>end</code> 的最短距离。</p></li><li><p>总结：算法整体采用贪心与动态规划的思路。与 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>Prim</mtext></mrow><annotation encoding="application/x-tex">\text{Prim}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord text"><span class="mord">Prim</span></span></span></span></span> 算法仔细比对可知，其中的贪心过程几乎一致，即每次选择加入 SPT 集合的点均为当前局面 <code>V-SPT</code> 集合中距离起点最近的点。而动态规划的过程体现在，在求解出集合 <code>V-SPT</code> 中到集合 <code>STP</code> 最短距离的点 <code>vex</code> 之后，利用该点对「在 <code>V-SPT</code> 集合且和 vex 点有连边的点 <code>i</code>」更新 <code>d[i]</code> 的过程。更新前的状态都是在之前的子结构下的最优解。</p></li><li><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></p></li></ul><p>思路二：堆优化。点数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>1.5</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">1\le n \le 1.5 \times 10^5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1.5</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span>，边数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>m</mi><mo>≤</mo><mn>1.5</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">1 \le m \le 1.5 \times 10^5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7804em;vertical-align:-0.136em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1.5</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span></p><ul><li><p>思路：根据数据量，我们采用邻接表的方式存储「点多边少」的稀疏图。如果采用上述朴素 Dijkstra 算法进行求解必然会因为点数过多而超时，因此我们利用数据结构「堆」进行时间开销上的优化。不难发现朴素 Dijkstra 算法在迭代过程中主要有三部分：</p><ol><li>选择距离起点最近的点 <code>vex</code>。因为需要枚举所有的顶点，因此总的时间复杂度为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></li><li>将该点加入 <code>SPT</code> 集合。因为只是简单的打个标记，因此总的时间复杂度为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li><li>利用该点更新 <code>V-SPT</code> 集合中和该点相连的点到起点的最短距离。因为此时枚举的是该点所有的连边，而邻接表的图存储方式无法进行重边的删除，因此最坏情况下会枚举所有的边，时间复杂度为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span></li></ol></li><li><p>时间复杂度：</p></li></ul></blockquote><p>朴素版C++：</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><br><span class="hljs-keyword">using</span> ll = <span class="hljs-type">long</span> <span class="hljs-type">long</span>;<br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">dijkstra_ori</span><span class="hljs-params">(std::vector&lt;std::vector&lt;<span class="hljs-type">int</span>&gt;&gt;&amp; g, <span class="hljs-type">int</span> start, <span class="hljs-type">int</span> end)</span> </span>{<br>    <span class="hljs-type">int</span> n = g.<span class="hljs-built_in">size</span>() - <span class="hljs-number">1</span>;<br>    <span class="hljs-function">std::vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">d</span><span class="hljs-params">(n + <span class="hljs-number">1</span>, INT_MAX &gt;&gt; <span class="hljs-number">1</span>)</span></span>;<br>    <span class="hljs-function">std::vector&lt;<span class="hljs-type">bool</span>&gt; <span class="hljs-title">SPT</span><span class="hljs-params">(n + <span class="hljs-number">1</span>, <span class="hljs-literal">false</span>)</span></span>;<br>    <br>    <span class="hljs-comment">// update start vex</span><br>    d[start] = <span class="hljs-number">0</span>;<br>    SPT[start] = <span class="hljs-literal">true</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-keyword">if</span> (!SPT[i] &amp;&amp; g[start][i] != INT_MAX &gt;&gt; <span class="hljs-number">1</span>) {<br>            d[i] = std::<span class="hljs-built_in">min</span>(d[i], d[start] + g[start][i]);<br>        }<br>    }<br>    <br>    <span class="hljs-comment">// update remain n-1 vex</span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> k = <span class="hljs-number">0</span>; k &lt; n - <span class="hljs-number">1</span>; k++) {<br>        <span class="hljs-type">int</span> vex = <span class="hljs-number">-1</span>;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>            <span class="hljs-keyword">if</span> (!SPT[i] &amp;&amp; (vex == <span class="hljs-number">-1</span> || d[i] &lt; d[vex])) {<br>                vex = i;<br>            }<br>        }<br>        SPT[vex] = <span class="hljs-literal">true</span>;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>            <span class="hljs-keyword">if</span> (!SPT[i] &amp;&amp; g[vex][i] != INT_MAX &gt;&gt; <span class="hljs-number">1</span>) {<br>                d[i] = std::<span class="hljs-built_in">min</span>(d[i], d[vex] + g[vex][i]);<br>            }<br>        }<br>    }<br>    <br>    <span class="hljs-keyword">return</span> d[end] == INT_MAX &gt;&gt; <span class="hljs-number">1</span> ? <span class="hljs-number">-1</span> : d[end];<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-type">int</span> n, m;<br>    cin &gt;&gt; n &gt;&gt; m;<br>    <br>    vector&lt;vector&lt;<span class="hljs-type">int</span>&gt;&gt; <span class="hljs-built_in">g</span>(n + <span class="hljs-number">1</span>, <span class="hljs-built_in">vector</span>&lt;<span class="hljs-type">int</span>&gt;(n + <span class="hljs-number">1</span>, INT_MAX &gt;&gt; <span class="hljs-number">1</span>));<br>    <br>    <span class="hljs-keyword">while</span> (m--) {<br>        <span class="hljs-type">int</span> u, v, w;<br>        cin &gt;&gt; u &gt;&gt; v &gt;&gt; w;<br>        g[u][v] = <span class="hljs-built_in">min</span>(g[u][v], w);<br>    }<br>    <br>    cout &lt;&lt; <span class="hljs-built_in">dijkstra_ori</span>(g, <span class="hljs-number">1</span>, n) &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    std::ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    std::cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    std::cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><p>朴素版Python：</p><figure class="highlight python"><figcaption><span>[]</span></figcaption><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">import</span> heapq<br><span class="hljs-keyword">from</span> collections <span class="hljs-keyword">import</span> defaultdict<br><span class="hljs-keyword">from</span> typing <span class="hljs-keyword">import</span> <span class="hljs-type">List</span>, <span class="hljs-type">Tuple</span><br><span class="hljs-keyword">import</span> math<br><span class="hljs-keyword">from</span> itertools <span class="hljs-keyword">import</span> combinations<br><br>II = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">int</span>(<span class="hljs-built_in">input</span>())<br>FI = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">float</span>(<span class="hljs-built_in">input</span>())<br>MII = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">tuple</span>(<span class="hljs-built_in">map</span>(<span class="hljs-built_in">int</span>, <span class="hljs-built_in">input</span>().split()))<br>LII = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">list</span>(<span class="hljs-built_in">map</span>(<span class="hljs-built_in">int</span>, <span class="hljs-built_in">input</span>().split()))<br><br><br><span class="hljs-keyword">def</span> <span class="hljs-title function_">dijkstra_ori</span>(<span class="hljs-params">g: <span class="hljs-type">List</span>[<span class="hljs-type">List</span>[<span class="hljs-built_in">int</span>]], start: <span class="hljs-built_in">int</span>, end: <span class="hljs-built_in">int</span></span>) -&gt; <span class="hljs-built_in">int</span>:<br>    n = <span class="hljs-built_in">len</span>(g) - <span class="hljs-number">1</span><br>    d = [<span class="hljs-number">10</span> ** <span class="hljs-number">5</span>] * (n + <span class="hljs-number">1</span>)<br>    SPT = [<span class="hljs-literal">False</span>] * (n + <span class="hljs-number">1</span>)<br>    <br>    d[start] = <span class="hljs-number">0</span><br>    SPT[start] = <span class="hljs-literal">True</span><br>    <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">1</span>, n + <span class="hljs-number">1</span>):<br>        <span class="hljs-keyword">if</span> <span class="hljs-keyword">not</span> SPT[i] <span class="hljs-keyword">and</span> g[start][i] != <span class="hljs-number">10</span> ** <span class="hljs-number">5</span>:<br>            d[i] = <span class="hljs-built_in">min</span>(d[i], d[start] + g[start][i])<br>    <br>    <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(n - <span class="hljs-number">1</span>):<br>        vex = -<span class="hljs-number">1</span><br>        <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">1</span>, n + <span class="hljs-number">1</span>):<br>            <span class="hljs-keyword">if</span> <span class="hljs-keyword">not</span> SPT[i] <span class="hljs-keyword">and</span> (vex == -<span class="hljs-number">1</span> <span class="hljs-keyword">or</span> d[i] &lt; d[vex]):<br>                vex = i<br>        SPT[vex] = <span class="hljs-literal">True</span><br>        <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">1</span>, n + <span class="hljs-number">1</span>):<br>            <span class="hljs-keyword">if</span> <span class="hljs-keyword">not</span> SPT[i] <span class="hljs-keyword">and</span> g[vex][i] != <span class="hljs-number">10</span> ** <span class="hljs-number">5</span>:<br>                d[i] = <span class="hljs-built_in">min</span>(d[i], d[vex] + g[vex][i])<br>    <br>    <span class="hljs-keyword">return</span> -<span class="hljs-number">1</span> <span class="hljs-keyword">if</span> d[end] == <span class="hljs-number">10</span> ** <span class="hljs-number">5</span> <span class="hljs-keyword">else</span> d[end]<br><br><br><span class="hljs-keyword">def</span> <span class="hljs-title function_">solve</span>() -&gt; <span class="hljs-literal">None</span>:<br>    n, m = MII()<br>    g = [[<span class="hljs-number">10</span> ** <span class="hljs-number">5</span>] * (n + <span class="hljs-number">1</span>) <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(n + <span class="hljs-number">1</span>)]<br>    <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(m):<br>        u, v, w = MII()<br>        g[u][v] = <span class="hljs-built_in">min</span>(g[u][v], w)<br>    <span class="hljs-built_in">print</span>(dijkstra_ori(g, <span class="hljs-number">1</span>, n))<br><br><br><span class="hljs-keyword">if</span> __name__ == <span class="hljs-string">'__main__'</span>:<br>    T = <span class="hljs-number">1</span><br>    <span class="hljs-comment"># T = II()</span><br>    <span class="hljs-keyword">while</span> T: solve(); T -= <span class="hljs-number">1</span><br></code></pre></td></tr></tbody></table></figure><p>堆优化版C++：</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br></pre></td><td class="code"><pre><code class="hljs cpp">```<br><br>堆优化版Python：<br><br>```python<br><br></code></pre></td></tr></tbody></table></figure><h3 id="【最短路】Floyd求最短路">【最短路】Floyd求最短路</h3><p><a href="https://www.acwing.com/problem/content/856/">https://www.acwing.com/problem/content/856/</a></p><blockquote><p>题意：给定一个稠密有向图，可能存在重边与自环，给出多个询问，需要给出每一个询问的两个点之前的最短路径长度</p><p>思路：我们采用动态规划的思路。在此使用多阶段决策的方法，即每一个路径状态为选择 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>→</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">1\to k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个点的情况下的最短路径长度</p><ul><li>状态表示：<code>f[k][i][j]</code> 表示在前 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个顶点中进行选择（中转），<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 号点到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> 号点的最短路径长度</li><li>状态转移：对于第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个顶点，我们可以选择中转，也可以不中转。<ul><li>对于不选择中转的情况：<code>f[k][i][j] = f[k-1][i][j]</code></li><li>对于可选择中转的情况：<code>f[k][i][j] = f[k-1][i][k] + f[k-1][k][j]</code></li><li>在其中取最小值即可，但是有一个注意点：对于第二种情况，选择是有一个约束的：即如果选择了 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 号点进行转移的话，那么 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 号点到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 号点以及 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 号点到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> 号点都是需要有路径可达的，从而可以选择最小距离</li></ul></li><li>初始化：即选择 0 个站点进行中转时，即 <code>f[0][i][j]</code> 的情况中，<ul><li>如果 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 号点与 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> 号点自环，则取 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></li><li>如果 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 号点与 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> 号点之间有边，则取重边的最小值</li><li>如果 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 号点与 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span></span></span></span> 号点之间无边，则初始化为正无穷</li></ul></li><li>答案状态：对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> 号点到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span> 号点之间的最小路径长度，就是 <code>f[n][a][b]</code></li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>3</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^3)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></li><li>空间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>3</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^3)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></li></ul><p>空间优化推导：我们尝试优化掉记忆数组的第一维度</p><ul><li><p>对于不选择的情况：由于决策局面 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 是从前往后枚举，故当前状态 <code>f[k][i][j]</code> 可以<strong>直接依赖于已经更新出来且不会被当前状态之后的状态再次覆盖的最优子结构 <code>f[i][j]</code></strong>。即上一个局面的选择情况，就是不选择第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个顶点的情况</p></li><li><p>对于选择的情况：如果删除第一维度，我们担心的是当前状态 <code>f[k][i][j]</code> 依赖的两个状态 <code>f[i][k]</code> 与 <code>f[k][j]</code> 会不会被后续覆盖掉，即<strong>我们不确定 <code>f[i][k]</code> 与 <code>f[k][j]</code> 是否是当前第 k 个局面的最优子结构</strong>。尝试推导：</p><blockquote><p>为了确定 <code>f[i][k]</code> 与 <code>f[k][j]</code> 是否是当前第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个局面的最优子结构，其实就是确定对于当前第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个局面，这两个状态会不会在当前状态 <code>f[i][j]</code> 之后被更新覆盖，那么我们就看这两个状态是从哪里转移过来进行更新的。如果 <code>f[i][k]</code> 与 <code>f[k][j]</code> 这两个状态的转移会依赖于当前状态之后的状态，那么删除第一维度就是错误的，反之就是成立的。</p><p>尝试推导 <code>f[i][k]</code> 与 <code>f[k][j]</code> 从何转移更新：利用我们未删除维度时正确的状态转移方程进行推演</p><p>我们知道：<code>f[k][i][k] = min(f[k-1][i][k], f[k-1][i][k] + f[k-1][k][k])</code>，其中的 <code>f[k-1][k][k]</code> 就是一个自环的路径长度，由于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mi>l</mi><mi>o</mi><mi>y</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">floyd</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal" style="margin-right:0.03588em;">oy</span><span class="mord mathnormal">d</span></span></span></span> 算法的约束条件是没有负环，因此 <code>f[k-1][k][k]</code> 一定大于零，故 <code>f[k][i][k]</code> 一定取前者，即 <code>f[k][i][k] = f[k-1][i][k]</code></p><p>同理可知：</p><p><code>f[k][k][j] = f[k-1][k][j]</code></p></blockquote><p>基于上述推导我们可以知道，当前第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个决策局面中的 <code>f[k][i][k]</code> 与 <code>f[k][k][j]</code> 是依赖于上一个决策局面 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 的，也就是说这<strong>两个状态一定是早于当前状态 <code>f[i][j]</code> 被更新覆盖的</strong>，故 <code>f[i][k]</code> 与 <code>f[k][j]</code> 就是当前第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个局面的最优子结构，证毕，可以进行维度的删除</p></li><li><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>3</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^3)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></p></li><li><p>空间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></p></li></ul></blockquote><p>不优化空间</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">210</span>, INF = <span class="hljs-number">0x3f3f3f3f</span>;<br><br><span class="hljs-type">int</span> n, m, Q;<br><span class="hljs-type">int</span> f[N][N][N];<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; m &gt;&gt; Q;<br>    <br>    <span class="hljs-comment">// init</span><br>    <span class="hljs-built_in">memset</span>(f, INF, <span class="hljs-keyword">sizeof</span> f);<br>    <br>    <span class="hljs-comment">// add edges and generate base</span><br>    <span class="hljs-keyword">while</span> (m--) {<br>        <span class="hljs-type">int</span> a, b, w;<br>        cin &gt;&gt; a &gt;&gt; b &gt;&gt; w;<br>        <span class="hljs-keyword">if</span> (a == b) <span class="hljs-keyword">continue</span>;                           <span class="hljs-comment">// 重边就不赋值</span><br>        <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (f[<span class="hljs-number">0</span>][a][b] == INF) f[<span class="hljs-number">0</span>][a][b] = w;     <span class="hljs-comment">// 第一次加边则直接赋值</span><br>        <span class="hljs-keyword">else</span> f[<span class="hljs-number">0</span>][a][b] = <span class="hljs-built_in">min</span>(f[<span class="hljs-number">0</span>][a][b], w);           <span class="hljs-comment">// 再次赋边权就取最小值</span><br>    }<br>    <br>    <span class="hljs-comment">// generate base again</span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++)<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">1</span>; j &lt;= n; j++)<br>            <span class="hljs-keyword">if</span> (i == j) <br>                f[<span class="hljs-number">0</span>][i][j] = <span class="hljs-number">0</span>;                         <span class="hljs-comment">// 自环取边权为 0</span><br>    <br>    <span class="hljs-comment">// dp </span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> k = <span class="hljs-number">1</span>; k &lt;= n; k++)<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++)<br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">1</span>; j &lt;= n; j++) {<br>                <span class="hljs-comment">// 不选第k个顶点</span><br>                f[k][i][j] = f[k - <span class="hljs-number">1</span>][i][j];<br>                <br>                <span class="hljs-comment">// 选择第k个顶点</span><br>                <span class="hljs-keyword">if</span> (f[k - <span class="hljs-number">1</span>][i][k] != INF &amp;&amp; f[k - <span class="hljs-number">1</span>][k][j] != INF)<br>                    f[k][i][j] = <span class="hljs-built_in">min</span>(f[k][i][j], f[k - <span class="hljs-number">1</span>][i][k] + f[k - <span class="hljs-number">1</span>][k][j]);<br>            }<br><br>    <span class="hljs-comment">// query</span><br>    <span class="hljs-keyword">while</span> (Q--) {<br>        <span class="hljs-type">int</span> a, b;<br>        cin &gt;&gt; a &gt;&gt; b;<br>        <span class="hljs-keyword">if</span> (f[n][a][b] == INF) cout &lt;&lt; <span class="hljs-string">"impossible\n"</span>;<br>        <span class="hljs-keyword">else</span> cout &lt;&lt; f[n][a][b] &lt;&lt; <span class="hljs-string">"\n"</span>;<br>    }<br>    <br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><p>优化空间</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">210</span>, INF = <span class="hljs-number">0x3f3f3f3f</span>;<br><br><span class="hljs-type">int</span> n, m, Q;<br><span class="hljs-type">int</span> f[N][N];<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; m &gt;&gt; Q;<br>    <br>    <span class="hljs-comment">// init</span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++)<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">1</span>; j &lt;= n; j++)<br>            <span class="hljs-keyword">if</span> (i == j) f[i][j] = <span class="hljs-number">0</span>;<br>            <span class="hljs-keyword">else</span> f[i][j] = INF;<br>            <br>    <span class="hljs-comment">// base</span><br>    <span class="hljs-keyword">while</span> (m--) {<br>        <span class="hljs-type">int</span> a, b, w;<br>        cin &gt;&gt; a &gt;&gt; b &gt;&gt; w;<br>        <span class="hljs-keyword">if</span> (a == b) <span class="hljs-keyword">continue</span>;<br>        <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (f[a][b] == INF) f[a][b] = w;<br>        <span class="hljs-keyword">else</span> f[a][b] = <span class="hljs-built_in">min</span>(f[a][b], w);<br>    }<br>    <br>    <span class="hljs-comment">// dp</span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> k = <span class="hljs-number">1</span>; k &lt;= n; k++)<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++)<br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">1</span>; j &lt;= n; j++)<br>                <span class="hljs-keyword">if</span> (f[i][k] != INF &amp;&amp; f[k][j] != INF)<br>                    f[i][j] = <span class="hljs-built_in">min</span>(f[i][j], f[i][k] + f[k][j]);<br>    <br>    <span class="hljs-comment">// query</span><br>    <span class="hljs-keyword">while</span> (Q--) {<br>        <span class="hljs-type">int</span> a, b;<br>        cin &gt;&gt; a &gt;&gt; b;<br>        <span class="hljs-keyword">if</span> (f[a][b] == INF) cout &lt;&lt; <span class="hljs-string">"impossible\n"</span>;<br>        <span class="hljs-keyword">else</span> cout &lt;&lt; f[a][b] &lt;&lt; <span class="hljs-string">"\n"</span>;<br>    }<br>    <br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【最短路】关闭分部的可行集合数目">【最短路】关闭分部的可行集合数目</h3><p><a href="https://leetcode.cn/problems/number-of-possible-sets-of-closing-branches/">https://leetcode.cn/problems/number-of-possible-sets-of-closing-branches/</a></p><blockquote><p>标签：二进制枚举、最短路</p><p>题意：给定一个含有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个顶点的无向图，如何删点可以使得剩余的图中顶点两两可达且最大距离不超过 maxDistance？返回所有删点的方案数。</p><p>思路：由于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 的数据范围只有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>→</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">1 \to 10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span></span></span></span>，我们可以直接枚举所有的删点方案。那么如何检查一个方案的合法性呢？直接使用最短路算法检查「所有顶点到每一个顶点」的最远距离即可。这里我们采用朴素 dijkstra 算法。</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mn>2</mn><mi>n</mi></msup><mo>×</mo><msup><mi>n</mi><mn>3</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(2^n \times n^3)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> - 其中枚举需要 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mn>2</mn><mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(2^n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>、计算所有顶点到某个顶点的最远距离需要 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>、检查所有顶点需要 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><figcaption><span>[]</span></figcaption><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span> {<br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">numberOfSets</span><span class="hljs-params">(<span class="hljs-type">int</span> n, <span class="hljs-type">int</span> maxDistance, vector&lt;vector&lt;<span class="hljs-type">int</span>&gt;&gt;&amp; roads)</span> </span>{<br>        vector&lt;vector&lt;<span class="hljs-type">int</span>&gt;&gt; <span class="hljs-built_in">g</span>(n, <span class="hljs-built_in">vector</span>&lt;<span class="hljs-type">int</span>&gt;(n, INT_MAX &gt;&gt; <span class="hljs-number">1</span>));<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; r: roads) {<br>            <span class="hljs-type">int</span> u = r[<span class="hljs-number">0</span>], v = r[<span class="hljs-number">1</span>], w = r[<span class="hljs-number">2</span>];<br>            g[u][v] = g[v][u] = <span class="hljs-built_in">min</span>(g[u][v], w);<br>        }<br><br>        <span class="hljs-keyword">auto</span> get_max_dist = [&amp;](<span class="hljs-type">int</span> mask, <span class="hljs-type">int</span> v) {<br>            vector&lt;<span class="hljs-type">bool</span>&gt; <span class="hljs-built_in">SPT</span>(n);<br>            <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">d</span><span class="hljs-params">(n, INT_MAX)</span></span>;<br>            <br>            d[v] = <span class="hljs-number">0</span>;<br>            SPT[v] = <span class="hljs-literal">true</span>;<br>            <br>            <span class="hljs-type">int</span> cnt = <span class="hljs-number">0</span>;<br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>                <span class="hljs-keyword">if</span> (mask &amp; (<span class="hljs-number">1</span> &lt;&lt; i) &amp;&amp; !SPT[i]) {<br>                    cnt++;<br>                    d[i] = <span class="hljs-built_in">min</span>(d[i], d[v] + g[v][i]);<br>                }<br>            }<br><br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> k = <span class="hljs-number">1</span>; k &lt;= cnt - <span class="hljs-number">1</span>; k++) {<br>                <span class="hljs-type">int</span> vex = <span class="hljs-number">-1</span>;<br>                <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>                    <span class="hljs-keyword">if</span> (mask &amp; (<span class="hljs-number">1</span> &lt;&lt; i) &amp;&amp; !SPT[i] &amp;&amp; (vex == <span class="hljs-number">-1</span> || d[i] &lt; d[vex])) {<br>                        vex = i;<br>                    }<br>                }<br>                SPT[vex] = <span class="hljs-literal">true</span>;<br>                <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>                    <span class="hljs-keyword">if</span> (mask &amp; (<span class="hljs-number">1</span> &lt;&lt; i) &amp;&amp; !SPT[i]) {<br>                        d[i] = <span class="hljs-built_in">min</span>(d[i], d[vex] + g[vex][i]);<br>                    }<br>                }<br>            }<br>            <br>            <span class="hljs-type">int</span> max_dist = <span class="hljs-number">-1</span>;<br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>                <span class="hljs-keyword">if</span> (mask &amp; (<span class="hljs-number">1</span> &lt;&lt; i)) {<br>                    max_dist = <span class="hljs-built_in">max</span>(max_dist, d[i]);<br>                }<br>            }<br>            <br>            <span class="hljs-keyword">return</span> max_dist;<br>        };<br><br>        <span class="hljs-type">int</span> res = <span class="hljs-number">0</span>;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> mask = <span class="hljs-number">0</span>; mask &lt; <span class="hljs-number">1</span> &lt;&lt; n; mask++) {<br>            <span class="hljs-type">bool</span> ok = <span class="hljs-literal">true</span>;<br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>                <span class="hljs-keyword">if</span> (mask &amp; (<span class="hljs-number">1</span> &lt;&lt; i) &amp;&amp; <span class="hljs-built_in">get_max_dist</span>(mask, i) &gt; maxDistance) {<br>                    ok = <span class="hljs-literal">false</span>;<br>                    <span class="hljs-keyword">break</span>;<br>                }<br>            }<br>            res += ok;<br>        }<br><br>        <span class="hljs-keyword">return</span> res;<br>    }<br>};<br></code></pre></td></tr></tbody></table></figure><figure class="highlight python"><figcaption><span>[]</span></figcaption><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span>:<br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">numberOfSets</span>(<span class="hljs-params">self, n: <span class="hljs-built_in">int</span>, maxDistance: <span class="hljs-built_in">int</span>, roads: <span class="hljs-type">List</span>[<span class="hljs-type">List</span>[<span class="hljs-built_in">int</span>]]</span>) -&gt; <span class="hljs-built_in">int</span>:<br>        g = [[<span class="hljs-number">10</span> ** <span class="hljs-number">6</span> <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(n)] <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(n)]<br>        <span class="hljs-keyword">for</span> u, v, w <span class="hljs-keyword">in</span> roads:<br>            g[u][v] = g[v][u] = <span class="hljs-built_in">min</span>(g[u][v], w)<br>        <br>        <span class="hljs-keyword">def</span> <span class="hljs-title function_">get_max_dist</span>(<span class="hljs-params">mask: <span class="hljs-built_in">int</span>, v: <span class="hljs-built_in">int</span></span>):<br>            SPT = [<span class="hljs-literal">False</span> <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(n)]<br>            d = [<span class="hljs-number">10</span> ** <span class="hljs-number">6</span> <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(n)]<br><br>            SPT[v] = <span class="hljs-literal">True</span><br>            d[v] = <span class="hljs-number">0</span><br><br>            cnt = <span class="hljs-number">0</span><br>            <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(n):<br>                <span class="hljs-keyword">if</span> mask &amp; (<span class="hljs-number">1</span> &lt;&lt; i) <span class="hljs-keyword">and</span> <span class="hljs-keyword">not</span> SPT[i]:<br>                    cnt += <span class="hljs-number">1</span><br>                    d[i] = <span class="hljs-built_in">min</span>(d[i], d[v] + g[v][i])<br>            <br>            <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(cnt - <span class="hljs-number">1</span>):<br>                vex = -<span class="hljs-number">1</span><br>                <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(n):<br>                    <span class="hljs-keyword">if</span> mask &amp; (<span class="hljs-number">1</span> &lt;&lt; i) <span class="hljs-keyword">and</span> <span class="hljs-keyword">not</span> SPT[i] <span class="hljs-keyword">and</span> (vex == -<span class="hljs-number">1</span> <span class="hljs-keyword">or</span> d[i] &lt; d[vex]):<br>                        vex = i<br>                SPT[vex] = <span class="hljs-literal">True</span><br>                <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(n):<br>                    <span class="hljs-keyword">if</span> mask &amp; (<span class="hljs-number">1</span> &lt;&lt; i) <span class="hljs-keyword">and</span> <span class="hljs-keyword">not</span> SPT[i]:<br>                        d[i] = <span class="hljs-built_in">min</span>(d[i], d[vex] + g[vex][i])<br>                <br>            max_dist = -<span class="hljs-number">1</span><br>            <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(n):<br>                <span class="hljs-keyword">if</span> mask &amp; (<span class="hljs-number">1</span> &lt;&lt; i):<br>                    max_dist = <span class="hljs-built_in">max</span>(max_dist, d[i])<br>            <br>            <span class="hljs-keyword">return</span> max_dist<br><br>        res = <span class="hljs-number">0</span><br>        <span class="hljs-keyword">for</span> mask <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">1</span> &lt;&lt; n):<br>            ok = <span class="hljs-literal">True</span><br>            <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(n):<br>                <span class="hljs-keyword">if</span> mask &amp; (<span class="hljs-number">1</span> &lt;&lt; i) <span class="hljs-keyword">and</span> get_max_dist(mask, i) &gt; maxDistance:<br>                    ok = <span class="hljs-literal">False</span><br>                    <span class="hljs-keyword">break</span><br>            res += ok<br><br>        <span class="hljs-keyword">return</span> res<br></code></pre></td></tr></tbody></table></figure><h3 id="【思维-遍历】图的遍历">【思维/遍历】图的遍历</h3><p><a href="https://www.luogu.com.cn/problem/P3916">https://www.luogu.com.cn/problem/P3916</a></p><blockquote><p>题意：给定一个有向图，求解每一个点可以到达的编号最大的点</p><p>思路：如果从正向考虑，很显然的一个暴力方法就是对于每一个点都跑一遍 dfs 或者 bfs 获取可达的最大点编号，时间复杂度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>，如果想要在遍历的过程中同时更新其余的点，那只有起点到最大点之间的点可以被更新，可以通过递归时记录路径点进行，时间复杂度几乎不变。我们尝试反向考虑：<strong>反向建边</strong>。既然正向考虑时需要标记的点为最大点与起点的路径，那不如直接从最大值点开始遍历搜索，在将所有的边全部反向以后，从最大值点开始遍历图，这样就可以在线性时间复杂度内解决问题</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n+m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span></p></blockquote><p>bfs 代码</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;queue&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;cstring&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;vector&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">100010</span>;<br><br><span class="hljs-type">int</span> n, m;<br>vector&lt;<span class="hljs-type">int</span>&gt; g[N], <span class="hljs-built_in">res</span>(N);<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">bfs</span><span class="hljs-params">(<span class="hljs-type">int</span> now)</span> </span>{<br>    queue&lt;<span class="hljs-type">int</span>&gt; q;<br><br>    res[now] = now;<br>    q.<span class="hljs-built_in">push</span>(now);<br>    <br>    <span class="hljs-keyword">while</span> (q.<span class="hljs-built_in">size</span>()) {<br>        <span class="hljs-type">int</span> h = q.<span class="hljs-built_in">front</span>();<br>        q.<span class="hljs-built_in">pop</span>();<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; ch: g[h]) {<br>            <span class="hljs-keyword">if</span> (!res[ch]) {<br>                res[ch] = now;<br>                q.<span class="hljs-built_in">push</span>(ch);<br>            }<br>        }<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; m;<br>    <span class="hljs-keyword">while</span> (m--) {<br>        <span class="hljs-type">int</span> a, b;<br>        cin &gt;&gt; a &gt;&gt; b;<br>        g[b].<span class="hljs-built_in">push_back</span>(a);<br>    }<br><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = n; i &gt;= <span class="hljs-number">1</span>; i--) {<br>        <span class="hljs-keyword">if</span> (!res[i]) {<br>            <span class="hljs-built_in">bfs</span>(i);<br>        }<br>    }<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        cout &lt;&lt; res[i] &lt;&lt; <span class="hljs-string">' '</span>;<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><p>dfs 代码</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;queue&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;cstring&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;vector&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">100010</span>;<br><br><span class="hljs-type">int</span> n, m, val;<br>vector&lt;<span class="hljs-type">int</span>&gt; g[N], <span class="hljs-built_in">res</span>(N);<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">dfs</span><span class="hljs-params">(<span class="hljs-type">int</span> now)</span> </span>{<br>    res[now] = val;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; ch: g[now]) {<br>        <span class="hljs-keyword">if</span> (!res[ch]) {<br>            <span class="hljs-built_in">dfs</span>(ch);<br>        }<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; m;<br>    <span class="hljs-keyword">while</span> (m--) {<br>        <span class="hljs-type">int</span> a, b;<br>        cin &gt;&gt; a &gt;&gt; b;<br>        g[b].<span class="hljs-built_in">push_back</span>(a);<br>    }<br><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = n; i &gt;= <span class="hljs-number">1</span>; i--) {<br>        <span class="hljs-keyword">if</span> (!res[i]) {<br>            val = i;<br>            <span class="hljs-built_in">dfs</span>(i);<br>        }<br>    }<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        cout &lt;&lt; res[i] &lt;&lt; <span class="hljs-string">' '</span>;<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【遍历-并查集】孤立点数量">【遍历/并查集】孤立点数量</h3><p><a href="https://www.acwing.com/problem/content/5560/">https://www.acwing.com/problem/content/5560/</a></p><blockquote><p>题意：给定一个无向图，可能不连通，没有重边和自环。现在需要给图中的每一条无向边定向，要求所有的边定完向以后 0 入度的点尽可能的少，给出最少的 0 入度点的数量</p><p>思路：我们知道对于一棵树而言，n 个结点一共有 n-1 条边，也就可以贡献 n-1 个入度，因此至少有一个点的入度为 0。而如果不是一棵树，就会有至少 n 条边，也就至少可以贡献 n 个入度，那么 n 个结点就至少全都有入度了。显然的，一个图至少含有 n 条边时就一定有环。有了上述思路以后就可以发现，这道题本质上就是在判断连通分量是否含有环，如果有环，那么该连通分量定向边以后就不会产生 0 入度的顶点，反之，如果没有环，那么定向边以后最少产生 1 个 0 入度的点。</p><ul><li><p>算法一：遍历图。我们采用 dfs 遍历的方式即可解决问题。一边遍历一边打标记，遇到已经打过标记的非父结点就表明当前连通分量有环。我们使用 C++ 实现</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n+m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span></p></li><li><p>算法二：并查集。由于没有重边，因此在判断每一个连通分量是否含有环时，可以直接通过该连通分量中点和边的数量关系得到结果。我们使用 Python 和 JavaScript 实现</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></li></ul></blockquote><p>C++</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-comment">// 实现 dfs 算法</span><br><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;cstring&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;vector&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;queue&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;stack&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;algorithm&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;unordered_map&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;set&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">100010</span>;<br><br><span class="hljs-type">int</span> n, m;<br>vector&lt;<span class="hljs-type">int</span>&gt; G[N];<br><span class="hljs-type">bool</span> vis[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">dfs</span><span class="hljs-params">(<span class="hljs-type">int</span> fa, <span class="hljs-type">int</span> now, <span class="hljs-type">bool</span>&amp; hasLoop)</span> </span>{<br>    vis[now] = <span class="hljs-literal">true</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; ch: G[now]) {<br>        <span class="hljs-keyword">if</span> (ch != fa) {<br>            <span class="hljs-keyword">if</span> (vis[ch]) hasLoop = <span class="hljs-literal">true</span>;<br>            <span class="hljs-keyword">else</span> <span class="hljs-built_in">dfs</span>(now, ch, hasLoop);<br>        }<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; m;<br>    <span class="hljs-keyword">while</span> (m--) {<br>        <span class="hljs-type">int</span> a, b;<br>        cin &gt;&gt; a &gt;&gt; b;<br>        G[a].<span class="hljs-built_in">push_back</span>(b);<br>        G[b].<span class="hljs-built_in">push_back</span>(a);<br>    }<br>    <br>    <span class="hljs-type">int</span> res = <span class="hljs-number">0</span>;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-keyword">if</span> (!vis[i]) {<br>            <span class="hljs-type">bool</span> hasLoop = <span class="hljs-literal">false</span>;<br>            <span class="hljs-built_in">dfs</span>(<span class="hljs-number">-1</span>, i, hasLoop);<br>            <span class="hljs-keyword">if</span> (!hasLoop) res++;<br>        }<br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><p>Python</p><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-comment"># 实现 dsu 算法</span><br><br>p = [_ <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">100010</span>)]<br><br><br><span class="hljs-keyword">def</span> <span class="hljs-title function_">Find</span>(<span class="hljs-params">x: <span class="hljs-built_in">int</span></span>) -&gt; <span class="hljs-built_in">int</span>:<br>    <span class="hljs-keyword">if</span> x != p[x]: p[x] = Find(p[x])<br>    <span class="hljs-keyword">return</span> p[x]<br><br><br><span class="hljs-keyword">def</span> <span class="hljs-title function_">solve</span>() -&gt; <span class="hljs-literal">None</span>:<br>    n, m = <span class="hljs-built_in">map</span>(<span class="hljs-built_in">int</span>, <span class="hljs-built_in">input</span>().split())<br><br>    edgeNum = [<span class="hljs-number">0</span>] * (n + <span class="hljs-number">1</span>) <span class="hljs-comment"># 每个点的连边数</span><br><br>    <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(m):<br>        u, v = <span class="hljs-built_in">map</span>(<span class="hljs-built_in">int</span>, <span class="hljs-built_in">input</span>().split())<br>        edgeNum[u] += <span class="hljs-number">1</span><br>        edgeNum[v] += <span class="hljs-number">1</span><br>        p[Find(u)] = Find(v)<br><br>    union = {}<br><br>    <span class="hljs-keyword">class</span> <span class="hljs-title class_">node</span>:<br>        <span class="hljs-keyword">def</span> <span class="hljs-title function_">__init__</span>(<span class="hljs-params">self</span>):<br>            <span class="hljs-variable language_">self</span>.v = <span class="hljs-variable language_">self</span>.e = <span class="hljs-number">0</span><br><br>    <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">1</span>, n + <span class="hljs-number">1</span>):<br>        nowp = Find(i)<br>        <span class="hljs-keyword">if</span> nowp <span class="hljs-keyword">not</span> <span class="hljs-keyword">in</span> union: union[nowp] = node()<br><br>        union[nowp].v += <span class="hljs-number">1</span><br>        union[nowp].e += edgeNum[i]<br><br>    res = <span class="hljs-number">0</span><br><br>    <span class="hljs-keyword">for</span> comp <span class="hljs-keyword">in</span> union:<br>        <span class="hljs-keyword">if</span> union[comp].e &gt;&gt; <span class="hljs-number">1</span> == union[comp].v - <span class="hljs-number">1</span>:<br>            res += <span class="hljs-number">1</span><br><br>    <span class="hljs-built_in">print</span>(res)<br><br><br><span class="hljs-keyword">if</span> __name__ == <span class="hljs-string">"__main__"</span>:<br>    solve()<br></code></pre></td></tr></tbody></table></figure><p>JavaScript</p><figure class="highlight javascript"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br></pre></td><td class="code"><pre><code class="hljs javascript"><span class="hljs-comment">// 实现 dsu 算法</span><br><br><span class="hljs-keyword">const</span> readline = <span class="hljs-built_in">require</span>(<span class="hljs-string">'readline'</span>);<br><span class="hljs-keyword">const</span> rl = readline.<span class="hljs-title function_">createInterface</span>({<br>    <span class="hljs-attr">input</span>: process.<span class="hljs-property">stdin</span>,<br>    <span class="hljs-attr">output</span>: process.<span class="hljs-property">stdout</span>,<br>});<br><br><span class="hljs-keyword">let</span> n = <span class="hljs-literal">null</span>;<br><span class="hljs-keyword">let</span> m = <span class="hljs-literal">null</span>;<br><span class="hljs-keyword">let</span> p = [], edgeNum = [];<br><br>rl.<span class="hljs-title function_">on</span>(<span class="hljs-string">'line'</span>, <span class="hljs-function"><span class="hljs-params">line</span> =&gt;</span> {<br>    <span class="hljs-keyword">const</span> [a, b] = line.<span class="hljs-title function_">split</span>(<span class="hljs-string">' '</span>).<span class="hljs-title function_">map</span>(<span class="hljs-function"><span class="hljs-params">i</span> =&gt;</span> <span class="hljs-title class_">Number</span>(i));<br>    <span class="hljs-keyword">if</span> (n === <span class="hljs-literal">null</span>) {<br>        n = a;<br>        m = b;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">let</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>            p[i] = i;<br>            edgeNum[i] = <span class="hljs-number">0</span>;<br>        }<br>    } <span class="hljs-keyword">else</span> {<br>        edgeNum[a]++;<br>        edgeNum[b]++;<br>        p[<span class="hljs-title class_">Find</span>(a)] = <span class="hljs-title class_">Find</span>(b);<br>    }<br>});<br><br>rl.<span class="hljs-title function_">on</span>(<span class="hljs-string">'close'</span>, <span class="hljs-function">() =&gt;</span> {<br>    <span class="hljs-keyword">const</span> res = <span class="hljs-title function_">solve</span>();<br>    <span class="hljs-variable language_">console</span>.<span class="hljs-title function_">log</span>(res);<br>});<br><br><span class="hljs-keyword">function</span> <span class="hljs-title function_">Find</span>(<span class="hljs-params">x</span>) {<br>    <span class="hljs-keyword">if</span> (x != p[x]) p[x] = <span class="hljs-title class_">Find</span>(p[x]);<br>    <span class="hljs-keyword">return</span> p[x];<br>}<br><br><span class="hljs-keyword">function</span> <span class="hljs-title function_">solve</span>(<span class="hljs-params"></span>) {<br>    <span class="hljs-keyword">let</span> res = <span class="hljs-number">0</span>;<br><br>    <span class="hljs-comment">// 自定义结构体</span><br>    <span class="hljs-keyword">class</span> <span class="hljs-title class_">Node</span> {<br>        <span class="hljs-title function_">constructor</span>(<span class="hljs-params"></span>) {<br>            <span class="hljs-variable language_">this</span>.<span class="hljs-property">v</span> = <span class="hljs-number">0</span>;<br>            <span class="hljs-variable language_">this</span>.<span class="hljs-property">e</span> = <span class="hljs-number">0</span>;<br>        }<br>    }<br><br>    <span class="hljs-comment">/*</span><br><span class="hljs-comment">        另一种结构体定义方法</span><br><span class="hljs-comment">        function Node() {</span><br><span class="hljs-comment">            this.v = 0;</span><br><span class="hljs-comment">            this.e = 0;</span><br><span class="hljs-comment">        }</span><br><span class="hljs-comment">    */</span><br><br>    <span class="hljs-comment">// 哈希</span><br>    <span class="hljs-keyword">let</span> union = <span class="hljs-keyword">new</span> <span class="hljs-title class_">Map</span>();<br><br>    <span class="hljs-keyword">for</span> (<span class="hljs-keyword">let</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-keyword">let</span> nowp = <span class="hljs-title class_">Find</span>(i); <span class="hljs-comment">// 当前结点的祖宗结点 nowp</span><br>        <span class="hljs-keyword">if</span> (!union.<span class="hljs-title function_">has</span>(nowp)) union.<span class="hljs-title function_">set</span>(nowp, <span class="hljs-keyword">new</span> <span class="hljs-title class_">Node</span>());<br><br>        union.<span class="hljs-title function_">get</span>(nowp).<span class="hljs-property">v</span> += <span class="hljs-number">1</span>;<br>        union.<span class="hljs-title function_">get</span>(nowp).<span class="hljs-property">e</span> += edgeNum[i];<br>    }<br><br>    <span class="hljs-comment">// 判断</span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-keyword">let</span> i <span class="hljs-keyword">of</span> union.<span class="hljs-title function_">keys</span>()) {<br>        <span class="hljs-keyword">if</span> (union.<span class="hljs-title function_">get</span>(i).<span class="hljs-property">e</span> &gt;&gt; <span class="hljs-number">1</span> === union.<span class="hljs-title function_">get</span>(i).<span class="hljs-property">v</span> - <span class="hljs-number">1</span>) {<br>            res++;<br>        }<br>    }<br><br>    <span class="hljs-keyword">return</span> res;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【LCA】树的直径">【LCA】树的直径</h3><p><a href="https://www.acwing.com/problem/content/5563/">https://www.acwing.com/problem/content/5563/</a></p><blockquote><p>题意：给定一棵树，初始时含有 4 个结点分别为 1 到 4，其中 1 号为根结点，2 到 4 均为根结点的叶子结点。现在进行 Q 次操作，每次指定一个已经存在的结点向其插入两个新结点作为叶节点。现在需要在每次操作以后输出这棵树的直径。我们定义<strong>树的直径</strong>为：树中距离最远的两个点之间的距离。</p><p>思路一：暴力搜索。</p><ul><li><p>我们将树重构为无向图，对于每一个状态的无向图，首先从任意一个已存在的结点 A 开始搜索到距离他最远的点 B，然后从 B 点出发搜索到离他最远的点 C，则 B 与 C 之间的距离就是当前状态的树的直径。由于每一个状态的树都要遍历两遍树，于是时间复杂度就是平方阶</p></li><li><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>q</mi><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(qn)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></li></ul><p>思路二：最近公共祖先 LCA。</p><ul><li><p><strong>从树的直径出发</strong>。我们知道，树的直径由直径的两个结点之间的距离决定，因此我们着眼于这两个结点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> 展开。不妨设当前局面直径的两个结点已知为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>，现在插入两个叶子结点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">L_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">L_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>。是否改变了树的直径大小取决于新插入的两个结点对于当前树的影响情况。如果 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">L_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 或 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>L</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">L_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 可以替代 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 或 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>，则树的直径就会改变。很显然新插入的两个叶子结点对于直径的两个端点影响是同效果的，因此我们统称新插入的叶子结点为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span>。</p></li><li><p><strong>什么时候树的直径会改变</strong>？对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> 来说，直径是否改变取决于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">L</span></span></span></span> 能否替代 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 或 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>，一共有六种情况。我们记 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>dist</mtext><mo stretchy="false">(</mo><mi>A</mi><mo separator="true">,</mo><mi>L</mi><mo stretchy="false">)</mo><mo>=</mo><mi>d</mi><mi>a</mi></mrow><annotation encoding="application/x-tex">\text{dist}(A,L)=da</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">dist</span></span><span class="mopen">(</span><span class="mord mathnormal">A</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">L</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">a</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>dist</mtext><mo stretchy="false">(</mo><mi>B</mi><mo separator="true">,</mo><mi>L</mi><mo stretchy="false">)</mo><mo>=</mo><mi>d</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">\text{dist}(B,L)=db</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">dist</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">L</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">b</span></span></span></span>，当前树的直径为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mi>e</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">res</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">res</span></span></span></span>，六种情况如下：</p><ol><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>max</mtext><mo stretchy="false">(</mo><mi>d</mi><mi>a</mi><mo separator="true">,</mo><mi>d</mi><mi>b</mi><mo stretchy="false">)</mo><mo>≤</mo><mtext>res</mtext></mrow><annotation encoding="application/x-tex">\text{max}(da, db) \le \text{res}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">max</span></span><span class="mopen">(</span><span class="mord mathnormal">d</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord text"><span class="mord">res</span></span></span></span></span>，交换 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> 得到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span> 种</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>min</mtext><mo stretchy="false">(</mo><mi>d</mi><mi>a</mi><mo separator="true">,</mo><mi>d</mi><mi>b</mi><mo stretchy="false">)</mo><mo>≥</mo><mtext>res</mtext></mrow><annotation encoding="application/x-tex">\text{min}(da,db) \ge \text{res}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">min</span></span><span class="mopen">(</span><span class="mord mathnormal">d</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord text"><span class="mord">res</span></span></span></span></span>，交换 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> 得到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span> 种</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>max</mtext><mo stretchy="false">(</mo><mi>d</mi><mi>a</mi><mo separator="true">,</mo><mi>d</mi><mi>b</mi><mo stretchy="false">)</mo><mo>&gt;</mo><mi>r</mi><mi>e</mi><mi>s</mi><mo separator="true">,</mo><mtext>min</mtext><mo stretchy="false">(</mo><mi>d</mi><mi>a</mi><mo separator="true">,</mo><mi>d</mi><mi>b</mi><mo stretchy="false">)</mo><mo>&lt;</mo><mtext>res</mtext></mrow><annotation encoding="application/x-tex">\text{max}(da,db) &gt;res,\text{min}(da,db) &lt; \text{res}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">max</span></span><span class="mopen">(</span><span class="mord mathnormal">d</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">res</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord text"><span class="mord">min</span></span><span class="mopen">(</span><span class="mord mathnormal">d</span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord text"><span class="mord">res</span></span></span></span></span>，交换 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> 得到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span>​ 种</li></ol><p>如图：我们只需要在其中的最大值严格超过当前树的直径 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>res</mtext></mrow><annotation encoding="application/x-tex">\text{res}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord text"><span class="mord">res</span></span></span></span></span> 时更新<strong>直径对应的结点</strong>以及<strong>直径的长度</strong>即可</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202403282344777.jpg" alt="六种情况"></p></li><li><p><strong>如何快速计算树上任意两个点之间的距离</strong>？我们可以使用最近公共祖先 LCA 算法。则树上任意两点 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo separator="true">,</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x,y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> 之间的距离 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>dist</mtext><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{dist}(x,y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">dist</span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span> 为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>dist</mtext><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mtext>dist</mtext><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>r</mi><mi>o</mi><mi>o</mi><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mtext>dist</mtext><mo stretchy="false">(</mo><mi>y</mi><mo separator="true">,</mo><mi>r</mi><mi>o</mi><mi>o</mi><mi>t</mi><mo stretchy="false">)</mo><mo>−</mo><mn>2</mn><mo>×</mo><mtext>dist</mtext><mo stretchy="false">(</mo><mtext>lca</mtext><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>r</mi><mi>o</mi><mi>o</mi><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{dist}(x,y) = \text{dist}(x,root) + \text{dist}(y,root) - 2 \times \text{dist}(\text{lca}(x,y),root)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">dist</span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">dist</span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">roo</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">dist</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">roo</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">dist</span></span><span class="mopen">(</span><span class="mord text"><span class="mord">lca</span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">roo</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span></span></p></li><li><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>q</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(q \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></li></ul></blockquote><p>暴力搜索代码</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;cstring&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;vector&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;queue&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;stack&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;algorithm&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;unordered_map&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;set&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">500010</span>;<br><br>vector&lt;<span class="hljs-type">int</span>&gt; g[N];<br><span class="hljs-type">int</span> d[N];<br><span class="hljs-type">bool</span> vis[N];<br>pair&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt; res;     <span class="hljs-comment">// first 为最远距离；second 为对应结点编号</span><br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">dfs</span><span class="hljs-params">(<span class="hljs-type">int</span> pre, <span class="hljs-type">int</span> now)</span> </span>{<br>    <span class="hljs-keyword">if</span> (vis[now]) <span class="hljs-keyword">return</span>;<br>    <br>    vis[now] = <span class="hljs-literal">true</span>;<br>    <br>    <span class="hljs-keyword">if</span> (pre != <span class="hljs-number">-1</span>) {<br>        d[now] = d[pre] + <span class="hljs-number">1</span>;<br>        <span class="hljs-keyword">if</span> (d[now] &gt; res.first) {<br>            res = {d[now], now};<br>        }<br>    }<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; ch: g[now]) {<br>        <span class="hljs-built_in">dfs</span>(now, ch);<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-comment">// init</span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">2</span>; i &lt;= <span class="hljs-number">4</span>; i++) {<br>        g[<span class="hljs-number">1</span>].<span class="hljs-built_in">push_back</span>(i);<br>        g[i].<span class="hljs-built_in">push_back</span>(<span class="hljs-number">1</span>);<br>    }<br>    <br>    <span class="hljs-type">int</span> now = <span class="hljs-number">4</span>;<br>    <br>    <span class="hljs-type">int</span> Q;<br>    cin &gt;&gt; Q;<br>    <span class="hljs-keyword">while</span> (Q--) {<br>        <span class="hljs-type">int</span> id;<br>        cin &gt;&gt; id;<br>        <br>        g[id].<span class="hljs-built_in">push_back</span>(++now);<br>        g[now].<span class="hljs-built_in">push_back</span>(id);<br>        <br>        g[id].<span class="hljs-built_in">push_back</span>(++now);<br>        g[now].<span class="hljs-built_in">push_back</span>(id);<br>        <br>        res = {<span class="hljs-number">-1</span>, <span class="hljs-number">-1</span>};<br>        <br>        <span class="hljs-comment">// 第一趟</span><br>        <span class="hljs-built_in">memset</span>(vis, <span class="hljs-literal">false</span>, <span class="hljs-keyword">sizeof</span> vis);<br>        <span class="hljs-built_in">memset</span>(d, <span class="hljs-number">0</span>, <span class="hljs-keyword">sizeof</span> d);<br>        d[<span class="hljs-number">1</span>] = <span class="hljs-number">0</span>;<br>        <span class="hljs-built_in">dfs</span>(<span class="hljs-number">-1</span>, <span class="hljs-number">1</span>);<br>        <br>        <span class="hljs-comment">// 第二趟</span><br>        <span class="hljs-built_in">memset</span>(vis, <span class="hljs-literal">false</span>, <span class="hljs-keyword">sizeof</span> vis);<br>        <span class="hljs-built_in">memset</span>(d, <span class="hljs-number">0</span>, <span class="hljs-keyword">sizeof</span> d);<br>        d[res.second] = <span class="hljs-number">0</span>;<br>        <span class="hljs-built_in">dfs</span>(<span class="hljs-number">-1</span>, res.second);<br>        <br>        cout &lt;&lt; res.first &lt;&lt; <span class="hljs-string">"\n"</span>;<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><p>LCA 代码</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br><span class="line">82</span><br><span class="line">83</span><br><span class="line">84</span><br><span class="line">85</span><br><span class="line">86</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;cstring&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;vector&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;queue&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;stack&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;algorithm&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;unordered_map&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;set&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">1000010</span>, M = <span class="hljs-number">20</span>;<br><br><span class="hljs-type">int</span> d[N];        <span class="hljs-comment">// d[i] 表示 i 号点到根结点的距离</span><br><span class="hljs-type">int</span> to[N][M];    <span class="hljs-comment">// to[i][j] 表示 i 号点向上跳 2^j 步后到达的结点编号</span><br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">lca</span><span class="hljs-params">(<span class="hljs-type">int</span> a, <span class="hljs-type">int</span> b)</span> </span>{<br>    <span class="hljs-keyword">if</span> (d[a] &lt; d[b]) <span class="hljs-built_in">swap</span>(a, b);<br><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> k = M - <span class="hljs-number">1</span>; k &gt;= <span class="hljs-number">0</span>; k--)<br>        <span class="hljs-keyword">if</span> (d[to[a][k]] &gt;= d[b])<br>            a = to[a][k];<br><br>    <span class="hljs-keyword">if</span> (a == b) <span class="hljs-keyword">return</span> a;<br><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> k = M - <span class="hljs-number">1</span>; k &gt;= <span class="hljs-number">0</span>; k--)<br>        <span class="hljs-keyword">if</span> (to[a][k] != to[b][k])<br>            a = to[a][k], b = to[b][k];<br><br>    <span class="hljs-keyword">return</span> to[a][<span class="hljs-number">0</span>];<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">dist</span><span class="hljs-params">(<span class="hljs-type">int</span> a, <span class="hljs-type">int</span> b)</span> </span>{<br>    <span class="hljs-keyword">return</span> d[a] + d[b] - <span class="hljs-number">2</span> * d[<span class="hljs-built_in">lca</span>(a, b)];<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-type">int</span> Q;<br>    cin &gt;&gt; Q;<br><br>    <span class="hljs-comment">// init lca</span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">2</span>; i &lt;= <span class="hljs-number">4</span>; i++) {<br>        d[i] = <span class="hljs-number">1</span>;<br>        to[i][<span class="hljs-number">0</span>] = <span class="hljs-number">1</span>;<br>    }<br><br>    <span class="hljs-type">int</span> A = <span class="hljs-number">2</span>, B = <span class="hljs-number">4</span>, now = <span class="hljs-number">4</span>, res = <span class="hljs-number">2</span>;<br><br>    <span class="hljs-keyword">while</span> (Q--) {<br>        <span class="hljs-type">int</span> fa;<br>        cin &gt;&gt; fa;<br><br>        <span class="hljs-type">int</span> L1 = ++now, L2 = ++now;<br><br>        <span class="hljs-comment">// upd lca</span><br>        d[L1] = d[fa] + <span class="hljs-number">1</span>;<br>        d[L2] = d[fa] + <span class="hljs-number">1</span>;<br>        to[L1][<span class="hljs-number">0</span>] = fa;<br>        to[L2][<span class="hljs-number">0</span>] = fa;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> k = <span class="hljs-number">1</span>; k &lt;= M - <span class="hljs-number">1</span>; k++) {<br>            to[L1][k] = to[ to[L1][k<span class="hljs-number">-1</span>] ][ k<span class="hljs-number">-1</span> ];<br>            to[L2][k] = to[ to[L2][k<span class="hljs-number">-1</span>] ][ k<span class="hljs-number">-1</span> ];<br>        }<br><br>        <span class="hljs-type">int</span> da = <span class="hljs-built_in">dist</span>(A, L1), db = <span class="hljs-built_in">dist</span>(B, L1);<br><br>        <span class="hljs-keyword">if</span> (<span class="hljs-built_in">max</span>(da, db) &lt;= res) res = res;<br>        <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (<span class="hljs-built_in">min</span>(da, db) &gt;= res) {<br>            <span class="hljs-keyword">if</span> (da &gt; db) res = da, B = L1;<br>            <span class="hljs-keyword">else</span> res = db, A = L1;<br>        } <span class="hljs-keyword">else</span> {<br>            <span class="hljs-keyword">if</span> (da &gt; db) res = da, B = L1;<br>            <span class="hljs-keyword">else</span> res = db, A = L1;<br>        }<br><br>        cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure>]]></content>
+    <content type="html"><![CDATA[<h3 id="贪心">贪心</h3><p>大胆猜测，小心求证（不会证也没事，做下一题吧）。证明方法总结了以下几种</p><ul><li>反证法：假设取一种方案比贪心方案更好，得出相反的结论</li><li>边界法：从边界开始考虑，因为满足边界条件更加容易枚举，从而进行后续的贪心</li><li>直觉法：遵循社会法则（）</li></ul><h3 id="1-green-gold-dog-array-and-permutation">1. green_gold_dog, array and permutation</h3><p><a href="https://codeforces.com/contest/1867/problem/A">https://codeforces.com/contest/1867/problem/A</a></p><blockquote><p>题意：给定n个数a[i]，其中可能有重复数，请构造一个n个数的排列b[i]，使得c[i]=a[i]-b[i]中，c[i]的不同数字的个数最多</p><p>思路：思路比较好想，就是最大的数匹配最小的数，次大的数匹配次小的数，以此类推。起初我想通过将原数列的拷贝数列降序排序后，创建一个哈希表来记录每一个数的排位，最终通过原数列的数作为键，通过哈希表直接输出排位，但是问题是原数列中的数可能会有重复的，那么在哈希的时候如果有重复的数，那么后来再次出现的数就会顶替掉原来的数的排位，故错误。</p><p>正确思路：为了<strong>保证原数列中每个数的唯一性</strong>，我们可以给原数列每一个数赋一个下标，排序后以下标进行唯一的一一对应的索引。那么就是：首先赋下标，接着以第一关键词进行排序，然后最大的数（其实是下标）匹配最小的结果以此类推</p><p>模拟一个样例：<br>5<br>8 7 4 5 5 9</p><p>最终的答案应该是<br>2 3 6 4 5 1</p><p>首先对每一个数赋予一个下标作为为唯一性索引<br>8 7 4 5 5 9<br>1 2 3 4 5 6（替身）</p><p>接着将上述数列按照第一关键词进行排序<br>9 8 7 5 5 4<br>6 1 2 4 5 3（替身）</p><p>对每一个数进行赋值匹配<br>9 8 7 5 5 4<br>6 1 2 4 5 3（替身）<br>1 2 3 4 5 6（想要输出的结果）</p><p>按照替身进行排序<br>8 7 4 5 5 9<br>1 2 3 4 5 6（替身）（排序后）<br>2 3 6 4 5 1（想要输出的结果）</p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span></span><br><span class="hljs-function"></span>{<br>    <span class="hljs-type">int</span> n; cin &gt;&gt; n;<br>    <br>    <span class="hljs-comment">// 第一关键词是原数列，第二关键词是赋予的唯一下标</span><br>    vector&lt;pair&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt;&gt; <span class="hljs-built_in">a</span>(n + <span class="hljs-number">1</span>);<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i ++)<br>    {<br>        cin &gt;&gt; a[i].first;<br>        a[i].second = i;<br>    }<br>    <br>    <span class="hljs-comment">// 按照第一关键词排序</span><br>    <span class="hljs-built_in">sort</span>(a.<span class="hljs-built_in">begin</span>() + <span class="hljs-number">1</span>, a.<span class="hljs-built_in">end</span>(), [&amp;](pair&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt; x, pair&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt; y) {<br>        <span class="hljs-keyword">return</span> x.first &gt; y.first;<br>    });<br>    <br>    <span class="hljs-comment">// 以下标作为原数列的替身，每一个替身对应一个升序的最终排名</span><br>    vector&lt;pair&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt;&gt; <span class="hljs-built_in">res</span>(n + <span class="hljs-number">1</span>);<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i ++)<br>    {<br>        res[i].first = a[i].second;<br>        res[i].second = i;<br>    }<br>    <br>    <span class="hljs-comment">// 通过下标，还原原数列的排序</span><br>    <span class="hljs-built_in">sort</span>(res.<span class="hljs-built_in">begin</span>() + <span class="hljs-number">1</span>, res.<span class="hljs-built_in">end</span>());<br>    <br>    <span class="hljs-comment">// 输出第二关键词</span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i ++)<br>        cout &lt;&lt; res[i].second &lt;&lt; <span class="hljs-string">" \n"</span>[i == n];<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span></span><br><span class="hljs-function"></span>{<br>    <span class="hljs-type">int</span> T; cin &gt;&gt; T;<br>    <span class="hljs-keyword">while</span> (T --) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="2-Good-Kid">2. Good Kid</h3><p><a href="https://codeforces.com/contest/1873/problem/B">https://codeforces.com/contest/1873/problem/B</a></p><blockquote><p>题意：对于一个数列，现在可以选择其中的一个数使其+1，问如何选择这个数，可以使得修改后的数列中的所有数之积的值最大</p><p>思路：其实就是选择n-1个数的乘积值加一倍，关键在于选哪一个n-1个数的乘积值，逆向思维就是对于n个数，去掉那个最小值，那么剩下来的n-1个数之积就会最大了，于是就是选择最小的数+1，最终数列之积就是答案了。</p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-keyword">typedef</span> <span class="hljs-type">long</span> <span class="hljs-type">long</span> ll;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span></span><br><span class="hljs-function"></span>{<br>    <span class="hljs-type">int</span> n; cin &gt;&gt; n;<br>    <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">a</span><span class="hljs-params">(n)</span></span>;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i ++)<br>        cin &gt;&gt; a[i];<br>    <br>    <span class="hljs-built_in">sort</span>(a.<span class="hljs-built_in">begin</span>(), a.<span class="hljs-built_in">end</span>());<br>    <br>    ll res = a[<span class="hljs-number">0</span>] + <span class="hljs-number">1</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt; n; i ++)<br>        res *= a[i];<br>    cout &lt;&lt; res &lt;&lt; endl; <br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span></span><br><span class="hljs-function"></span>{<br>    <span class="hljs-type">int</span> T; cin &gt;&gt; T;<br>    <span class="hljs-keyword">while</span> (T --) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="3-1D-Eraser">3. 1D Eraser</h3><p><a href="https://codeforces.com/contest/1873/problem/D">https://codeforces.com/contest/1873/problem/D</a></p><blockquote><p>题意：给定一个只有B和W两种字符的字符串，现在有一种操作可以将一个指定大小的区间（大小为k）中所有的字符全部变为W，问对于这个字符串，至少需要几次上述操作才可以将字符串全部变为W字符</p><p>思路：我们直接遍历一边字符串即可，在遍历到B字符的时候，指针向后移动k位，如果是W字符，指针向后移动1位即可，最终统计一下遇到B字符的次数即可。</p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-keyword">typedef</span> <span class="hljs-type">long</span> <span class="hljs-type">long</span> ll;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span></span><br><span class="hljs-function"></span>{<br>    <span class="hljs-type">int</span> n, k;<br>    cin &gt;&gt; n &gt;&gt; k;<br>    string s;<br>    cin &gt;&gt; s;<br>    <br>    <span class="hljs-type">int</span> res = <span class="hljs-number">0</span>;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n;)<br>    {<br>        <span class="hljs-keyword">if</span> (s[i] == <span class="hljs-string">'B'</span>)<br>        {<br>            res ++;<br>            i += k;<br>        }<br>        <span class="hljs-keyword">else</span> i++;<br>    }<br>    cout &lt;&lt; res &lt;&lt; endl;<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span></span><br><span class="hljs-function"></span>{<br>    <span class="hljs-type">int</span> T; cin &gt;&gt; T;<br>    <span class="hljs-keyword">while</span> (T --) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="4-ABBC-or-BACB">4. ABBC or BACB</h3><p><a href="https://codeforces.com/contest/1873/problem/G">https://codeforces.com/contest/1873/problem/G</a></p><blockquote><p>题意：现在有一个只有A和B两种字符的字符串，有如下两种操作：第一种，可以将AB转化为BC，第二种，可以将BA转化为CB。问最多可以执行上述操作几次</p><p>思路：其实一开始想到了之前做过的翻转数字为-1，最后的逻辑是将数字向左右移动，于是这道题就有了突破口了。上述两种操作就可以看做将A和B两种字符交换位置，并且B保持不变，A变为另一种字符C。由于C是无法执行上述操作得到，因此就可以理解为B走过的路就不可以再走了，那么就是说对于一个字符串，最终都会变成B走过的路C。并且只要有B，那么一个方向上所有的A都会被利用转化为C（直到遇到下一个B停止），那么我们就可以将B看做一个独立的个体，他可以选择一个方向上的所有的A并且执行操作将该方向的A全部转化为C，那么在左和右的抉择中，就是要选择A数量最多的方向进行操作，但是如果B的足够多，那就不需要考虑哪个方向了，所有的A都可以获得一次操作机会。现在就转向考虑B需要多少呢，</p><p>答案是两种：如果B的数量小于A“块”的数量，其实就是有一块A无法被转化，那么为了让被转化的A数量尽可能的多，就选择A块数量最少的那一块不被转化。如果B的数量&gt;=A块的数量，那么可以保证所有的A都可以被转化为C块，从而最终的答案就是A字符的数量</p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-keyword">typedef</span> <span class="hljs-type">long</span> <span class="hljs-type">long</span> ll;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span></span><br><span class="hljs-function"></span>{<br>    string s; cin &gt;&gt; s;<br>    <span class="hljs-type">int</span> cntb = <span class="hljs-number">0</span>, blocka = <span class="hljs-number">0</span>, cnta = <span class="hljs-number">0</span>; <span class="hljs-comment">// B字符的数量，A连续区间的数量，A字符的个数 </span><br>    <br>    vector&lt;<span class="hljs-type">int</span>&gt; a; <span class="hljs-comment">// 每一个A连续区间中A字符的数量 </span><br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; s.<span class="hljs-built_in">size</span>();)<br>    {<br>        <span class="hljs-keyword">if</span> (s[i] == <span class="hljs-string">'A'</span>)<br>        {<br>            <span class="hljs-type">int</span> sum = <span class="hljs-number">0</span>;<br>            <span class="hljs-keyword">while</span> (s[i] == <span class="hljs-string">'A'</span>) cnta ++, sum ++, i ++;<br>            a.<span class="hljs-built_in">emplace_back</span>(sum);<br>            blocka ++;<br>        }<br>        <span class="hljs-keyword">else</span><br>        {<br>            cntb ++;<br>            i ++;<br>        }<br>    }<br>    <br>    <span class="hljs-keyword">if</span> (cntb &gt;= blocka) cout &lt;&lt; cnta &lt;&lt; endl;<br>    <span class="hljs-keyword">else</span><br>    {<br>        <span class="hljs-built_in">sort</span>(a.<span class="hljs-built_in">begin</span>(), a.<span class="hljs-built_in">end</span>());<br>        cout &lt;&lt; cnta - a[<span class="hljs-number">0</span>] &lt;&lt; endl;<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span></span><br><span class="hljs-function"></span>{<br>    <span class="hljs-type">int</span> T; cin &gt;&gt; T;<br>    <span class="hljs-keyword">while</span> (T --) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="5-Smilo-and-Monsters">5. Smilo and Monsters</h3><p><a href="https://codeforces.com/contest/1891/problem/C">https://codeforces.com/contest/1891/problem/C</a></p><blockquote><p>题意：给定一个序列，其中每一个元素代表一个怪物群中的怪物的数量。现在有两种操作：</p><ol><li>选择一个怪物群，杀死其中的一只怪物。同时累加值 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>+</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x+=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">x</span><span class="mord">+</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></li><li>选择一个怪物群，通过之前积累的累加值，一次性杀死当前怪物群中的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 只怪物，同时累加值 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 归零</li></ol><p>现在询问杀死所有怪物的最小操作次数</p><p>思路：一开始我是分奇偶进行判断的，但是那只是局部最优解，全局最优解的计算需要我们“纵观全局”。</p><p>我们可以先做一个假设：假设当前只有一个怪物群，为了最小化操作次数，最优解肯定是先杀死一半的怪物（假设数量为n，则获得累计值x=n），然后无论剩余n个还是n+1个，我们都使用累计值x一次性杀死n个怪物。</p><p>现在我们回到原题，有很多个怪物群，易证，最终的最优解也一点是最大化利用操作2，和特殊情况类似，能够利用的一次性杀死的次数就是怪物总数的一半。区别在于：此时不止一个怪物群。那么我们就要考虑累加值如何使用。很容易想到先将怪物群按照数量大小进行排序，关键就在于将累加值从小到大进行使用还是从大到小进行使用。可以发现，对于一个确定的累加值，由于操作2的次数也会算在答案中。那么如果从最小的怪物群开始使用，就会导致操作2的次数变多。因此我们需要从最大值开始进行操作2。</p><p>那么最终的答案就是：</p><ul><li>首先根据怪物数量的总和计算出最终的累加值s（s=sum/2）</li><li>接着我们将怪物群按照数量进行升序排序</li><li>然后我们从怪物数量最大的开始进行操作2，即一次性杀死当前怪物群，没进行一次，答案数量+1</li><li>最后将答案累加上无法一次性杀死的怪物群中怪物的总数</li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    n = <span class="hljs-built_in">read</span>(); s = <span class="hljs-number">0</span>; res = <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        a[i] = <span class="hljs-built_in">read</span>();<br>        s += a[i];<br>    }<br> <br>    s &gt;&gt;= <span class="hljs-number">1</span>;<br> <br>    <span class="hljs-built_in">sort</span>(a + <span class="hljs-number">1</span>, a + n + <span class="hljs-number">1</span>);<br> <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = n; i &gt;= <span class="hljs-number">1</span>; i--) {<br>        <span class="hljs-keyword">if</span> (s &gt;= a[i]) {<br>            s -= a[i], a[i] = <span class="hljs-number">0</span>, res++;<br>        } <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (s) {<br>            a[i] -= s;<br>            s = <span class="hljs-number">0</span>;<br>            res++;<br>        }<br>    }<br> <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        res += a[i];<br>    }<br> <br>    <span class="hljs-built_in">printf</span>(<span class="hljs-string">"%lld\n"</span>, res);<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="6-通关">6. 通关</h3><p><a href="https://www.lanqiao.cn/problems/5889/learning/?contest_id=145">https://www.lanqiao.cn/problems/5889/learning/?contest_id=145</a></p><blockquote><p>声明：谨以此题记录我的第一道正式图论题（存图技巧抛弃了y总的纯数组，转而使用动态数组<code>vector</code>进行建图）e.g.</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-keyword">struct</span> <span class="hljs-title class_">Node</span> {<br><span class="hljs-type">int</span> id;        <span class="hljs-comment">// 当前结点的编号</span><br><span class="hljs-type">int</span> a;    <span class="hljs-comment">// 属性1</span><br><span class="hljs-type">int</span> b;    <span class="hljs-comment">// 属性2</span><br>};<br>vector&lt;Node&gt; G[N];<br></code></pre></td></tr></tbody></table></figure><p>题意：给定一棵树，每个结点代表一个关卡，每个关卡拥有两个属性：通关奖励值<code>val</code>和可通关的最低价值<code>cost</code>。现在从根节点开始尝试通关，每一个结点下面的关卡必须要当前结点通过了之后才能继续闯关。问应该如何选择通关规划使得通过的关卡数最多？</p><p>思路：一开始想到的是直接莽bfs，对每一层结点按照cost升序排序进行闯关，然后wa麻了。最后一看正解原来是还不够贪。正确的闯关思路应该是始终选择当前可以闯过的cost最小的关卡，而非限定在了一层上。因此我们最终的闯关顺序是：对于所有解锁的关卡，选择cost最小的关卡进行通关，如果连cost最小的关卡都无法通过就直接结束闯关。那么我们应该如何进行代码的编写呢？分析可得，解锁的关卡都是当前可通过的关卡的所有子结点，而快速获得当前cost最小值的操作可以通过一个堆来维护。</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">100010</span>;<br><span class="hljs-keyword">typedef</span> <span class="hljs-type">long</span> <span class="hljs-type">long</span> ll;<br><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">Node</span> {<br>    <span class="hljs-type">int</span> id;<br>    <span class="hljs-type">int</span> val;<br>    <span class="hljs-type">int</span> cost;<br>    <span class="hljs-comment">// 有趣的重载</span><br>    <span class="hljs-type">bool</span> <span class="hljs-keyword">operator</span>&lt; (<span class="hljs-type">const</span> Node&amp; t) <span class="hljs-type">const</span> {<br>        <span class="hljs-keyword">return</span> <span class="hljs-keyword">this</span>-&gt;cost &gt; t.cost;<br>    }<br>};<br><br><span class="hljs-type">int</span> n, p;<br>ll res;<br>vector&lt;Node&gt; G[N];<br>priority_queue&lt;Node&gt; q;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">bfs</span><span class="hljs-params">()</span> </span>{<br>    q.<span class="hljs-built_in">push</span>(G[<span class="hljs-number">0</span>][<span class="hljs-number">0</span>]);<br><br>    <span class="hljs-keyword">while</span> (q.<span class="hljs-built_in">size</span>()) {<br>        Node now = q.<span class="hljs-built_in">top</span>();<br>        q.<span class="hljs-built_in">pop</span>();<br>        <span class="hljs-keyword">if</span> (p &gt;= now.cost) {<br>            res++;<br>            p += now.val;<br>            <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; child: G[now.id]) {<br>                q.<span class="hljs-built_in">push</span>(child);<br>            }<br>        } <span class="hljs-keyword">else</span> {<br>            <span class="hljs-keyword">break</span>;<br>        }<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; p;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-type">int</span> fa, v, c;<br>        cin &gt;&gt; fa &gt;&gt; v &gt;&gt; c;<br>        G[fa].<span class="hljs-built_in">push_back</span>({i, v, c});<br>    }<br>    <span class="hljs-built_in">bfs</span>();<br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="7-串门">7. 串门</h3><p><a href="https://www.lanqiao.cn/problems/5890/learning/?contest_id=145">https://www.lanqiao.cn/problems/5890/learning/?contest_id=145</a></p><blockquote><p>题意：给定一棵无向树，边权为正。现在询问在访问到每一个结点的情况下最小化行走的路径长度</p><p>思路：首先我们不管路径长度的长短，我们直接开始串门。可以发现，我们一点会有访问的起点和终点，那么在起点到终点的这条路上，可能会有很多个分叉，对于每一个分叉，我们都需要进入再返回来确保分叉中的结点也被访问到，那么最终的路径长度就是总路径长度的2倍 - 起点到终点的距离，知道了这个性质后。我们可以发现，路径的总长度是一个定值，我们只有最大化起点到终点距离才能确保行走路径长度的最小值。那么问题就转化为了求解一棵树中，最长路径长度的问题。即求解树的直径的问题。</p><p>树的直径：首先我们反向考虑，假设知道了直径的一个端点，那么树的直径长度就是这棵树以当前结点为根的深度。那么我们就需要先确定一个直径的端点。不难发现，对于树中的任意一个结点，距离它最远的一个（可能是两个）结点一定是树的直径的端点。那么问题就迎刃而解了。</p><p>为了降低代码量，我们可以设置一个dist数组来记录当前结点到根的距离。</p><ul><li>在确定直径端点的时候，选择任意一个结点为根节点，然后维护dist数组，最终dist[i~n]中的最大值对应的下标maxi就是直径的一个端点。</li><li>接着在计算直径长度的时候，以maxi为树根，再来一次上述的维护dist数组的过程，最终dist[1~n]中的最大值就是树的直径的长度</li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">100010</span>;<br><span class="hljs-keyword">typedef</span> <span class="hljs-type">long</span> <span class="hljs-type">long</span> ll;<br><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">Node</span> {<br>    <span class="hljs-type">int</span> id;<br>    ll w;<br>};<br><br><span class="hljs-type">int</span> n;<br>vector&lt;Node&gt; G[N];<br><span class="hljs-type">bool</span> st[N];<br>ll sum, d, dist[N];<br><br><span class="hljs-comment">/**</span><br><span class="hljs-comment"> * @note 计算当前所有子结点到当前点的距离</span><br><span class="hljs-comment"> * @param now 当前点的编号</span><br><span class="hljs-comment"> * @param x 上一个点到当前点的距离</span><br><span class="hljs-comment"> */</span><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">dfs</span><span class="hljs-params">(<span class="hljs-type">int</span> now, ll x)</span> </span>{<br>    <span class="hljs-keyword">if</span> (!st[now]) {<br>        st[now] = <span class="hljs-literal">true</span>;<br>        dist[now] = x;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; G[now].<span class="hljs-built_in">size</span>(); i++) {<br>            <span class="hljs-type">int</span> ch = G[now][i].id;<br>            <span class="hljs-built_in">dfs</span>(ch, x + G[now][i].w);<br>        }<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n - <span class="hljs-number">1</span>; i++) {<br>        <span class="hljs-type">int</span> a, b, w;<br>        cin &gt;&gt; a &gt;&gt; b &gt;&gt; w;<br>        sum += w;<br>        G[a].<span class="hljs-built_in">push_back</span>({b, w});<br>        G[b].<span class="hljs-built_in">push_back</span>({a, w});<br>    }<br><br>    <span class="hljs-comment">// 以1号点为根，计算所有点到1号点的距离</span><br>    <span class="hljs-built_in">dfs</span>(<span class="hljs-number">1</span>, <span class="hljs-number">0</span>);<br><br>    <span class="hljs-built_in">memset</span>(st, <span class="hljs-number">0</span>, <span class="hljs-keyword">sizeof</span> st);<br><br>    <span class="hljs-comment">// 到1号点最远的那个点的编号 maxi，即直径的一个端点</span><br>    <span class="hljs-type">int</span> maxi = <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-keyword">if</span> (dist[i] &gt; dist[maxi]) {<br>            maxi = i;<br>        }<br>    }<br><br>    <span class="hljs-comment">// 以直径的一个端点 maxi 为根，计算所有点到 maxi 点的距离</span><br>    <span class="hljs-built_in">dfs</span>(maxi, <span class="hljs-number">0</span>);<br><br>    <span class="hljs-comment">// 找到直径长度</span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        d = <span class="hljs-built_in">max</span>(d, dist[i]);<br>    }<br><br>    cout &lt;&lt; sum * <span class="hljs-number">2</span> - d &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="8-Codeforces-rating">8. Codeforces rating</h3><p><a href="https://vijos.org/d/nnu_contest/p/1532">https://vijos.org/d/nnu_contest/p/1532</a></p><blockquote><p>题意：现在已知一个初始值R，现在给定了n个P值，选择其中的合适的值，使得最终的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>R</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mn>3</mn><mi mathvariant="normal">/</mi><mn>4</mn><mi>R</mi><mo>+</mo><mn>1</mn><mi mathvariant="normal">/</mi><mn>4</mn><mi>P</mi></mrow><annotation encoding="application/x-tex">R'=3/4 R + 1/4 P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">3/4</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1/4</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> 最大</p><p>思路：首先一点就是我们一定要选择P比当前R大的值，接下来就是选择合适的P使得最终迭代出来的R’最大。首先我们知道，对于筛选出来的比当前R大的P集合，任意选择其中的一个P，都会让R增大，但是不管增加多少都是不会超过选择的P。那么显然，如果筛选出来的P集合是一个递增序列，那么就可以让R不断的增加。但是这一定是最大的吗？我们不妨反证一下，现在我们有两个P，分别为x，y，其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>&lt;</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x&lt;y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>。</p><p>那么按照上述思路，首先就是</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>R</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mi>R</mi><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>x</mi><mo stretchy="false">(</mo><msup><mi>R</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>&lt;</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R'=\frac{3}{4}R+\frac{1}{4}x(R'&lt;x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8019em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">x</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></p><p>接着就是</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mi>R</mi><mn>1</mn><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mo>=</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><msup><mi>R</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>y</mi><mo stretchy="false">(</mo><msubsup><mi>R</mi><mn>1</mn><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mo>&lt;</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>9</mn><mn>16</mn></mfrac><mi>R</mi><mo>+</mo><mfrac><mn>3</mn><mn>16</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>y</mi></mrow><annotation encoding="application/x-tex">R''_1=\frac{3}{4}R'+\frac{1}{4}y(R''_1&lt;y)=\frac{9}{16}R+\frac{3}{16}x+\frac{1}{4}y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0489em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-2.453em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-2.453em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">16</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">9</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">16</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span></span></p><p><strong>反之</strong>，首先选择一个较大的P=y，再选择一个较小的P=x，即首先就是</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>R</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mi>R</mi><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>y</mi><mo stretchy="false">(</mo><msup><mi>R</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>&lt;</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R'=\frac{3}{4}R+\frac{1}{4}y(R'&lt;y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8019em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span></span></p><p>此时我们还不一点可以继续选择x，因为此时的R’可能已经超过了x的值，那么我们按照最优的情况计算，可以选择x，那么就是</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mi>R</mi><mn>2</mn><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mo>=</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><msup><mi>R</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>x</mi><mo stretchy="false">(</mo><msubsup><mi>R</mi><mn>2</mn><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mo>&lt;</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>9</mn><mn>16</mn></mfrac><mi>R</mi><mo>+</mo><mfrac><mn>3</mn><mn>16</mn></mfrac><mi>y</mi><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>x</mi></mrow><annotation encoding="application/x-tex">R''_2=\frac{3}{4}R'+\frac{1}{4}x(R''_2&lt;x)=\frac{9}{16}R+\frac{3}{16}y+\frac{1}{4}x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0489em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-2.453em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">x</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-2.453em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">16</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">9</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">16</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">x</span></span></span></span></span></p><p>可以发现</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mi>R</mi><mn>2</mn><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mo>−</mo><msubsup><mi>R</mi><mn>1</mn><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mo>=</mo><mfrac><mn>1</mn><mn>16</mn></mfrac><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">)</mo><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">R''_2-R''_1=\frac{1}{16}(x-y)&lt;0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0489em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-2.453em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0489em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-2.453em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">16</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p><p>即会使得最终的答案变小。因此最优的策略就是按照增序进行叠加计算</p><p>==注意点：==</p><p>四舍五入的语句</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs cpp">cout &lt;&lt; (<span class="hljs-type">int</span>)(res + <span class="hljs-number">0.5</span>) &lt;&lt; <span class="hljs-string">"\n"</span>;<br></code></pre></td></tr></tbody></table></figure></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; k;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        cin &gt;&gt; a[i];<br>    }<br><br>    <span class="hljs-built_in">sort</span>(a, a + n);<br><br>    <span class="hljs-type">int</span> i = <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">for</span> (; i &lt; n; i++) {<br>        <span class="hljs-keyword">if</span> (a[i] &gt; k) {<br>            <span class="hljs-keyword">break</span>;<br>        }<br>    }<br><br>    <span class="hljs-type">double</span> res = k;<br><br>    <span class="hljs-keyword">for</span> (; i &lt; n; i++) {<br>        res = (res * <span class="hljs-number">3.0</span> + a[i] * <span class="hljs-number">1.0</span>) / <span class="hljs-number">4.0</span>;<br>    }<br><br>    cout &lt;&lt; (<span class="hljs-type">int</span>)(res + <span class="hljs-number">0.5</span>) &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="9-货运公司">9. 货运公司</h3><p><a href="https://www.acwing.com/problem/content/5380/">https://www.acwing.com/problem/content/5380/</a></p><blockquote><p>题意：给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个运货订单，每个订单需要运送一定量的货物，同时给予一定量的报酬。现在有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个卡车进行运送，每个卡车有运送容量。需要合理规划卡车的装配，使得最终获得的总报酬最大。有以下约束条件：</p><ul><li><strong>每一个订单</strong>只能用一辆货车运送</li><li><strong>每一辆货车</strong>只能运送不超过最大容量的货物</li><li><strong>每一辆货车</strong>只能运送一次</li><li><strong>每一辆货车</strong>在装得下的情况下只能装运一个订单</li></ul><p>思路：一开始尝试采用 01 背包的思路，但是题目中货物是否可以被运送是取决于卡车的装载的。思考无果后考虑贪心。<strong>考虑一般情况</strong></p><ul><li><p>为了让总收益最大，很显然报酬越多，就越优先选择该订单</p></li><li><p>在报酬一致时，货物量越少，就越优先选择该订单</p></li><li><p>按照上述规则对订单排序后，我们需要安排合理的货车进行运输。我们优先将运输量小的货车安排给合适的订单。因为运输量大的货车既可以满足当前订单，同样也满足后续的订单，那么为了最大化增加利润，就需要在运输量小的货车满足当前订单时，保留运输量大的货车用于后续的订单。我们可以画一个图进一步理解为什么这么安排货车进行运输：</p>  <img src="D:/BaiduSyncdisk/_images/typora-user-images/202402012332958.png" alt="image-20231223222627134" style="zoom: 50%;">  <div style="text-align: center">图例：我们按照上述红笔的需要进行货车的选择进行运输</div><p>如果红色序号顺序变掉了，则就不能将上述图例中的 6 个订单全部运输了</p></li></ul><p>时间复杂度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(nk)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">nk</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">1010</span>;<br><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">Item</span> {<br>    <span class="hljs-type">int</span> id, weight, money;<br>    <span class="hljs-type">bool</span> <span class="hljs-keyword">operator</span>&lt; (<span class="hljs-type">const</span> Item&amp; t) <span class="hljs-type">const</span> {<br>        <span class="hljs-keyword">if</span> (money != t.money) <span class="hljs-keyword">return</span> money &gt; t.money;<br>        <span class="hljs-keyword">else</span> <span class="hljs-keyword">return</span> weight &lt; t.weight;<br>    }<br>} item[N];<br><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">Car</span> {<br>    <span class="hljs-type">int</span> id, v;<br>    <span class="hljs-type">bool</span> <span class="hljs-keyword">operator</span>&lt; (<span class="hljs-type">const</span> Car&amp; t) <span class="hljs-type">const</span> {<br>        <span class="hljs-keyword">return</span> v &lt; t.v;<br>    }<br>} car[N];<br><br>pair&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt; res[N];<br><span class="hljs-type">bool</span> vis[N];<br><span class="hljs-type">int</span> cnt, sum;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-type">int</span> n;<br>    cin &gt;&gt; n;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        item[i].id = i;<br>        cin &gt;&gt; item[i].weight &gt;&gt; item[i].money;<br>    }<br><br>    <span class="hljs-type">int</span> k;<br>    cin &gt;&gt; k;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= k; i++) {<br>        car[i].id = i;<br>        cin &gt;&gt; car[i].v;<br>    }<br><br>    <span class="hljs-built_in">sort</span>(item + <span class="hljs-number">1</span>, item + n + <span class="hljs-number">1</span>);<br>    <span class="hljs-built_in">sort</span>(car + <span class="hljs-number">1</span>, car + k + <span class="hljs-number">1</span>);<br><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">1</span>; j &lt;= k; j++) {<br>            <span class="hljs-keyword">if</span> (!vis[j] &amp;&amp; car[j].v &gt;= item[i].weight) {<br>                sum += item[i].money;<br>                vis[j] = <span class="hljs-literal">true</span>;<br>                res[cnt++] = {item[i].id, car[j].id};<br>                <span class="hljs-keyword">break</span>;<br>            }<br>        }<br>    }<br><br>    cout &lt;&lt; cnt &lt;&lt; <span class="hljs-string">" "</span> &lt;&lt; sum &lt;&lt; <span class="hljs-string">"\n"</span>;<br><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; cnt; i++) {<br>        cout &lt;&lt; res[i].first &lt;&lt; <span class="hljs-string">" "</span> &lt;&lt; res[i].second &lt;&lt; <span class="hljs-string">"\n"</span>;<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="10-部分背包问题">10. 部分背包问题</h3><p><a href="https://www.luogu.com.cn/problem/P2240">https://www.luogu.com.cn/problem/P2240</a></p><blockquote><ul><li>题意：给定一组物品，每一个物品有一定的重量与价值，现在给定一个背包总承重，需要从中选择合适的物品进行装载，使得最终装载的物品的总价值最大。现在规定每一个物品可以进行随意分割，按照单位重量计算价值</li><li>思路：<strong>考虑一般情况</strong>。很显然，如果不说可以随意分割，那就是一个 01 背包，既然说了可以随意分割，那就直接贪心选择即可，按照单位重量的价值降序进行选择</li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">110</span>;<br><br><span class="hljs-type">int</span> n, W;<br><span class="hljs-type">double</span> res;<br><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">Item</span> {<br>    <span class="hljs-type">double</span> w;<br>    <span class="hljs-type">double</span> val;<br>    <br>    <span class="hljs-type">bool</span> <span class="hljs-keyword">operator</span>&lt; (<span class="hljs-type">const</span> Item&amp; t) <span class="hljs-type">const</span> {<br>        <span class="hljs-keyword">return</span> <span class="hljs-keyword">this</span>-&gt;val / <span class="hljs-keyword">this</span>-&gt;w &gt; t.val / t.w;<br>    }<br>} a[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; W;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) cin &gt;&gt; a[i].w &gt;&gt; a[i].val;    <br>    <br>    <span class="hljs-built_in">sort</span>(a, a + n);<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        <span class="hljs-keyword">if</span> (W &gt; a[i].w) {<br>            W -= a[i].w;<br>            res += a[i].val;<br>        } <span class="hljs-keyword">else</span> {<br>            res += W * a[i].val / a[i].w;<br>            <span class="hljs-keyword">break</span>;<br>        }<br>    }<br>    <br>    cout &lt;&lt; fixed &lt;&lt; <span class="hljs-built_in">setprecision</span>(<span class="hljs-number">2</span>) &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    cout &lt;&lt; fixed &lt;&lt; <span class="hljs-built_in">setprecision</span>(<span class="hljs-number">2</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="11-排队接水">11. 排队接水</h3><p><a href="https://www.luogu.com.cn/problem/P1223">https://www.luogu.com.cn/problem/P1223</a></p><blockquote><ul><li><p>题意：贪心经典，排队接水，每一个人都有一个打水时间，现在询问如何排队能够使得平均等待时间最短</p></li><li><p>思路：<strong>考虑一般情况</strong>。注意问的是平均等待时间，如果是打完水总花费时间那就是一个定值，即所有人打水的时间总和。现在为了让平均等待时间最短，我们的第一反应应该是，在人多的时候让打水时间短的先打，人少的时候让打水时间长的再打，那么很显然就是按照打水时间的升序进行排队。为了证明这个结论，我们采用数学归纳法，如下，对于一个排队序列来说，我们将其中的两个顺序进行颠倒，计算结果得到让打水时间短的排在前面更优，进而推广到一般情况，于是按照打水时间升序进行打水的策略是最优的。证毕</p>  <img src="D:/BaiduSyncdisk/_images/typora-user-images/202402012332959.png" alt="demo" style="zoom: 67%;"></li><li><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">1010</span>;<br><br><span class="hljs-type">int</span> n;<br><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">Item</span> {<br>    <span class="hljs-type">int</span> time;<br>    <span class="hljs-type">int</span> i;<br>    <br>    <span class="hljs-type">bool</span> <span class="hljs-keyword">operator</span>&lt; (<span class="hljs-type">const</span> Item&amp; t) <span class="hljs-type">const</span> {<br>        <span class="hljs-keyword">return</span> <span class="hljs-keyword">this</span>-&gt;time &lt; t.time;<br>    }<br>} a[N]; <br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        cin &gt;&gt; a[i].time;<br>        a[i].i = i;<br>    }<br>    <br>    <span class="hljs-built_in">sort</span>(a + <span class="hljs-number">1</span>, a + n + <span class="hljs-number">1</span>);<br>    <br>    <span class="hljs-type">double</span> sum = <span class="hljs-number">0</span>;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        sum += a[i].time * (n - i);<br>        cout &lt;&lt; a[i].i &lt;&lt; <span class="hljs-string">" \n"</span>[i == n];<br>    }<br>    <br>    cout &lt;&lt; fixed &lt;&lt; <span class="hljs-built_in">setprecision</span>(<span class="hljs-number">2</span>) &lt;&lt; sum / n &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="12-线段覆盖">12. 线段覆盖</h3><p><a href="https://www.luogu.com.cn/problem/P1803">https://www.luogu.com.cn/problem/P1803</a></p><blockquote><ul><li>题意：给定一个数轴和数轴上的 n 个线段，现在需要从中选出尽可能多的 k 个线段，使得线段之间两两不相交，问 k 最大是多少</li><li>思路：绞尽脑汁想着各种情况进行分类讨论，什么包含、相交、相离，但是都忽视了最重要的一点，那就是一开始应该选择线段。即<strong>考虑边界</strong>，对于最左侧我们应该选择哪一条线段呢？答案是最左侧的右端点最靠左的那一条线段，这样子后续的线段才能有更多的摆放空间</li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">1000010</span>;<br><br><span class="hljs-type">int</span> n;<br><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">Item</span> {<br>    <span class="hljs-type">int</span> l, r;<br>    <span class="hljs-type">bool</span> <span class="hljs-keyword">operator</span>&lt; (<span class="hljs-type">const</span> Item&amp; t) <span class="hljs-type">const</span> {<br>        <span class="hljs-keyword">return</span> <span class="hljs-keyword">this</span>-&gt;r &lt; t.r;<br>    }<br>} a[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) cin &gt;&gt; a[i].l &gt;&gt; a[i].r;<br>    <br>    <span class="hljs-built_in">sort</span>(a + <span class="hljs-number">1</span>, a + n + <span class="hljs-number">1</span>);<br>    <br>    <span class="hljs-type">int</span> right = <span class="hljs-number">-1</span>, res = <span class="hljs-number">0</span>;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-keyword">if</span> (a[i].l &gt;= right) {<br>            res++;<br>            right = a[i].r;<br>        }<br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="13-小A的糖果">13. 小A的糖果</h3><p><a href="https://www.luogu.com.cn/problem/P3817">https://www.luogu.com.cn/problem/P3817</a></p><blockquote><ul><li>题意：给定一个序列，现在每次可以选择序列中的某个数进行减一操作，问最少需要减多少可以使得序列中相邻两个元素之和不超过规定的数值 x</li><li>思路：同 T12 的思路展开，<strong>考虑边界</strong>，我们直接从任意的一个边界开始思考，假设从左边界开始考虑。为了让左边界满足，那么 a[1] 与 a[2] 之和就需要满足条件，为了后续可以尽可能少的进行减一操作，很显然我们最好从两个数的右边的数开始下手，于是贪心的结果就有了</li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">100010</span>;<br><br><span class="hljs-type">int</span> n, x;<br><span class="hljs-type">int</span> a[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; x;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) cin &gt;&gt; a[i];<br>    <br>    <span class="hljs-type">int</span> res = <span class="hljs-number">0</span>;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">2</span>; i &lt;= n; i++) {<br>        <span class="hljs-keyword">if</span> (a[i] + a[i - <span class="hljs-number">1</span>] &gt; x) {<br>            <span class="hljs-type">int</span> eat = a[i] + a[i - <span class="hljs-number">1</span>] - x;<br>            res += eat;<br>            a[i] = a[i] &gt;= eat ? a[i] - eat : <span class="hljs-number">0</span>;<br>        }<br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="15-陶陶摘苹果（升级版）">15. 陶陶摘苹果（升级版）</h3><p><a href="https://www.luogu.com.cn/problem/P1478">https://www.luogu.com.cn/problem/P1478</a></p><blockquote><ul><li>题意：现在有 n 个苹果，含有高度和消耗体力值两个属性，现在你有一个最大高度以及总体力值，每次只能摘高度值不超过最大高度的苹果且会消耗相应的体力值。问最多可以摘多少个苹果</li><li>思路：既然是想要摘的苹果尽可能多，那么高度都是浮云，体力值才是关键。为了摘得尽可能多的苹果，自然是消耗体力越少越优先摘。于是按照消耗体力值升序排序后，扫描一遍即可计数</li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">5010</span>;<br><br><span class="hljs-type">int</span> n, s;<br><span class="hljs-type">int</span> aa, bb;<br><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">Apple</span> {<br>    <span class="hljs-type">int</span> h, cost;<br>    <span class="hljs-type">bool</span> <span class="hljs-keyword">operator</span>&lt; (<span class="hljs-type">const</span> Apple&amp; t) <span class="hljs-type">const</span> {<br>        <span class="hljs-keyword">return</span> <span class="hljs-keyword">this</span>-&gt;cost &lt; t.cost;<br>    }<br>} a[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; s;<br>    cin &gt;&gt; aa &gt;&gt; bb;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) cin &gt;&gt; a[i].h &gt;&gt; a[i].cost;<br>    <br>    <span class="hljs-built_in">sort</span>(a + <span class="hljs-number">1</span>, a + n + <span class="hljs-number">1</span>);<br>    <br>    <span class="hljs-type">int</span> cnt = <span class="hljs-number">0</span>;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-keyword">if</span> (s &gt;= a[i].cost &amp;&amp; a[i].h &lt;= aa + bb) {<br>            cnt++;<br>            s -= a[i].cost;<br>        }<br>    }<br>    <br>    cout &lt;&lt; cnt &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="16-NOIP2018-提高组-铺设道路">16. [NOIP2018 提高组] 铺设道路</h3><p><a href="https://www.luogu.com.cn/problem/P5019">https://www.luogu.com.cn/problem/P5019</a></p><blockquote><ul><li><p>题意：给定一个序列，每次可以选择一个连续的区间使得其中的所有数 -1，前提是区间中的数都要 &gt; 1，问最少选择几次区间可以将所有的数全部都减到 0</p></li><li><p>思路：很显然的一个贪心思路就是，对于一个数，左右两侧均呈现递减的趋势，那么这个区间的选择数就是这个最大的数的值。但是这不符合代码逻辑，我们尝试<strong>从端点开始考虑</strong>。为了更好的理解，我们从递推的角度进行推演。假设填平前 i 个数花费的选择次数为 f[i]，那么对于第一个数 a[1]，我们一定需要花费 a[1] 次选择来将其减到零，于是 f[1] = a[1]；对于后续的数而言，可以发现</p><ul><li>如果 a[i] &gt; a[i-1]，那么之前的选择没有完全波及当前的数，还需要一定的次数来将其减到零，于是 <code>f[i] = f[i-1] + a[i] - a[i-1]</code></li><li>如果 a[i] <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≤</mo></mrow><annotation encoding="application/x-tex">\le</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≤</span></span></span></span> a[i-1]，那么当前的数值一定会被之前的选择“波及”而直接减到零，于是 <code>f[i] = f[i-1]</code></li></ul><p>从递推的角度理解之后我们直接滚动一遍数组记录増势即可</p></li><li><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">100010</span>;<br><br><span class="hljs-type">int</span> n, a[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) cin &gt;&gt; a[i];<br>    <br>    <span class="hljs-type">int</span> res = a[<span class="hljs-number">1</span>];<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">2</span>; i &lt;= n; i++)<br>        <span class="hljs-keyword">if</span> (a[i] &gt; a[i - <span class="hljs-number">1</span>])<br>            res += a[i] - a[i - <span class="hljs-number">1</span>];<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>} <br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="17-USACO1-3-混合牛奶-Mixing-Milk">17. [USACO1.3] 混合牛奶 Mixing Milk</h3><p><a href="https://www.luogu.com.cn/problem/P1208">https://www.luogu.com.cn/problem/P1208</a></p><blockquote><ul><li>题意：现在需要收集 need 容量的牛奶，同时有 n 个农户提供奶源，每一个商户的奶源有一定的数量与单价，问如何采购可以在满足需求 need 的情况下成本最低化</li><li>思路：很显然单价越低越优先采购</li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">2000010</span>;<br><br><span class="hljs-type">int</span> need, n;<br><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">Milk</span> {<br>    <span class="hljs-type">int</span> per, tot;<br>    <span class="hljs-type">bool</span> <span class="hljs-keyword">operator</span>&lt; (<span class="hljs-type">const</span> Milk&amp; t) <span class="hljs-type">const</span> {<br>        <span class="hljs-keyword">return</span> per &lt; t.per;<br>    }<br>} a[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; need &gt;&gt; n;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) cin &gt;&gt; a[i].per &gt;&gt; a[i].tot;<br>    <br>    <span class="hljs-built_in">sort</span>(a + <span class="hljs-number">1</span>, a + n + <span class="hljs-number">1</span>);<br>    <br>    <span class="hljs-type">int</span> res = <span class="hljs-number">0</span>;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-keyword">if</span> (need &gt;= a[i].tot) {<br>            need -= a[i].tot;<br>            res += a[i].tot * a[i].per;<br>        } <span class="hljs-keyword">else</span> {<br>            res += need * a[i].per;<br>            need = <span class="hljs-number">0</span>;<br>        }<br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="18-NOIP2007-普及组-纪念品分组">18. [NOIP2007 普及组] 纪念品分组</h3><p><a href="https://www.luogu.com.cn/problem/P1094">https://www.luogu.com.cn/problem/P1094</a></p><blockquote><ul><li>题意：给定 n 个数，问如何分组可以使得在每一组之和不超过定值 lim 且每一组最多两个数的情况下，分得的组数最少</li><li>思路：对序列没有要求，显然直接排序然后<strong>从一端开始考虑</strong>。首先有一个概念就是，为了尽可能的减少分得的组数，那么一定想要每一组分完后距离 lim 的差距越小越好。我们将 n 个数降序排序后，从最大值开始考虑，对于当前的数，为了找到一个数与当前数之和尽可能接近 lim，从贪心的角度来说一定是从当前数之后开始寻找那个配对的数从而使得两数之和与 lim 的差距最小，那么时间复杂度就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>。但是由于题目说了最多只有两个数，我们不妨换一种寻找配对数的策略，可以从原始的寻找方法得到启发，可以发现，原始的寻找配对数的方法中，如果找到的那个配对数可以与当前数之和不超过 lim 且之和尽可能的大，那么这个寻找到的配对数也一定可以和当前数的下一个数进行配对。由于我们一组只能有两个数，因此我们不用考虑两数组合之后比 lim 小的数尽可能的小，只需要考虑两个数能否组合即可，因此我们直接双指针进行寻找配对数操作即可</li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">30010</span>;<br><br><span class="hljs-type">int</span> lim, n;<br><span class="hljs-type">int</span> a[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; lim &gt;&gt; n;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) cin &gt;&gt; a[i];<br>    <br>    <span class="hljs-built_in">sort</span>(a + <span class="hljs-number">1</span>, a + n + <span class="hljs-number">1</span>, [&amp;](<span class="hljs-type">int</span> x, <span class="hljs-type">int</span> y) {<br>        <span class="hljs-keyword">return</span> x &gt; y;<br>    });<br>    <br>    <span class="hljs-type">int</span> res = <span class="hljs-number">0</span>, l = <span class="hljs-number">1</span>, r = n;<br>    <br>    <span class="hljs-keyword">while</span> (l &lt; r) {<br>        <span class="hljs-keyword">if</span> (a[l] + a[r] &lt;= lim) r--;<br>        l++, res++;<br>    }<br>    <br>    <span class="hljs-keyword">if</span> (l == r) res++;<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>} <br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="19-跳跳！">19. 跳跳！</h3><p><a href="https://www.luogu.com.cn/problem/P4995">https://www.luogu.com.cn/problem/P4995</a></p><blockquote><ul><li><p>题意：给定 n 个数，每一个数代表一个高度，现在有一只青蛙从地面（高度为 0）开始起跳，每一个数字只能被跳上去一次。问如何设定跳数顺序 h[]，可以使得下式尽可能的大</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mo stretchy="false">(</mo><msub><mi>h</mi><mi>i</mi></msub><mo>−</mo><msub><mi>h</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\sum_{i=1}^n (h_i-h_{i-1})^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></p></li><li><p>思路：题目意思其实就是让每一次跳跃的两个高度之差尽可能的大。那么一个贪心的思路就是从地面跳到最高，再从最高跳到次低，接着从次低跳到次高，以此类推得到的结果就是最优解，为了证明我们可以用数学归纳法，假设当前只有两个高度可以选择，那么肯定是先跳高的再跳低的；假设当前有三个高度可以选择，那么可以证明先从地面跳到最高处，再跳到最低处，最后跳到第二高的方法得到的值最大</p></li><li><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">310</span>;<br><br><span class="hljs-type">int</span> n, a[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) cin &gt;&gt; a[i];<br><br>    <span class="hljs-built_in">sort</span>(a + <span class="hljs-number">1</span>, a + n + <span class="hljs-number">1</span>, [&amp;](<span class="hljs-type">int</span> x, <span class="hljs-type">int</span> y) {<br>        <span class="hljs-keyword">return</span> x &gt; y;<br>    });<br><br>    <span class="hljs-type">int</span> res = a[<span class="hljs-number">1</span>] * a[<span class="hljs-number">1</span>], l = <span class="hljs-number">1</span>, r = n;<br>    <br>    <span class="hljs-keyword">while</span> (l &lt; r) {<br>        res += (a[l] - a[r]) * (a[l] - a[r]);<br>        l++;<br>        <span class="hljs-keyword">if</span> (l &lt; r) {<br>            res += (a[l] - a[r]) * (a[l] - a[r]);<br>            r--;<br>        }<br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="20-安排时间">20. 安排时间</h3><p><a href="https://www.acwing.com/problem/content/5466/">https://www.acwing.com/problem/content/5466/</a></p><blockquote><ul><li>题意：给定 n 个时间段处理餐厅事务，m 个用餐预定信息，含有三个信息分别为订单发起时间，订单需要准备的时间，客人来用餐时间。问如何安排每一个时间段，可以让所有的用户得到正常的服务。如果不能全部满足则输出 -1</li><li>思路：玄学贪心，先到的用户先做它的菜</li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">110</span>;<br><br><span class="hljs-type">int</span> n, m;<br><span class="hljs-type">int</span> res[N];<br><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">Item</span> {<br>    <span class="hljs-type">int</span> id;<br>    <span class="hljs-type">int</span> begi, arri, time;<br>    <span class="hljs-type">bool</span> <span class="hljs-keyword">operator</span>&lt; (<span class="hljs-type">const</span> Item&amp; t) <span class="hljs-type">const</span> {<br>        <span class="hljs-keyword">return</span> arri &lt; t.arri;<br>    }<br>} a[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; m;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= m; i++) {<br>        cin &gt;&gt; a[i].begi &gt;&gt; a[i].arri &gt;&gt; a[i].time;<br>        a[i].id = i;<br>        res[a[i].arri] = m + <span class="hljs-number">1</span>; <span class="hljs-comment">// 接客 </span><br>    }<br>    <br>    <span class="hljs-built_in">sort</span>(a + <span class="hljs-number">1</span>, a + m + <span class="hljs-number">1</span>);<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= m; i++) {<br>        <span class="hljs-type">int</span> t = a[i].time;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = a[i].begi; j &lt;= a[i].arri - <span class="hljs-number">1</span>; j++) {<br>            <span class="hljs-keyword">if</span> (t &gt; <span class="hljs-number">0</span> &amp;&amp; !res[j]) {<br>                res[j] = a[i].id; <span class="hljs-comment">// 做菜 </span><br>                t--;<br>            }<br>        }<br>        <br>        <span class="hljs-keyword">if</span> (t) {<br>            cout &lt;&lt; <span class="hljs-number">-1</span> &lt;&lt; <span class="hljs-string">"\n"</span>;<br>            <span class="hljs-keyword">return</span>;<br>        }<br>    }<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) cout &lt;&lt; res[i] &lt;&lt; <span class="hljs-string">" \n"</span>[i == n];<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="21-NOIP2004-提高组-合并果子">21. [NOIP2004 提高组] 合并果子</h3><p><a href="https://www.luogu.com.cn/problem/P1090">https://www.luogu.com.cn/problem/P1090</a></p><blockquote><ul><li>题意：给定 n 个数，现在需要将其两两合并为最终的一个数，每次合并的代价是两个数之和，问如何对这些数进行合并可以使得总代价最小</li><li>思路：很显然一共需要合并 n-1 次。那么我们从三个物品开始考虑，然后使用数学归纳法推导到 n 个数。对于三个物品，很显然先将两个物品进行合并，再将这个结果与最大的数进行合并方案时最优的。那么推广到 n 个数，我们只需要每次合并当前局面中最小的两个数即可</li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">10010</span>;<br><br><span class="hljs-type">int</span> n, a[N];<br>priority_queue&lt;<span class="hljs-type">int</span>, vector&lt;<span class="hljs-type">int</span>&gt;, greater&lt;<span class="hljs-type">int</span>&gt;&gt; q;<br><span class="hljs-type">int</span> res;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-type">int</span> x;<br>        cin &gt;&gt; x;<br>        q.<span class="hljs-built_in">push</span>(x);<br>    }<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n - <span class="hljs-number">1</span>; i++) {<br>        <span class="hljs-type">int</span> a = q.<span class="hljs-built_in">top</span>(); q.<span class="hljs-built_in">pop</span>();<br>        <span class="hljs-type">int</span> b = q.<span class="hljs-built_in">top</span>(); q.<span class="hljs-built_in">pop</span>();<br>        <br>        res += a + b;<br>        <br>        q.<span class="hljs-built_in">push</span>(a + b);<br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>} <br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="22-删数问题">22. 删数问题</h3><p><a href="https://www.luogu.com.cn/problem/P1106">https://www.luogu.com.cn/problem/P1106</a></p><blockquote><ul><li>题意：给定一个数字字符串长度为 n，现在需要选择其中的 m 位数，使得这 m 位组成的数最小是多少 (m=n-k)</li><li>思路：<strong>我们从一端考虑问题</strong>。为了组成的数尽可能的小，我们希望从首位开始就尽可能的小，同时如果有重复的数字，就选择最左侧的，剩余的数字留给后续的位数继续选择。有一个需要注意的点就是选择每一位的数字时，范围的约束，我们假设需要从 10 位的数字串中选择 3 位，那么我们第一次选择时需要在 0-7位中选择一位，第二次选择在 上次选择的下一个位置到第8位中选择一位，以此类推进行贪心选择</li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br>string s, res;<br><span class="hljs-type">int</span> k;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; s &gt;&gt; k;<br>    <span class="hljs-type">int</span> n = s.<span class="hljs-built_in">size</span>();<br>    <br>    <span class="hljs-comment">// 特判保留的数字个数 n-k 为负数的情况 </span><br>    <span class="hljs-keyword">if</span> (n - k &lt;= <span class="hljs-number">0</span>) {<br>        cout &lt;&lt; <span class="hljs-number">0</span> &lt;&lt; <span class="hljs-string">"\n"</span>;<br>        <span class="hljs-keyword">return</span>;<br>    }<br>    <br>    <span class="hljs-comment">// 循环 n-k 次选择保留的数 </span><br>    <span class="hljs-type">int</span> l = <span class="hljs-number">0</span>, r = k;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n - k; i++) {<br>        <span class="hljs-type">int</span> idx = <span class="hljs-number">-1</span>;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = l; j &lt;= r; j++)<br>            <span class="hljs-keyword">if</span> (idx == <span class="hljs-number">-1</span> || s[j] &lt; s[idx])<br>                idx = j;<br>        <br>        res += s[idx];<br>        <br>        l = idx + <span class="hljs-number">1</span>, r++;<br>    }<br>    <br>    <span class="hljs-comment">// 删除前导零 </span><br>    <span class="hljs-type">int</span> i = <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">while</span> (res[i] == <span class="hljs-string">'0'</span>) i++;<br>    <br>    <span class="hljs-comment">// 输出结果 </span><br>    <span class="hljs-keyword">if</span> (i == res.<span class="hljs-built_in">size</span>()) <br>        cout &lt;&lt; <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">else</span><br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = i; j &lt; res.<span class="hljs-built_in">size</span>(); j++) <br>            cout &lt;&lt; res[j];<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="23-截断数组">23. 截断数组</h3><p><a href="https://www.acwing.com/problem/content/5483/">https://www.acwing.com/problem/content/5483/</a></p><blockquote><p>题意：定义平衡数组为奇偶数数量相同的数组，现在给定一个平衡数组，和一个总成本数，最多可以将原数组截成多少截子平衡数组，且截断代价总和不超过总成本，截断代价计算公式为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><msub><mi>a</mi><mi>i</mi></msub><mo>−</mo><msub><mi>a</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|a_i-a_{i+1}|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mord">∣</span></span></span></span></p><p>思路：我们直接枚举所有的可以被截断的位置，统计所有的代价值，然后根据成本总数按降序枚举代价值即可</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;queue&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;cstring&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;vector&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;unordered_map&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;algorithm&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">110</span>;<br><br><span class="hljs-type">int</span> n, b;<br><span class="hljs-type">int</span> a[N], odd[N], even[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; b;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        cin &gt;&gt; a[i];<br>        <span class="hljs-keyword">if</span> (a[i] % <span class="hljs-number">2</span> == <span class="hljs-number">0</span>) {<br>            even[i] = even[i - <span class="hljs-number">1</span>] + <span class="hljs-number">1</span>;<br>            odd[i] = odd[i - <span class="hljs-number">1</span>];<br>        } <span class="hljs-keyword">else</span> {<br>            even[i] = even[i - <span class="hljs-number">1</span>];<br>            odd[i] = odd[i - <span class="hljs-number">1</span>] + <span class="hljs-number">1</span>;<br>        }<br>    }<br>    <br>    vector&lt;<span class="hljs-type">int</span>&gt; c;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">2</span>; i &lt;= n - <span class="hljs-number">2</span>; i += <span class="hljs-number">2</span>) {<br>        <span class="hljs-keyword">if</span> (odd[i] == even[i]) {<br>            c.<span class="hljs-built_in">push_back</span>(<span class="hljs-built_in">abs</span>(a[i] - a[i + <span class="hljs-number">1</span>]));<br>        }<br>    }<br>    <br>    <span class="hljs-type">int</span> res = <span class="hljs-number">0</span>;<br>    <span class="hljs-built_in">sort</span>(c.<span class="hljs-built_in">begin</span>(), c.<span class="hljs-built_in">end</span>());<br>    <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; x: c) {<br>        <span class="hljs-keyword">if</span> (b &gt;= x) {<br>            b -= x;<br>            res++;<br>        }<br>    }<br>    <br>    cout &lt;&lt; res;<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="24-修改后的最大二进制字符串">24. 修改后的最大二进制字符串</h3><p><a href="https://leetcode.cn/problems/maximum-binary-string-after-change/description/">https://leetcode.cn/problems/maximum-binary-string-after-change/description/</a></p><blockquote><p>题意：给定一个仅由01组成的字符串，现在可以执行下面两种操作任意次，使得最终的字符串对应的十进制数值最大，给出最终的字符串</p><ol><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>00</mn><mo>→</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">00 \to 10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">00</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span></span></span></span></li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10</mn><mo>→</mo><mn>01</mn></mrow><annotation encoding="application/x-tex">10 \to 01</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">01</span></span></span></span></li></ol><p>思路：</p><ul><li>有点像之前做的翻转转化为平移的问题，事实确实如此。首先需要确定一点就是：如果字符串起始有连续的1，则保留，因为两种操作都是针对0的。接着就是对从第一个0开始的尾串处理的思路</li><li>对于尾串而言，我们自然希望1越多、越靠前越好，但是第二种操作只能将1向后移，第一种操作又必须要连续的两个0，因此我们就产生了这样的贪心思路：将所有的1通过操作二移动到串尾，接着对尾串的开头连续的0串执行第二种操作，这样就可以得到最大数值的二进制串</li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span> {<br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-function">string <span class="hljs-title">maximumBinaryString</span><span class="hljs-params">(string binary)</span> </span>{<br>        <span class="hljs-type">int</span> n = binary.<span class="hljs-built_in">size</span>();<br><br>        string res;<br><br>        <span class="hljs-comment">// 前缀1全部加入答案</span><br>        <span class="hljs-type">int</span> i = <span class="hljs-number">0</span>;<br>        <span class="hljs-keyword">for</span> (i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>            <span class="hljs-keyword">if</span> (binary[i] == <span class="hljs-string">'0'</span>) {<br>                <span class="hljs-keyword">break</span>;<br>            }<br>            <span class="hljs-keyword">else</span> {<br>                res += <span class="hljs-string">'1'</span>;<br>            }<br>        }<br><br>        <span class="hljs-comment">// 非前缀1的部分一定可以操作为 00...011..1，进而转化为 11...11011...1</span><br>        <span class="hljs-type">int</span> zero = <span class="hljs-built_in">count</span>(binary.<span class="hljs-built_in">begin</span>() + i, binary.<span class="hljs-built_in">end</span>(), <span class="hljs-string">'0'</span>);<br>        <span class="hljs-keyword">if</span> (zero &gt; <span class="hljs-number">0</span>) {<br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">0</span>; j &lt; zero - <span class="hljs-number">1</span>; j++) res += <span class="hljs-string">'1'</span>;<br>            res += <span class="hljs-string">'0'</span>;<br>        }<br><br>        <span class="hljs-keyword">while</span> (res.<span class="hljs-built_in">size</span>() &lt; n) res += <span class="hljs-string">'1'</span>;<br><br>        <span class="hljs-keyword">return</span> res;<br>    }<br>};<br></code></pre></td></tr></tbody></table></figure><h3 id="25-子序列最大优雅度">25. 子序列最大优雅度</h3><p><a href="https://leetcode.cn/problems/maximum-elegance-of-a-k-length-subsequence/">https://leetcode.cn/problems/maximum-elegance-of-a-k-length-subsequence/</a></p><blockquote><p>标签：反悔贪心</p><p>题意：给定长度为 n 的物品序列，现在需要从中寻找一个长度为 k 的子序列使得子序列的总价值最大，给出最终的最大价值。一个序列的价值定义为每个物品的 <strong>本身价值之和</strong> + <strong>物品种类数的平方</strong></p><p>思路：我们采用反悔贪心的思路。首先按照物品本身价值进行降序排序并选择前 k 个物品作为集合 S，然后按顺序枚举剩下的物品进行替换决策。我们记每个物品有一个「本身价值」，集合 S 有一个「种类价值」。那么就会遇到下面两种情况：</p><ul><li>第 i 个物品对应的种类<strong>存在</strong>集合 S 中。此时该物品「一定不选」，因为无论当前物品替代集合中的 S 哪一个物品，本身价值的贡献会下降、种类价值的贡献保持不变。</li><li>第 i 个物品对应的种类<strong>不在</strong>集合 S 中。此时该物品一定选择吗？不一定。有两种情况：<ul><li>集合 S 没有重复种类的物品。此时该物品「一定不选」，同样的，无论当前物品替代集合中的 S 哪一个物品，本身价值的贡献会下降、种类价值的贡献保持不变。</li><li>集合 S 存在重复种类的物品。此时该物品「一定选」，因为一定可以提升种类价值的贡献，至于减去本身价值的损失后是否使得总贡献增加，并不具备可能的性质，只能每次替换物品时<strong>更新全局答案</strong>进行记录。</li></ul></li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><figcaption><span>[]</span></figcaption><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span> {<br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-function"><span class="hljs-type">long</span> <span class="hljs-type">long</span> <span class="hljs-title">findMaximumElegance</span><span class="hljs-params">(std::vector&lt;std::vector&lt;<span class="hljs-type">int</span>&gt;&gt;&amp; items, <span class="hljs-type">int</span> k)</span> </span>{<br>        <span class="hljs-keyword">using</span> ll = <span class="hljs-type">long</span> <span class="hljs-type">long</span>;<br><br>        std::<span class="hljs-built_in">sort</span>(items.<span class="hljs-built_in">rbegin</span>(), items.<span class="hljs-built_in">rend</span>());<br><br>        ll t = <span class="hljs-number">0</span>, cat = <span class="hljs-number">0</span>, res = <span class="hljs-number">0</span>;<br>        std::unordered_map&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt; cnt;<br>        std::stack&lt;ll&gt; stk;<br><br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; items.<span class="hljs-built_in">size</span>(); i++) {<br>            ll v = items[i][<span class="hljs-number">0</span>], c = items[i][<span class="hljs-number">1</span>];<br>            <span class="hljs-keyword">if</span> (i &lt; k) {<br>                <span class="hljs-comment">// 前 k 个物品</span><br>                t += v;<br>                cnt[c]++;<br>                cat += cnt[c] == <span class="hljs-number">1</span>;<br>                <span class="hljs-keyword">if</span> (cnt[c] &gt; <span class="hljs-number">1</span>) {<br>                    stk.<span class="hljs-built_in">push</span>(v);<br>                }<br>            } <span class="hljs-keyword">else</span> {<br>                <span class="hljs-comment">// 后 n-k 个物品</span><br>                <span class="hljs-keyword">if</span> (cnt[c]) {<br>                    <span class="hljs-comment">// 已经出现过的物品种类</span><br>                    <span class="hljs-keyword">continue</span>;<br>                } <span class="hljs-keyword">else</span> {<br>                    <span class="hljs-comment">// 没有出现过的物品种类</span><br>                    <span class="hljs-keyword">if</span> (stk.<span class="hljs-built_in">size</span>()) {<br>                        cat++;<br>                        t -= stk.<span class="hljs-built_in">top</span>();<br>                        t += v;<br><br>                        stk.<span class="hljs-built_in">pop</span>();<br>                        cnt[c]++;<br>                    }<br>                }<br>            }<br>            <span class="hljs-comment">// 更新全局答案</span><br>            res = <span class="hljs-built_in">max</span>(res, t + cat * cat);<br>        }<br><br>        <span class="hljs-keyword">return</span> res;<br>    }<br>};<br></code></pre></td></tr></tbody></table></figure><figure class="highlight python"><figcaption><span>[]</span></figcaption><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span>:<br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">findMaximumElegance</span>(<span class="hljs-params">self, items: <span class="hljs-type">List</span>[<span class="hljs-type">List</span>[<span class="hljs-built_in">int</span>]], k: <span class="hljs-built_in">int</span></span>) -&gt; <span class="hljs-built_in">int</span>:<br>        <span class="hljs-keyword">from</span> collections <span class="hljs-keyword">import</span> defaultdict<br>        items.sort(reverse=<span class="hljs-literal">True</span>)<br><br>        t, cat, res = <span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">0</span><br>        cnt, stk = defaultdict(<span class="hljs-built_in">int</span>), []<br><br>        <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-built_in">len</span>(items)):<br>            v, c = items[i][<span class="hljs-number">0</span>], items[i][<span class="hljs-number">1</span>]<br>            <span class="hljs-keyword">if</span> i &lt; k:<br>                t += v<br>                cnt[c] += <span class="hljs-number">1</span><br>                cat += cnt[c] == <span class="hljs-number">1</span><br>                <span class="hljs-keyword">if</span> cnt[c] &gt; <span class="hljs-number">1</span>:<br>                    stk.append(v)<br>            <span class="hljs-keyword">else</span>:<br>                <span class="hljs-keyword">if</span> cnt[c]:<br>                    <span class="hljs-keyword">continue</span><br>                <span class="hljs-keyword">else</span>:<br>                    <span class="hljs-keyword">if</span> <span class="hljs-built_in">len</span>(stk):<br>                        t -= stk[-<span class="hljs-number">1</span>]<br>                        t += v<br>                        cat += <span class="hljs-number">1</span><br>                        stk.pop()<br>                        cnt[c] += <span class="hljs-number">1</span><br>            res = <span class="hljs-built_in">max</span>(res, t + cat**<span class="hljs-number">2</span>)<br><br>        <span class="hljs-keyword">return</span> res<br></code></pre></td></tr></tbody></table></figure><h3 id="【按位贪心-分类讨论】">【按位贪心/分类讨论】</h3><p><a href="https://www.acwing.com/problem/content/1000/">https://www.acwing.com/problem/content/1000/</a></p><blockquote><p>题意：给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个运算数和对应的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个操作，仅有 <code>&amp;, |, ^</code> 三种。现在需要选择一个数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mi>m</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">num \in [0,m]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mclose">]</span></span></span></span> 使得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">num</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span></span></span></span> 经过这 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个二元运算后得到的结果 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mi>e</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">res</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">res</span></span></span></span> 最大。给出最终的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mi>e</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">res</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">res</span></span></span></span>。</p><p>思路：首先不难发现这些运算操作在二进制位上是相互独立的，没有进位或结尾的情况，因此我们可以「按二进制位」单独考虑。由于运算数和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 均 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≤</mo><mn>1</mn><msup><mn>0</mn><mn>9</mn></msup></mrow><annotation encoding="application/x-tex">\le 10^9</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span></span></span></span>，因此我们枚举 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mn>30</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0,30]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">30</span><span class="mclose">]</span></span></span></span> 对应的二进制位。接下来我们单独考虑第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 位的情况，显然的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">num</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span></span></span></span> 的第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 位可以取 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 也可以取 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，对应的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mi>e</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">res</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">res</span></span></span></span> 的第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 位可以取 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 也可以取 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>。有以下 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">2^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 种运算情况：</p><table><thead><tr><th style="text-align:center"></th><th style="text-align:center">$\text{num 的第 i 位填 1 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> res 第 i 位的情况}$</th><th style="text-align:center">$\text{num 的第 i 位填 0 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> res 第 i 位的情况}$</th><th style="text-align:center">选择方案</th></tr></thead><tbody><tr><td style="text-align:center">情况一</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">1 \to 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0 \to 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></td><td style="text-align:center">num 填 0，res 填 0</td></tr><tr><td style="text-align:center">情况二</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">1 \to 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">0 \to 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></td><td style="text-align:center">num 填 0，res 填 1</td></tr><tr><td style="text-align:center">情况三</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">1 \to 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0 \to 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></td><td style="text-align:center">待定</td></tr><tr><td style="text-align:center">情况四</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">1 \to 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">0 \to 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></td><td style="text-align:center">num 填 0，res 填 1</td></tr></tbody></table><p>选择方案时我们需要考虑两个约束，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">num</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span></span></span></span> 不能超过 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mi>e</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">res</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">res</span></span></span></span> 尽可能大，因此有了上述表格第四列的贪心选择结果。之所以情况三待定是因为其对两种约束的相互矛盾的，并且其余情况没有增加 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">num</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span></span></span></span>，即对 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">num</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span></span></span></span> 上限的约束仅存在于情况三。假设有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 位是上述情况三，那么显然的其余位枚举顺序是随意的，因为答案是固定的。对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个情况三，从贪心的角度考虑，我们希望 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mi>e</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">res</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">res</span></span></span></span> 尽可能大，那么就有「高位能出 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 就出 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>」的结论。因此我们需要从高位到低位逆序枚举这 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个情况三，由于其他位枚举顺序随意，因此总体就是从高位到低位枚举。</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><br><span class="hljs-keyword">using</span> ll = <span class="hljs-type">long</span> <span class="hljs-type">long</span>;<br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-type">int</span> n, m;<br>    cin &gt;&gt; n &gt;&gt; m;<br>    <br>    vector&lt;pair&lt;string, <span class="hljs-type">int</span>&gt;&gt; <span class="hljs-built_in">a</span>(n);<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        cin &gt;&gt; a[i].first &gt;&gt; a[i].second;<br>    }<br>    <br>    <span class="hljs-type">int</span> res = <span class="hljs-number">0</span>, num = <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">30</span>; i &gt;= <span class="hljs-number">0</span>; i--) {<br>        <span class="hljs-type">int</span> x = <span class="hljs-number">1</span> &lt;&lt; i, y = <span class="hljs-number">0</span>;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">0</span>; j &lt; n; j++) {<br>            string op = a[j].first;<br>            <span class="hljs-type">int</span> t = a[j].second &amp; (<span class="hljs-number">1</span> &lt;&lt; i);<br>            <span class="hljs-keyword">if</span> (op == <span class="hljs-string">"AND"</span>) {<br>                x &amp;= t, y &amp;= t;<br>            } <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (op == <span class="hljs-string">"OR"</span>) {<br>                x |= t, y |= t;<br>            } <span class="hljs-keyword">else</span> {<br>                x ^= t, y ^= t;<br>            }<br>        }<br>        <span class="hljs-keyword">if</span> ((num | (<span class="hljs-number">1</span> &lt;&lt; i)) &lt;= m &amp;&amp; x &gt; y) {<br>            num |= <span class="hljs-number">1</span> &lt;&lt; i;<br>            res |= <span class="hljs-number">1</span> &lt;&lt; i;<br>        } <span class="hljs-keyword">else</span> {<br>            res += y;<br>        }<br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    std::ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    std::cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    std::cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure>]]></content>
     
     
     <categories>
@@ -1498,11 +1498,11 @@
   
   
   <entry>
-    <title>number-theory</title>
-    <link href="/Algorithm/number-theory/"/>
-    <url>/Algorithm/number-theory/</url>
+    <title>hashing</title>
+    <link href="/Algorithm/hashing/"/>
+    <url>/Algorithm/hashing/</url>
     
-    <content type="html"><![CDATA[<h3 id="数论">数论</h3><p>整数问题。</p><h3 id="【质数】Divide-and-Equalize">【质数】Divide and Equalize</h3><blockquote><p>题意：给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个数，问能否找到一个数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">num</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span></span></span></span>，使得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><msup><mi>m</mi><mi>n</mi></msup><mo>=</mo><msubsup><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>a</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">num^n = \prod_{i=1}^{n}a_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6644em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.104em;vertical-align:-0.2997em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∏</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8043em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p><p>原始思路：起初我的思路是二分，我们需要寻找一个数使得n个数相乘为原数组所有元素之积，那么我们预计算出所有数之积，并且在数组最大值和最小值之间进行二分，每次二分出来的数计算n次方进行笔比较即可。但是有一个很大的问题是，数据量是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mn>4</mn></msup></mrow><annotation encoding="application/x-tex">10^4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span>，而数字最大为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mn>6</mn></msup></mrow><annotation encoding="application/x-tex">10^6</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span></span></span></span>，最大之积为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mn>10</mn></msup></mrow><annotation encoding="application/x-tex">10^{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span></span></span></span></span></span></span></span> 吗？不是！最大之和才是，最大之积是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mrow><mn>6</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mn>4</mn></msup></mrow></msup></mrow><annotation encoding="application/x-tex">10^{6\times10^4}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9869em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9869em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">6</span><span class="mbin mtight">×</span><span class="mord mtight">1</span><span class="mord mtight"><span class="mord mtight">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></p><p>最终思路：我们可以将选数看做多个水池匀水的过程。现在每一个水池的水高都不定，很显然我们一定可以一个值使得所有的水池的高度一致，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>a</mi><mi>i</mi></msub></mrow><mi>n</mi></mfrac></mrow><annotation encoding="application/x-tex">\frac{\sum_{i=1}^{n}a_i}{n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.405em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.06em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.535em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mop op-symbol small-op mtight" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7385em;"><span style="top:-2.1786em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3214em;"><span></span></span></span></span></span></span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mtight"><span class="mord mathnormal mtight">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>。但是我们最终的数字是一个整数，显然不可以直接求和然后除以n，那么应该如何分配呢？我们知道水池之所以可以直接除以n，是因为水的最小分配单位是无穷小，可以随意分割；而对于整数而言，最小分配单位是什么呢？答案是<strong>质因子</strong>！为了通过分配最小单位使得最终的“水池高度一致”，我们需要让每一个“水池”获得的数值相同的质因子数量相同。于是我们只需要统计一下数组中所有数的质因子数量即可。如果对于每一种质因子的数量都可以均匀分配每一个数（即数量是n的整数倍），那么就一定可以找到这个数使得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><msup><mi>m</mi><mi>n</mi></msup><mo>=</mo><msubsup><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>a</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">num^n = \prod_{i=1}^{n}a_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6644em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.104em;vertical-align:-0.2997em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∏</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8043em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br><span class="hljs-type">int</span> n;<br>cin &gt;&gt; n;<br><br><span class="hljs-comment">// 统计所有数字的所有质因数</span><br>unordered_map&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt; m;<br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br><span class="hljs-type">int</span> x;<br>cin &gt;&gt; x;<br><br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> k = <span class="hljs-number">2</span>; k &lt;= x / k; k++) {<br><span class="hljs-keyword">if</span> (x % k == <span class="hljs-number">0</span>) {<br><span class="hljs-keyword">while</span> (x % k == <span class="hljs-number">0</span>) {<br>m[k]++;<br>x /= k;<br>}<br>}<br>}<br><br><span class="hljs-keyword">if</span> (x &gt; <span class="hljs-number">1</span>) {<br>m[x]++;<br>}<br>}<br><br><span class="hljs-comment">// 查看每一种质因数是否是n的整数倍</span><br><span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; x: m) {<br><span class="hljs-keyword">if</span> (x.second % n) {<br>cout &lt;&lt; <span class="hljs-string">"No\n"</span>;<br><span class="hljs-keyword">return</span>;<br>}<br>}<br><br>cout &lt;&lt; <span class="hljs-string">"Yes\n"</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【整除】Deja-Vu">【整除】Deja Vu</h3><p><a href="https://codeforces.com/contest/1891/problem/B">https://codeforces.com/contest/1891/problem/B</a></p><blockquote><p>题意：给点序列 a 和 b，对于 b 中的每一个元素 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>b</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">b_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，如果 a 中的元素 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">a_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 能够整除 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><msub><mi>b</mi><mi>i</mi></msub></msup></mrow><annotation encoding="application/x-tex">2^{b_i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>，则将 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">a_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 加上 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mrow><msub><mi>b</mi><mi>i</mi></msub><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">2^{b_i - 1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span>。给出最后的 a 序列</p><p>思路一：暴力枚举。</p><ul><li><p>我们很容易就想到暴力的做法，即两层循环，第一层枚举 b 中的元素，第二层枚举 a 中的元素，如果 a 中的元素能够整除 2 的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>b</mi><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">b^i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8247em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span></span> 次方，就将 a 中相应的元素加上一个值即可。但是时间复杂度肯定过不了，考虑优化。</p></li><li><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(nm)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">nm</span><span class="mclose">)</span></span></span></span></p></li></ul><p>思路二：整除优化。</p><ul><li><p>现在我们假设a中有一个数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">a_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><msub><mi>b</mi><mi>i</mi></msub></msup></mrow><annotation encoding="application/x-tex">2^{b_i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span> 的整数倍（其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>b</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">b_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是b序列中第一个枚举到的能够让 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">a_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 整除的数），那么就有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub><mo>=</mo><mi>k</mi><msup><mn>2</mn><msub><mi>b</mi><mi>i</mi></msub></msup><mo stretchy="false">(</mo><mi>k</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a_j = k2^{b_i}(k=1,2,...,)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0991em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">...</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span></span></span></span>，那么 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">a_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 就要加上 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mrow><msub><mi>b</mi><mi>i</mi></msub><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">2^{b_i-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span>，于是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">a_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 就变为了 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><msup><mn>2</mn><msub><mi>b</mi><mi>i</mi></msub></msup><mo>+</mo><msup><mn>2</mn><mrow><msub><mi>b</mi><mi>i</mi></msub><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mo stretchy="false">(</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mn>2</mn><mrow><msub><mi>b</mi><mi>i</mi></msub><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">k2^{b_i}+2^{b_i-1}=(2k+1)2^{b_i-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9324em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0991em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span>。此后 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">a_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 就一定是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>t</mi></msup><mo stretchy="false">(</mo><mi>t</mi><mo>∈</mo><mrow><mo fence="true">[</mo><mn>1</mn><mo separator="true">,</mo><msub><mi>b</mi><mi>i</mi></msub><mo>−</mo><mn>1</mn><mo fence="true">]</mo></mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">2^t(t\in \left[ 1,b_i-1 \right])</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0436em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose delimcenter" style="top:0em;">]</span></span><span class="mclose">)</span></span></span></span> 的倍数。因此我们需要做的就是首先找到b序列中第一个数x，能够在a中找到数是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>x</mi></msup></mrow><annotation encoding="application/x-tex">2^x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6644em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span></span> 的整数倍。这一步可以这样进行：对于 a中的每一个数，我们进行30次循环统计当前数是否是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">2^i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8247em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span></span> 的倍数，如果是就用哈希表记录当前的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>。最后我们在遍历寻找 x 时，只需要查看当前的 x 是否被哈希过即可。接着我们统计b序列中从x开始的严格降序序列c（由题意知，次序列的数量一定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≤</mo></mrow><annotation encoding="application/x-tex">\le</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≤</span></span></span></span> 30，因为b序列中数值的值域为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mn>1</mn><mtext>&nbsp;</mtext><mn>30</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[1~30]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">1</span><span class="mspace nobreak">&nbsp;</span><span class="mord">30</span><span class="mclose">]</span></span></span></span>）。最后我们再按照原来的思路，双重循环 a 序列和 c 序列即可。</p></li><li><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>30</mn><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(30n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">30</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br><span class="hljs-type">int</span> n, m;<br>cin &gt;&gt; n &gt;&gt; m;<br><br><span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">a</span><span class="hljs-params">(n + <span class="hljs-number">1</span>)</span>, <span class="hljs-title">b</span><span class="hljs-params">(m + <span class="hljs-number">1</span>)</span></span>;<br>unordered_map&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt; ha;<br><br>    <span class="hljs-comment">// 边读边哈希</span><br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>cin &gt;&gt; a[i];<br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">30</span>; j &gt;= <span class="hljs-number">1</span>; j--) {<br><span class="hljs-keyword">if</span> (a[i] % (<span class="hljs-number">1</span> &lt;&lt; j) == <span class="hljs-number">0</span>) {<br>ha[j]++;<br>}<br>}<br>}<br><br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= m; i++) {<br>cin &gt;&gt; b[i];<br>}<br><br><span class="hljs-comment">// 寻找b中第一个能够让a[j]整除的数b[flag]</span><br><span class="hljs-type">int</span> flag = <span class="hljs-number">-1</span>;<br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= m; i++) {<br><span class="hljs-keyword">if</span> (ha[b[i]]) {<br>flag = i;<br><span class="hljs-keyword">break</span>;<br>}<br>}<br><br>    <span class="hljs-comment">// 特判</span><br><span class="hljs-keyword">if</span> (flag == <span class="hljs-number">-1</span>) {<br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">1</span>; j &lt;= n; j++) {<br>cout &lt;&lt; a[j] &lt;&lt; <span class="hljs-string">" \n"</span>[j == n];<br>}<br><span class="hljs-keyword">return</span>;<br>}<br><br>    <span class="hljs-comment">// 寻找b中从flag开始的严格单调递减的序列c</span><br>vector&lt;<span class="hljs-type">int</span>&gt; c;<br>c.<span class="hljs-built_in">push_back</span>(b[flag]);<br><span class="hljs-keyword">for</span> (; flag &lt;= m; flag++) {<br><span class="hljs-keyword">if</span> (b[flag] &lt; c.<span class="hljs-built_in">back</span>()) {<br>c.<span class="hljs-built_in">push_back</span>(b[flag]);<br>}<br>}<br><br>    <span class="hljs-comment">// 暴力循环一遍即可</span><br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">1</span>; j &lt;= n; j++) {<br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> k = <span class="hljs-number">0</span>; k &lt; c.<span class="hljs-built_in">size</span>(); k++) {<br><span class="hljs-keyword">if</span> (a[j] % (<span class="hljs-number">1</span> &lt;&lt; c[k]) == <span class="hljs-number">0</span>) {<br>a[j] += <span class="hljs-number">1</span> &lt;&lt; (c[k] - <span class="hljs-number">1</span>);<br>}<br>}<br>}<br><br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">1</span>; j &lt;= n; j++) {<br>cout &lt;&lt; a[j] &lt;&lt; <span class="hljs-string">" \n"</span>[j == n];<br>}<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【组合数学】序列数量">【组合数学】序列数量</h3><p><a href="https://www.acwing.com/problem/content/5571/">https://www.acwing.com/problem/content/5571/</a></p><blockquote><p>题意：给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i(i=1,2,\cdots,n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">i</span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 个苹果，将其分给 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 个人，问一共有多少种分配方案，给出结果对 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mn>6</mn></msup><mo>+</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">10^6+3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span></span></span></span> 取模的结果</p><p>思路：整数分配问题。我们采用隔板法，隔板法相关例题见这篇博客：<a href="https://www.acwing.com/solution/content/241669/%E3%80%82%E4%B8%8B%E9%9D%A2%E5%BC%80%E5%A7%8B%E8%AE%B2%E8%A7%A3">https://www.acwing.com/solution/content/241669/。下面开始讲解</a></p><ul><li><p>利用<strong>隔板法</strong>推导结果。首先我们考虑当前局面，即只有 i 个苹果的情况下的方案数。于是题目就是 i 个苹果分给 m 个人，允许分到 0 个。于是借鉴上述链接中“少分型”的思路，先借 m 个苹果，那么此时局面中就有 i+m 个苹果，现在就等价于将 i+m 个苹果分给 m 个人，每人至少分得 1 个苹果。（分完以后每个人都还回去就行了），此时的隔板操作就是”标准型”，即 i+m 个苹果产生 i+m-1 个间隔，在其中插 m-1 块板，从而将划分出来的 m 个部分分给 m 个人。此时的划分方案就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>C</mi><mrow><mi>i</mi><mo>+</mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">C_{i+m-1}^{m-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1894em;vertical-align:-0.3352em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8542em;"><span style="top:-2.4231em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">+</span><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.1031em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3352em;"><span></span></span></span></span></span></span></span></span></span>，那么对于所有的 i，结果就是</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>C</mi><mrow><mi>i</mi><mo>+</mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msubsup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msubsup><mi>C</mi><mi>m</mi><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mo>+</mo><msubsup><mi>C</mi><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mo>+</mo><mo>⋯</mo><mo>+</mo><msubsup><mi>C</mi><mrow><mi>n</mi><mo>+</mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msubsup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msubsup><mi>C</mi><mi>m</mi><mn>1</mn></msubsup><mo>+</mo><msubsup><mi>C</mi><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mo>+</mo><mo>⋯</mo><mo>+</mo><msubsup><mi>C</mi><mrow><mi>n</mi><mo>+</mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow><mi>n</mi></msubsup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo stretchy="false">(</mo><mo>−</mo><msubsup><mi>C</mi><mi>m</mi><mn>0</mn></msubsup><mo>+</mo><msubsup><mi>C</mi><mi>m</mi><mn>0</mn></msubsup><mo stretchy="false">)</mo><mo>+</mo><msubsup><mi>C</mi><mi>m</mi><mn>1</mn></msubsup><mo>+</mo><msubsup><mi>C</mi><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mo>+</mo><mo>⋯</mo><mo>+</mo><msubsup><mi>C</mi><mrow><mi>n</mi><mo>+</mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow><mi>n</mi></msubsup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>−</mo><msubsup><mi>C</mi><mi>m</mi><mn>0</mn></msubsup><mo>+</mo><mo stretchy="false">(</mo><msubsup><mi>C</mi><mi>m</mi><mn>0</mn></msubsup><mo>+</mo><msubsup><mi>C</mi><mi>m</mi><mn>1</mn></msubsup><mo>+</mo><msubsup><mi>C</mi><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mo>+</mo><mo>⋯</mo><mo>+</mo><msubsup><mi>C</mi><mrow><mi>n</mi><mo>+</mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>−</mo><mn>1</mn><mo>+</mo><mo stretchy="false">(</mo><msubsup><mi>C</mi><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow><mn>1</mn></msubsup><mo>+</mo><msubsup><mi>C</mi><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mo>+</mo><mo>⋯</mo><mo>+</mo><msubsup><mi>C</mi><mrow><mi>n</mi><mo>+</mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>−</mo><mn>1</mn><mo>+</mo><mo stretchy="false">(</mo><msubsup><mi>C</mi><mrow><mi>m</mi><mo>+</mo><mn>2</mn></mrow><mn>2</mn></msubsup><mo>+</mo><mo>⋯</mo><mo>+</mo><msubsup><mi>C</mi><mrow><mi>n</mi><mo>+</mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>−</mo><mn>1</mn><mo>+</mo><msubsup><mi>C</mi><mrow><mi>n</mi><mo>+</mo><mi>m</mi></mrow><mi>n</mi></msubsup></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\sum_{i=1}^n C_{i+m-1}^{m-1} &amp;= C_{m}^{m-1} + C_{m+1}^{m-1} + \cdots + C_{n+m-1}^{m-1} \\&amp;= C_{m}^{1} + C_{m+1}^{2} + \cdots + C_{n+m-1}^{n} \\&amp;= (-C_{m}^{0}+C_{m}^{0})+C_{m}^{1} + C_{m+1}^{2} + \cdots + C_{n+m-1}^{n} \\&amp;= -C_{m}^{0}+(C_{m}^{0}+C_{m}^{1} + C_{m+1}^{2} + \cdots + C_{n+m-1}^{n}) \\&amp;= -1+(C_{m+1}^{1} + C_{m+1}^{2} + \cdots + C_{n+m-1}^{n}) \\&amp;= -1+(C_{m+2}^{2} + \cdots + C_{n+m-1}^{n}) \\&amp;= -1+C_{n+m}^n\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:12.3496em;vertical-align:-5.9248em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.4248em;"><span style="top:-8.4248em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.433em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">+</span><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3253em;"><span></span></span></span></span></span></span></span></span><span style="top:-5.983em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"></span></span><span style="top:-4.4589em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"></span></span><span style="top:-2.9348em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"></span></span><span style="top:-1.4107em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"></span></span><span style="top:0.1134em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"></span></span><span style="top:1.6134em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.9248em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.4248em;"><span style="top:-8.4248em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.4436em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3148em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.4436em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3148em;"><span></span></span></span></span></span></span></span></span><span style="top:-5.983em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.4589em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.9348em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">−</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-1.4107em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:0.1134em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">+</span><span class="mord mtight">2</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:1.6134em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mathnormal mtight">m</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.9248em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li><li><p>利用<strong>乘法逆元</strong>计算组合数。结果已经知道了，现在如何计算上述表达式呢？由于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">n\times m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 超过了常规递推的组合数计算方法内存空间，因此我们采用乘法逆元的思路计算。值得一提的是，本题在计算某个数关于 <code>1e6+3</code> 的乘法逆元时，不需要判断两者是否互质，因为 <code>1e6+3</code> 是一个质数，并且数据范围中的数均小于 <code>1e6+3</code>，因此两数一定互质，可以直接使用费马小定理计算乘法逆元</p></li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mn>1</mn><mi>e</mi><mn>6</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\log 1e6)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mord mathnormal">e</span><span class="mord">6</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><span class="hljs-keyword">using</span> ll = <span class="hljs-type">long</span> <span class="hljs-type">long</span>;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">7e5</span> + <span class="hljs-number">10</span>;<br><span class="hljs-type">const</span> <span class="hljs-type">int</span> mod = <span class="hljs-number">1e6</span> + <span class="hljs-number">3</span>;<br><br><span class="hljs-type">int</span> fact[N], infact[N];<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">qmi</span><span class="hljs-params">(<span class="hljs-type">int</span> a, <span class="hljs-type">int</span> b, <span class="hljs-type">int</span> p)</span> </span>{<br>    <span class="hljs-type">int</span> res = <span class="hljs-number">1</span> % p;<br>    <span class="hljs-keyword">while</span> (b) {<br>        <span class="hljs-keyword">if</span> (b &amp; <span class="hljs-number">1</span>) res = (ll)res * a % p;<br>        a = (ll)a * a % p;<br>        b &gt;&gt;= <span class="hljs-number">1</span>;<br>    }<br>    <span class="hljs-keyword">return</span> res;<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">init</span><span class="hljs-params">()</span> </span>{<br>    fact[<span class="hljs-number">0</span>] = <span class="hljs-number">1</span>, infact[<span class="hljs-number">0</span>] = <span class="hljs-number">1</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt; N; i++) {<br>        fact[i] = (ll)fact[i - <span class="hljs-number">1</span>] * i % mod;<br>        infact[i] = (ll)infact[i - <span class="hljs-number">1</span>] * <span class="hljs-built_in">qmi</span>(i, mod - <span class="hljs-number">2</span>, mod) % mod;<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-built_in">init</span>();<br>    <br>    <span class="hljs-type">int</span> n, m;<br>    cin &gt;&gt; n &gt;&gt; m;<br>    <br>    cout &lt;&lt; (ll)fact[n + m] * infact[n] % mod * infact[m] % mod - <span class="hljs-number">1</span>;<br>    <br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure>]]></content>
+    <content type="html"><![CDATA[<h3 id="哈希">哈希</h3><h3 id="【哈希】分组">【哈希】分组</h3><p><a href="https://www.acwing.com/problem/content/5182/">https://www.acwing.com/problem/content/5182/</a></p><blockquote><p>存储不想同组和想同组的人员信息：存入数组，数据类型为一对字符串</p><p>存储所有的组队信息：存入哈希表，数据类型为“键:字符串”“值:一对字符串”</p><p>想要知道最终的分组情况，只需要查询数组中的队员情况与想同组 or 不想同组的成员名字是否一致即可</p><p>时间复杂度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>，空间复杂度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mtext>&nbsp;</mtext><mi>l</mi><mi>e</mi><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mi>a</mi><mi>m</mi><mi>e</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></mrow><annotation encoding="application/x-tex">O(n\ len(name))_{max}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace">&nbsp;</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">e</span><span class="mord mathnormal">n</span><span class="mopen">(</span><span class="mord mathnormal">nam</span><span class="mord mathnormal">e</span><span class="mclose">)</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">ma</span><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;vector&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;unordered_map&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span></span><br><span class="hljs-function"></span>{<br>    <span class="hljs-type">int</span> x;<br>    cin &gt;&gt; x;<br>    <br>    vector&lt;pair&lt;string, string&gt;&gt; <span class="hljs-built_in">X</span>(x);<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; x; i ++)<br>        cin &gt;&gt; X[i].first &gt;&gt; X[i].second;<br>    <br>    <span class="hljs-type">int</span> y;<br>    cin &gt;&gt; y;<br>    <br>    vector&lt;pair&lt;string, string&gt;&gt; <span class="hljs-built_in">Y</span>(y);<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; y; i ++)<br>        cin &gt;&gt; Y[i].first &gt;&gt; Y[i].second;<br>        <br>    <span class="hljs-type">int</span> sum;<br>    cin &gt;&gt; sum;<br>    <br>    unordered_map&lt;string, pair&lt;string, string&gt;&gt; a;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; sum; i ++)<br>    {<br>        string s, t, p;<br>        cin &gt;&gt; s &gt;&gt; t &gt;&gt; p;<br>        a[s] = {t, p};<br>        a[t] = {s, p};<br>        a[p] = {s, t};<br>    }<br>    <br>    <span class="hljs-type">int</span> res = <span class="hljs-number">0</span>;<br>    <br>    <span class="hljs-comment">// 想同组 </span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; x; i ++)<br>    {<br>        string s = X[i].first, t = X[i].second;<br>        <span class="hljs-keyword">if</span> (a[s].first != t &amp;&amp; a[s].second != t)<br>            res ++;<br>    }<br>    <br>    <span class="hljs-comment">// 不想同组 </span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; y; i ++)<br>    {<br>        string s = Y[i].first, t = Y[i].second;<br>        <span class="hljs-keyword">if</span> (a[s].first == t || a[s].second == t)<br>            res ++; <br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; endl; <br>    <br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【哈希】海港">【哈希】海港</h3><p><a href="https://www.luogu.com.cn/problem/P2058">https://www.luogu.com.cn/problem/P2058</a></p><blockquote><ul><li>题意：给定 n 艘船只的到达时间、载客信息（载客人数和每一个客人的国籍），现在需要知道对于每一艘抵达的船只，前 24 小时中抵达的客人的国籍总数</li><li>思路：本题思路很简单，就是一个队列的应用以及哈希客人国籍的过程。由于船只抵达的时间是顺序增加的，故每抵达一艘船只，就对新来的客人国籍进行哈希，为了计算前 24 小时的情况，需要对船只抵达队列进行删减，即只保留 24 小时以内的船只抵达信息。对于删除的船只信息，需要将这些船只上的客人国籍信息从哈希表中删除，故每一艘船只的访问次数为 2。</li><li><code>unordered_map </code>补充：在进行哈希统计时。为了判断当前 24 小时内客人国籍数，在删除哈希记录时，为了判断是否将当前国籍的游客全部删除时，需要统计哈希表中某个国籍是否减为了 0，我用了 <code>.count(x)</code> 内置方法，但这是不正确的，因为我想要统计的是 值 是否为 0，而 <code>.count(x)</code> 统计的是哈希表中 x 这个 键 的个数，而 <code>unordered_map </code>中是没有重复的键的，故 <code>.count(x)</code> 方法只会返回 0 或 1，返回 0 就表示当前哈希表中没有 x 这个键，返回 1 就表示哈希表中有 x 这个键，但是有这个键不代表对应的值就存在，可能是 <code>x: 0</code> 的情况，即键存在，但是值记录为 0</li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>2</mn><mo>∑</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(2 \sum x_i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">2</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> - 即两倍的所有游客数</li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-comment">// #include &lt;bits/stdc++.h&gt;</span><br><span class="hljs-comment">// #define int long long</span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;unordered_map&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;stack&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;queue&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">1e5</span> + <span class="hljs-number">10</span>;<br><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">Ship</span> { <span class="hljs-type">int</span> idx, t; };<br><br><span class="hljs-type">int</span> n;<br>queue&lt;Ship&gt; q;<br>vector&lt;<span class="hljs-type">int</span>&gt; G[N];<br>unordered_map&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt; cnt;<br><span class="hljs-type">int</span> kind;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n;<br><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-type">int</span> t, num;<br>        cin &gt;&gt; t &gt;&gt; num;<br>        <br>        q.<span class="hljs-built_in">push</span>({i, t});<br>        <br>        <span class="hljs-comment">// 哈希</span><br>        <span class="hljs-keyword">while</span> (num--) {<br>            <span class="hljs-type">int</span> id;<br>            cin &gt;&gt; id;<br>            <br>            <span class="hljs-keyword">if</span> (!cnt[id]) kind++;<br>            cnt[id]++;<br>            G[i].<span class="hljs-built_in">push_back</span>(id);<br>        }<br>        <br>        <span class="hljs-comment">// 去哈希</span><br>        Ship h = q.<span class="hljs-built_in">front</span>();<br>        <span class="hljs-keyword">while</span> (t - h.t &gt;= <span class="hljs-number">86400</span>) {<br>            <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; id: G[h.idx]) {<br>                cnt[id]--;<br>                <span class="hljs-keyword">if</span> (!cnt[id]) kind--;<br>            }<br>            q.<span class="hljs-built_in">pop</span>();<br>            h = q.<span class="hljs-built_in">front</span>();<br>        }<br>        <br>        cout &lt;&lt; kind &lt;&lt; <span class="hljs-string">"\n"</span>;<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【哈希】Cities-and-States-S">【哈希】Cities and States S</h3><p><a href="https://www.luogu.com.cn/problem/P3405">https://www.luogu.com.cn/problem/P3405</a></p><blockquote><p>题意：给定 n 个字符串，每一个字符串归属一个集合，现在需要统计字符串与集合名相反相等的对数</p><p>思路：很显然的哈希计数。难点有两个，如何哈希？如何计数？哈希可以采用扩展字符的方法进行，即第一个字符乘某一个较大的质数再加上第二个字符。此处采用一种较为巧妙的方法，直接将两个字符串与集合名加起来进行唯一性哈希，降低编码难度。计数有两种方式，第一种就是全部哈希结束之后，再遍历哈希表进行统计，最后将结果除二即可。第二种就是边哈希边计数，遇到相反相等的就直接计数，这样就不会重复计数了，也很巧妙</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;algorithm&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;unordered_map&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;stack&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;queue&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;set&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">int</span> n;<br>unordered_map&lt;string, <span class="hljs-type">int</span>&gt; a;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n;<br>    <br>    <span class="hljs-type">int</span> res = <span class="hljs-number">0</span>;<br>    <br>    <span class="hljs-keyword">while</span> (n--) {<br>        string s, t;<br>        cin &gt;&gt; s &gt;&gt; t;<br>        s = s.<span class="hljs-built_in">substr</span>(<span class="hljs-number">0</span>, <span class="hljs-number">2</span>);<br>        res += a[t + <span class="hljs-string">" "</span> + s] * (s != t);<br>        a[s + <span class="hljs-string">" "</span> + t]++;<br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【哈希-枚举-思维】Torn-Lucky-Ticket">【哈希/枚举/思维】Torn Lucky Ticket</h3><p><a href="https://codeforces.com/contest/1895/problem/C">https://codeforces.com/contest/1895/problem/C</a></p><blockquote><p>题意: 给定一个长度为 n 的字符串数组 nums, 数组的每一个元素长度不超过 5 且仅由数字组成. 问能找到多少对 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo separator="true">,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i,j)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">)</span></span></span></span> 可以使得拼接后的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi><mi>s</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo>+</mo><mi>n</mi><mi>u</mi><mi>m</mi><mi>s</mi><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">nums[i]+nums[j]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">]</span></span></span></span> 长度为偶数且左半部分的数字之和与右半部分的数字之和相等.</p><p>思路:</p><ul><li><p>首先最暴力的做法就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 枚举,所有的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo separator="true">,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i,j)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">)</span></span></span></span> 然后 check 合法性.</p></li><li><p>尝试优化掉第二层的枚举循环. 对于第二层循环, 我们就是要寻找合适的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi><mi>s</mi><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">nums[j]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">]</span></span></span></span> 并和当前的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi><mi>s</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">nums[i]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span></span></span></span> 拼接. 显然我们可以通过扫描当前的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi><mi>s</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">nums[i]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span></span></span></span> 并 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 的计算出所有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>e</mi><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mi>u</mi><mi>m</mi><mi>s</mi><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>≤</mo><mi>l</mi><mi>e</mi><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mi>u</mi><mi>m</mi><mi>s</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">len(nums[j])\le len(nums[i])</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">e</span><span class="mord mathnormal">n</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">])</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">e</span><span class="mord mathnormal">n</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">])</span></span></span></span> 且可以和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi><mi>s</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">nums[i]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span></span></span></span> 匹配的字符串的 <code>[长度][数字和]</code> 信息, 只需要一个二维数组预存储每一个字符串的 <strong>长度</strong> <strong>数字和</strong> 信息即可.</p></li><li><p>那么对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>e</mi><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mi>u</mi><mi>m</mi><mi>s</mi><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>&gt;</mo><mi>l</mi><mi>e</mi><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mi>u</mi><mi>m</mi><mi>s</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">len(nums[j])&gt; len(nums[i])</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">e</span><span class="mord mathnormal">n</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">])</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">e</span><span class="mord mathnormal">n</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">])</span></span></span></span> 的情况如何统计呢. 显然此时我们没法 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 的检查 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi><mi>s</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo>+</mo><mi>n</mi><mi>u</mi><mi>m</mi><mi>s</mi><mo stretchy="false">[</mo><mi>j</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">nums[i]+nums[j]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mclose">]</span></span></span></span> 的合法性. 不妨换一个角度, 当我们枚举 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi><mi>s</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">nums[i]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span></span></span></span> 时:</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>统计右侧拼接长度更大的&nbsp;nums[j]&nbsp;的合法情况数</mtext><mtext>  </mtext><mo>⟺</mo><mtext>  </mtext><mtext>统计左侧拼接长度更小的&nbsp;nums[j]&nbsp;的合法情况数</mtext></mrow><annotation encoding="application/x-tex">\text{统计右侧拼接长度更大的 nums[j] 的合法情况数} \iff \text{统计左侧拼接长度更小的 nums[j] 的合法情况数}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord cjk_fallback">统计右侧拼接长度更大的</span><span class="mord">&nbsp;nums[j]&nbsp;</span><span class="mord cjk_fallback">的合法情况数</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">⟺</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord cjk_fallback">统计左侧拼接长度更小的</span><span class="mord">&nbsp;nums[j]&nbsp;</span><span class="mord cjk_fallback">的合法情况数</span></span></span></span></span></span></p><p>于是合法情况数就可以表示为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><mo fence="false" stretchy="true" minsize="1.2em" maxsize="1.2em">[</mo><msub><mtext>cond</mtext><mn>1</mn></msub><mo stretchy="false">(</mo><mi>n</mi><mi>u</mi><mi>m</mi><mi>s</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>+</mo><msub><mtext>cond</mtext><mn>2</mn></msub><mo stretchy="false">(</mo><mi>n</mi><mi>u</mi><mi>m</mi><mi>s</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo fence="false" stretchy="true" minsize="1.2em" maxsize="1.2em">]</mo></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \sum_{i=0}^{n-1}\big[\text{cond}_1(nums[i])+\text{cond}_2(nums[i])\big]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0788em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8011em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="delimsizing size1">[</span></span><span class="mord"><span class="mord text"><span class="mord">cond</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">])</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.2em;vertical-align:-0.35em;"></span><span class="mord"><span class="mord text"><span class="mord">cond</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mord mathnormal">s</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">])</span><span class="mord"><span class="delimsizing size1">]</span></span></span></span></span>, 其中第一种情况 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>cond</mtext><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\text{cond}_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">cond</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 就是统计右侧拼接长度更小的字符串数量, 第二种情况 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mtext>cond</mtext><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\text{cond}_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord text"><span class="mord">cond</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 就是统计左侧拼接长度更小的字符串数量. 这两步可以同时计算.</p></li></ul><p>时间复杂度: <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">from</span> typing <span class="hljs-keyword">import</span> <span class="hljs-type">List</span>, <span class="hljs-type">Tuple</span>, <span class="hljs-type">Dict</span>, <span class="hljs-type">Optional</span><br><span class="hljs-keyword">from</span> collections <span class="hljs-keyword">import</span> defaultdict, deque<br><span class="hljs-keyword">from</span> itertools <span class="hljs-keyword">import</span> combinations, permutations<br><span class="hljs-keyword">import</span> math, heapq, queue<br><br>II = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">int</span>(<span class="hljs-built_in">input</span>())<br>FI = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">float</span>(<span class="hljs-built_in">input</span>())<br>MII = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">tuple</span>(<span class="hljs-built_in">map</span>(<span class="hljs-built_in">int</span>, <span class="hljs-built_in">input</span>().split()))<br>LII = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">list</span>(<span class="hljs-built_in">map</span>(<span class="hljs-built_in">int</span>, <span class="hljs-built_in">input</span>().split()))<br>LSI = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">list</span>(<span class="hljs-built_in">map</span>(<span class="hljs-built_in">str</span>, <span class="hljs-built_in">input</span>().split()))<br><br><span class="hljs-keyword">def</span> <span class="hljs-title function_">solve</span>() -&gt; <span class="hljs-type">Optional</span>:<br>    n, nums = II(), LSI()<br>    <br>    f = [[<span class="hljs-number">0</span> <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">46</span>)] <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">6</span>)]<br>    <span class="hljs-keyword">for</span> num <span class="hljs-keyword">in</span> nums:<br>        m = <span class="hljs-built_in">len</span>(num)<br>        s = <span class="hljs-built_in">sum</span>([<span class="hljs-built_in">int</span>(c) <span class="hljs-keyword">for</span> c <span class="hljs-keyword">in</span> num])<br>        f[m][s] += <span class="hljs-number">1</span><br>    <br>    res = <span class="hljs-number">0</span><br>    <span class="hljs-keyword">for</span> num <span class="hljs-keyword">in</span> nums:<br>        m = <span class="hljs-built_in">len</span>(num)<br>        s = [<span class="hljs-number">0</span>] * (m + <span class="hljs-number">1</span>)<br>        <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(m - <span class="hljs-number">1</span>, -<span class="hljs-number">1</span>, -<span class="hljs-number">1</span>):<br>            s[i] = s[i + <span class="hljs-number">1</span>] + <span class="hljs-built_in">int</span>(num[i])<br>        <br>        <span class="hljs-comment"># cond1: now + right -&gt; len(now) &gt;= len(right)</span><br>        <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(m - <span class="hljs-number">1</span>, -<span class="hljs-number">1</span>, -<span class="hljs-number">1</span>):<br>            now_len, now_sum = i + <span class="hljs-number">1</span>, s[<span class="hljs-number">0</span>] - s[i + <span class="hljs-number">1</span>]<br>            r_len, r_sum = now_len - (m - <span class="hljs-number">1</span> - i), now_sum - s[i + <span class="hljs-number">1</span>]<br>            <span class="hljs-keyword">if</span> <span class="hljs-number">1</span> &lt;= r_len &lt;= now_len <span class="hljs-keyword">and</span> r_sum &gt;= <span class="hljs-number">0</span>:<br>                res += f[r_len][r_sum]<br>        <br>        <span class="hljs-comment"># cond2: left + now -&gt; len(left) &lt; len(now)</span><br>        <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(m):<br>            now_len, now_sum = m - i, s[i]<br>            l_len, l_sum = now_len - i, now_sum - (s[<span class="hljs-number">0</span>] - s[i])<br>            <span class="hljs-keyword">if</span> <span class="hljs-number">1</span> &lt;= l_len &lt; now_len <span class="hljs-keyword">and</span> l_sum &gt;= <span class="hljs-number">0</span>:<br>                res += f[l_len][l_sum]<br>    <br>    <span class="hljs-keyword">return</span> res<br><br><span class="hljs-keyword">if</span> __name__ == <span class="hljs-string">'__main__'</span>:<br>    OUTs = []<br>    N = <span class="hljs-number">1</span><br>    <span class="hljs-comment"># N = II()</span><br>    <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(N):<br>        OUTs.append(solve())<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">'\n'</span>.join(<span class="hljs-built_in">map</span>(<span class="hljs-built_in">str</span>, OUTs)))<br></code></pre></td></tr></tbody></table></figure>]]></content>
     
     
     <categories>
@@ -1517,11 +1517,11 @@
   
   
   <entry>
-    <title>greedy</title>
-    <link href="/Algorithm/greedy/"/>
-    <url>/Algorithm/greedy/</url>
+    <title>number-theory</title>
+    <link href="/Algorithm/number-theory/"/>
+    <url>/Algorithm/number-theory/</url>
     
-    <content type="html"><![CDATA[<h3 id="贪心">贪心</h3><p>大胆猜测，小心求证（不会证也没事，做下一题吧）。证明方法总结了以下几种</p><ul><li>反证法：假设取一种方案比贪心方案更好，得出相反的结论</li><li>边界法：从边界开始考虑，因为满足边界条件更加容易枚举，从而进行后续的贪心</li><li>直觉法：遵循社会法则（）</li></ul><h3 id="1-green-gold-dog-array-and-permutation">1. green_gold_dog, array and permutation</h3><p><a href="https://codeforces.com/contest/1867/problem/A">https://codeforces.com/contest/1867/problem/A</a></p><blockquote><p>题意：给定n个数a[i]，其中可能有重复数，请构造一个n个数的排列b[i]，使得c[i]=a[i]-b[i]中，c[i]的不同数字的个数最多</p><p>思路：思路比较好想，就是最大的数匹配最小的数，次大的数匹配次小的数，以此类推。起初我想通过将原数列的拷贝数列降序排序后，创建一个哈希表来记录每一个数的排位，最终通过原数列的数作为键，通过哈希表直接输出排位，但是问题是原数列中的数可能会有重复的，那么在哈希的时候如果有重复的数，那么后来再次出现的数就会顶替掉原来的数的排位，故错误。</p><p>正确思路：为了<strong>保证原数列中每个数的唯一性</strong>，我们可以给原数列每一个数赋一个下标，排序后以下标进行唯一的一一对应的索引。那么就是：首先赋下标，接着以第一关键词进行排序，然后最大的数（其实是下标）匹配最小的结果以此类推</p><p>模拟一个样例：<br>5<br>8 7 4 5 5 9</p><p>最终的答案应该是<br>2 3 6 4 5 1</p><p>首先对每一个数赋予一个下标作为为唯一性索引<br>8 7 4 5 5 9<br>1 2 3 4 5 6（替身）</p><p>接着将上述数列按照第一关键词进行排序<br>9 8 7 5 5 4<br>6 1 2 4 5 3（替身）</p><p>对每一个数进行赋值匹配<br>9 8 7 5 5 4<br>6 1 2 4 5 3（替身）<br>1 2 3 4 5 6（想要输出的结果）</p><p>按照替身进行排序<br>8 7 4 5 5 9<br>1 2 3 4 5 6（替身）（排序后）<br>2 3 6 4 5 1（想要输出的结果）</p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span></span><br><span class="hljs-function"></span>{<br>    <span class="hljs-type">int</span> n; cin &gt;&gt; n;<br>    <br>    <span class="hljs-comment">// 第一关键词是原数列，第二关键词是赋予的唯一下标</span><br>    vector&lt;pair&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt;&gt; <span class="hljs-built_in">a</span>(n + <span class="hljs-number">1</span>);<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i ++)<br>    {<br>        cin &gt;&gt; a[i].first;<br>        a[i].second = i;<br>    }<br>    <br>    <span class="hljs-comment">// 按照第一关键词排序</span><br>    <span class="hljs-built_in">sort</span>(a.<span class="hljs-built_in">begin</span>() + <span class="hljs-number">1</span>, a.<span class="hljs-built_in">end</span>(), [&amp;](pair&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt; x, pair&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt; y) {<br>        <span class="hljs-keyword">return</span> x.first &gt; y.first;<br>    });<br>    <br>    <span class="hljs-comment">// 以下标作为原数列的替身，每一个替身对应一个升序的最终排名</span><br>    vector&lt;pair&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt;&gt; <span class="hljs-built_in">res</span>(n + <span class="hljs-number">1</span>);<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i ++)<br>    {<br>        res[i].first = a[i].second;<br>        res[i].second = i;<br>    }<br>    <br>    <span class="hljs-comment">// 通过下标，还原原数列的排序</span><br>    <span class="hljs-built_in">sort</span>(res.<span class="hljs-built_in">begin</span>() + <span class="hljs-number">1</span>, res.<span class="hljs-built_in">end</span>());<br>    <br>    <span class="hljs-comment">// 输出第二关键词</span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i ++)<br>        cout &lt;&lt; res[i].second &lt;&lt; <span class="hljs-string">" \n"</span>[i == n];<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span></span><br><span class="hljs-function"></span>{<br>    <span class="hljs-type">int</span> T; cin &gt;&gt; T;<br>    <span class="hljs-keyword">while</span> (T --) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="2-Good-Kid">2. Good Kid</h3><p><a href="https://codeforces.com/contest/1873/problem/B">https://codeforces.com/contest/1873/problem/B</a></p><blockquote><p>题意：对于一个数列，现在可以选择其中的一个数使其+1，问如何选择这个数，可以使得修改后的数列中的所有数之积的值最大</p><p>思路：其实就是选择n-1个数的乘积值加一倍，关键在于选哪一个n-1个数的乘积值，逆向思维就是对于n个数，去掉那个最小值，那么剩下来的n-1个数之积就会最大了，于是就是选择最小的数+1，最终数列之积就是答案了。</p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-keyword">typedef</span> <span class="hljs-type">long</span> <span class="hljs-type">long</span> ll;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span></span><br><span class="hljs-function"></span>{<br>    <span class="hljs-type">int</span> n; cin &gt;&gt; n;<br>    <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">a</span><span class="hljs-params">(n)</span></span>;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i ++)<br>        cin &gt;&gt; a[i];<br>    <br>    <span class="hljs-built_in">sort</span>(a.<span class="hljs-built_in">begin</span>(), a.<span class="hljs-built_in">end</span>());<br>    <br>    ll res = a[<span class="hljs-number">0</span>] + <span class="hljs-number">1</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt; n; i ++)<br>        res *= a[i];<br>    cout &lt;&lt; res &lt;&lt; endl; <br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span></span><br><span class="hljs-function"></span>{<br>    <span class="hljs-type">int</span> T; cin &gt;&gt; T;<br>    <span class="hljs-keyword">while</span> (T --) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="3-1D-Eraser">3. 1D Eraser</h3><p><a href="https://codeforces.com/contest/1873/problem/D">https://codeforces.com/contest/1873/problem/D</a></p><blockquote><p>题意：给定一个只有B和W两种字符的字符串，现在有一种操作可以将一个指定大小的区间（大小为k）中所有的字符全部变为W，问对于这个字符串，至少需要几次上述操作才可以将字符串全部变为W字符</p><p>思路：我们直接遍历一边字符串即可，在遍历到B字符的时候，指针向后移动k位，如果是W字符，指针向后移动1位即可，最终统计一下遇到B字符的次数即可。</p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-keyword">typedef</span> <span class="hljs-type">long</span> <span class="hljs-type">long</span> ll;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span></span><br><span class="hljs-function"></span>{<br>    <span class="hljs-type">int</span> n, k;<br>    cin &gt;&gt; n &gt;&gt; k;<br>    string s;<br>    cin &gt;&gt; s;<br>    <br>    <span class="hljs-type">int</span> res = <span class="hljs-number">0</span>;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n;)<br>    {<br>        <span class="hljs-keyword">if</span> (s[i] == <span class="hljs-string">'B'</span>)<br>        {<br>            res ++;<br>            i += k;<br>        }<br>        <span class="hljs-keyword">else</span> i++;<br>    }<br>    cout &lt;&lt; res &lt;&lt; endl;<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span></span><br><span class="hljs-function"></span>{<br>    <span class="hljs-type">int</span> T; cin &gt;&gt; T;<br>    <span class="hljs-keyword">while</span> (T --) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="4-ABBC-or-BACB">4. ABBC or BACB</h3><p><a href="https://codeforces.com/contest/1873/problem/G">https://codeforces.com/contest/1873/problem/G</a></p><blockquote><p>题意：现在有一个只有A和B两种字符的字符串，有如下两种操作：第一种，可以将AB转化为BC，第二种，可以将BA转化为CB。问最多可以执行上述操作几次</p><p>思路：其实一开始想到了之前做过的翻转数字为-1，最后的逻辑是将数字向左右移动，于是这道题就有了突破口了。上述两种操作就可以看做将A和B两种字符交换位置，并且B保持不变，A变为另一种字符C。由于C是无法执行上述操作得到，因此就可以理解为B走过的路就不可以再走了，那么就是说对于一个字符串，最终都会变成B走过的路C。并且只要有B，那么一个方向上所有的A都会被利用转化为C（直到遇到下一个B停止），那么我们就可以将B看做一个独立的个体，他可以选择一个方向上的所有的A并且执行操作将该方向的A全部转化为C，那么在左和右的抉择中，就是要选择A数量最多的方向进行操作，但是如果B的足够多，那就不需要考虑哪个方向了，所有的A都可以获得一次操作机会。现在就转向考虑B需要多少呢，</p><p>答案是两种：如果B的数量小于A“块”的数量，其实就是有一块A无法被转化，那么为了让被转化的A数量尽可能的多，就选择A块数量最少的那一块不被转化。如果B的数量&gt;=A块的数量，那么可以保证所有的A都可以被转化为C块，从而最终的答案就是A字符的数量</p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-keyword">typedef</span> <span class="hljs-type">long</span> <span class="hljs-type">long</span> ll;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span></span><br><span class="hljs-function"></span>{<br>    string s; cin &gt;&gt; s;<br>    <span class="hljs-type">int</span> cntb = <span class="hljs-number">0</span>, blocka = <span class="hljs-number">0</span>, cnta = <span class="hljs-number">0</span>; <span class="hljs-comment">// B字符的数量，A连续区间的数量，A字符的个数 </span><br>    <br>    vector&lt;<span class="hljs-type">int</span>&gt; a; <span class="hljs-comment">// 每一个A连续区间中A字符的数量 </span><br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; s.<span class="hljs-built_in">size</span>();)<br>    {<br>        <span class="hljs-keyword">if</span> (s[i] == <span class="hljs-string">'A'</span>)<br>        {<br>            <span class="hljs-type">int</span> sum = <span class="hljs-number">0</span>;<br>            <span class="hljs-keyword">while</span> (s[i] == <span class="hljs-string">'A'</span>) cnta ++, sum ++, i ++;<br>            a.<span class="hljs-built_in">emplace_back</span>(sum);<br>            blocka ++;<br>        }<br>        <span class="hljs-keyword">else</span><br>        {<br>            cntb ++;<br>            i ++;<br>        }<br>    }<br>    <br>    <span class="hljs-keyword">if</span> (cntb &gt;= blocka) cout &lt;&lt; cnta &lt;&lt; endl;<br>    <span class="hljs-keyword">else</span><br>    {<br>        <span class="hljs-built_in">sort</span>(a.<span class="hljs-built_in">begin</span>(), a.<span class="hljs-built_in">end</span>());<br>        cout &lt;&lt; cnta - a[<span class="hljs-number">0</span>] &lt;&lt; endl;<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span></span><br><span class="hljs-function"></span>{<br>    <span class="hljs-type">int</span> T; cin &gt;&gt; T;<br>    <span class="hljs-keyword">while</span> (T --) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="5-Smilo-and-Monsters">5. Smilo and Monsters</h3><p><a href="https://codeforces.com/contest/1891/problem/C">https://codeforces.com/contest/1891/problem/C</a></p><blockquote><p>题意：给定一个序列，其中每一个元素代表一个怪物群中的怪物的数量。现在有两种操作：</p><ol><li>选择一个怪物群，杀死其中的一只怪物。同时累加值 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>+</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x+=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">x</span><span class="mord">+</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></li><li>选择一个怪物群，通过之前积累的累加值，一次性杀死当前怪物群中的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 只怪物，同时累加值 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 归零</li></ol><p>现在询问杀死所有怪物的最小操作次数</p><p>思路：一开始我是分奇偶进行判断的，但是那只是局部最优解，全局最优解的计算需要我们“纵观全局”。</p><p>我们可以先做一个假设：假设当前只有一个怪物群，为了最小化操作次数，最优解肯定是先杀死一半的怪物（假设数量为n，则获得累计值x=n），然后无论剩余n个还是n+1个，我们都使用累计值x一次性杀死n个怪物。</p><p>现在我们回到原题，有很多个怪物群，易证，最终的最优解也一点是最大化利用操作2，和特殊情况类似，能够利用的一次性杀死的次数就是怪物总数的一半。区别在于：此时不止一个怪物群。那么我们就要考虑累加值如何使用。很容易想到先将怪物群按照数量大小进行排序，关键就在于将累加值从小到大进行使用还是从大到小进行使用。可以发现，对于一个确定的累加值，由于操作2的次数也会算在答案中。那么如果从最小的怪物群开始使用，就会导致操作2的次数变多。因此我们需要从最大值开始进行操作2。</p><p>那么最终的答案就是：</p><ul><li>首先根据怪物数量的总和计算出最终的累加值s（s=sum/2）</li><li>接着我们将怪物群按照数量进行升序排序</li><li>然后我们从怪物数量最大的开始进行操作2，即一次性杀死当前怪物群，没进行一次，答案数量+1</li><li>最后将答案累加上无法一次性杀死的怪物群中怪物的总数</li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    n = <span class="hljs-built_in">read</span>(); s = <span class="hljs-number">0</span>; res = <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        a[i] = <span class="hljs-built_in">read</span>();<br>        s += a[i];<br>    }<br> <br>    s &gt;&gt;= <span class="hljs-number">1</span>;<br> <br>    <span class="hljs-built_in">sort</span>(a + <span class="hljs-number">1</span>, a + n + <span class="hljs-number">1</span>);<br> <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = n; i &gt;= <span class="hljs-number">1</span>; i--) {<br>        <span class="hljs-keyword">if</span> (s &gt;= a[i]) {<br>            s -= a[i], a[i] = <span class="hljs-number">0</span>, res++;<br>        } <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (s) {<br>            a[i] -= s;<br>            s = <span class="hljs-number">0</span>;<br>            res++;<br>        }<br>    }<br> <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        res += a[i];<br>    }<br> <br>    <span class="hljs-built_in">printf</span>(<span class="hljs-string">"%lld\n"</span>, res);<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="6-通关">6. 通关</h3><p><a href="https://www.lanqiao.cn/problems/5889/learning/?contest_id=145">https://www.lanqiao.cn/problems/5889/learning/?contest_id=145</a></p><blockquote><p>声明：谨以此题记录我的第一道正式图论题（存图技巧抛弃了y总的纯数组，转而使用动态数组<code>vector</code>进行建图）e.g.</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-keyword">struct</span> <span class="hljs-title class_">Node</span> {<br><span class="hljs-type">int</span> id;        <span class="hljs-comment">// 当前结点的编号</span><br><span class="hljs-type">int</span> a;    <span class="hljs-comment">// 属性1</span><br><span class="hljs-type">int</span> b;    <span class="hljs-comment">// 属性2</span><br>};<br>vector&lt;Node&gt; G[N];<br></code></pre></td></tr></tbody></table></figure><p>题意：给定一棵树，每个结点代表一个关卡，每个关卡拥有两个属性：通关奖励值<code>val</code>和可通关的最低价值<code>cost</code>。现在从根节点开始尝试通关，每一个结点下面的关卡必须要当前结点通过了之后才能继续闯关。问应该如何选择通关规划使得通过的关卡数最多？</p><p>思路：一开始想到的是直接莽bfs，对每一层结点按照cost升序排序进行闯关，然后wa麻了。最后一看正解原来是还不够贪。正确的闯关思路应该是始终选择当前可以闯过的cost最小的关卡，而非限定在了一层上。因此我们最终的闯关顺序是：对于所有解锁的关卡，选择cost最小的关卡进行通关，如果连cost最小的关卡都无法通过就直接结束闯关。那么我们应该如何进行代码的编写呢？分析可得，解锁的关卡都是当前可通过的关卡的所有子结点，而快速获得当前cost最小值的操作可以通过一个堆来维护。</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">100010</span>;<br><span class="hljs-keyword">typedef</span> <span class="hljs-type">long</span> <span class="hljs-type">long</span> ll;<br><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">Node</span> {<br>    <span class="hljs-type">int</span> id;<br>    <span class="hljs-type">int</span> val;<br>    <span class="hljs-type">int</span> cost;<br>    <span class="hljs-comment">// 有趣的重载</span><br>    <span class="hljs-type">bool</span> <span class="hljs-keyword">operator</span>&lt; (<span class="hljs-type">const</span> Node&amp; t) <span class="hljs-type">const</span> {<br>        <span class="hljs-keyword">return</span> <span class="hljs-keyword">this</span>-&gt;cost &gt; t.cost;<br>    }<br>};<br><br><span class="hljs-type">int</span> n, p;<br>ll res;<br>vector&lt;Node&gt; G[N];<br>priority_queue&lt;Node&gt; q;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">bfs</span><span class="hljs-params">()</span> </span>{<br>    q.<span class="hljs-built_in">push</span>(G[<span class="hljs-number">0</span>][<span class="hljs-number">0</span>]);<br><br>    <span class="hljs-keyword">while</span> (q.<span class="hljs-built_in">size</span>()) {<br>        Node now = q.<span class="hljs-built_in">top</span>();<br>        q.<span class="hljs-built_in">pop</span>();<br>        <span class="hljs-keyword">if</span> (p &gt;= now.cost) {<br>            res++;<br>            p += now.val;<br>            <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; child: G[now.id]) {<br>                q.<span class="hljs-built_in">push</span>(child);<br>            }<br>        } <span class="hljs-keyword">else</span> {<br>            <span class="hljs-keyword">break</span>;<br>        }<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; p;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-type">int</span> fa, v, c;<br>        cin &gt;&gt; fa &gt;&gt; v &gt;&gt; c;<br>        G[fa].<span class="hljs-built_in">push_back</span>({i, v, c});<br>    }<br>    <span class="hljs-built_in">bfs</span>();<br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="7-串门">7. 串门</h3><p><a href="https://www.lanqiao.cn/problems/5890/learning/?contest_id=145">https://www.lanqiao.cn/problems/5890/learning/?contest_id=145</a></p><blockquote><p>题意：给定一棵无向树，边权为正。现在询问在访问到每一个结点的情况下最小化行走的路径长度</p><p>思路：首先我们不管路径长度的长短，我们直接开始串门。可以发现，我们一点会有访问的起点和终点，那么在起点到终点的这条路上，可能会有很多个分叉，对于每一个分叉，我们都需要进入再返回来确保分叉中的结点也被访问到，那么最终的路径长度就是总路径长度的2倍 - 起点到终点的距离，知道了这个性质后。我们可以发现，路径的总长度是一个定值，我们只有最大化起点到终点距离才能确保行走路径长度的最小值。那么问题就转化为了求解一棵树中，最长路径长度的问题。即求解树的直径的问题。</p><p>树的直径：首先我们反向考虑，假设知道了直径的一个端点，那么树的直径长度就是这棵树以当前结点为根的深度。那么我们就需要先确定一个直径的端点。不难发现，对于树中的任意一个结点，距离它最远的一个（可能是两个）结点一定是树的直径的端点。那么问题就迎刃而解了。</p><p>为了降低代码量，我们可以设置一个dist数组来记录当前结点到根的距离。</p><ul><li>在确定直径端点的时候，选择任意一个结点为根节点，然后维护dist数组，最终dist[i~n]中的最大值对应的下标maxi就是直径的一个端点。</li><li>接着在计算直径长度的时候，以maxi为树根，再来一次上述的维护dist数组的过程，最终dist[1~n]中的最大值就是树的直径的长度</li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">100010</span>;<br><span class="hljs-keyword">typedef</span> <span class="hljs-type">long</span> <span class="hljs-type">long</span> ll;<br><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">Node</span> {<br>    <span class="hljs-type">int</span> id;<br>    ll w;<br>};<br><br><span class="hljs-type">int</span> n;<br>vector&lt;Node&gt; G[N];<br><span class="hljs-type">bool</span> st[N];<br>ll sum, d, dist[N];<br><br><span class="hljs-comment">/**</span><br><span class="hljs-comment"> * @note 计算当前所有子结点到当前点的距离</span><br><span class="hljs-comment"> * @param now 当前点的编号</span><br><span class="hljs-comment"> * @param x 上一个点到当前点的距离</span><br><span class="hljs-comment"> */</span><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">dfs</span><span class="hljs-params">(<span class="hljs-type">int</span> now, ll x)</span> </span>{<br>    <span class="hljs-keyword">if</span> (!st[now]) {<br>        st[now] = <span class="hljs-literal">true</span>;<br>        dist[now] = x;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; G[now].<span class="hljs-built_in">size</span>(); i++) {<br>            <span class="hljs-type">int</span> ch = G[now][i].id;<br>            <span class="hljs-built_in">dfs</span>(ch, x + G[now][i].w);<br>        }<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n - <span class="hljs-number">1</span>; i++) {<br>        <span class="hljs-type">int</span> a, b, w;<br>        cin &gt;&gt; a &gt;&gt; b &gt;&gt; w;<br>        sum += w;<br>        G[a].<span class="hljs-built_in">push_back</span>({b, w});<br>        G[b].<span class="hljs-built_in">push_back</span>({a, w});<br>    }<br><br>    <span class="hljs-comment">// 以1号点为根，计算所有点到1号点的距离</span><br>    <span class="hljs-built_in">dfs</span>(<span class="hljs-number">1</span>, <span class="hljs-number">0</span>);<br><br>    <span class="hljs-built_in">memset</span>(st, <span class="hljs-number">0</span>, <span class="hljs-keyword">sizeof</span> st);<br><br>    <span class="hljs-comment">// 到1号点最远的那个点的编号 maxi，即直径的一个端点</span><br>    <span class="hljs-type">int</span> maxi = <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-keyword">if</span> (dist[i] &gt; dist[maxi]) {<br>            maxi = i;<br>        }<br>    }<br><br>    <span class="hljs-comment">// 以直径的一个端点 maxi 为根，计算所有点到 maxi 点的距离</span><br>    <span class="hljs-built_in">dfs</span>(maxi, <span class="hljs-number">0</span>);<br><br>    <span class="hljs-comment">// 找到直径长度</span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        d = <span class="hljs-built_in">max</span>(d, dist[i]);<br>    }<br><br>    cout &lt;&lt; sum * <span class="hljs-number">2</span> - d &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="8-Codeforces-rating">8. Codeforces rating</h3><p><a href="https://vijos.org/d/nnu_contest/p/1532">https://vijos.org/d/nnu_contest/p/1532</a></p><blockquote><p>题意：现在已知一个初始值R，现在给定了n个P值，选择其中的合适的值，使得最终的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>R</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mn>3</mn><mi mathvariant="normal">/</mi><mn>4</mn><mi>R</mi><mo>+</mo><mn>1</mn><mi mathvariant="normal">/</mi><mn>4</mn><mi>P</mi></mrow><annotation encoding="application/x-tex">R'=3/4 R + 1/4 P</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">3/4</span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1/4</span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span></span></span></span> 最大</p><p>思路：首先一点就是我们一定要选择P比当前R大的值，接下来就是选择合适的P使得最终迭代出来的R’最大。首先我们知道，对于筛选出来的比当前R大的P集合，任意选择其中的一个P，都会让R增大，但是不管增加多少都是不会超过选择的P。那么显然，如果筛选出来的P集合是一个递增序列，那么就可以让R不断的增加。但是这一定是最大的吗？我们不妨反证一下，现在我们有两个P，分别为x，y，其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>&lt;</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x&lt;y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>。</p><p>那么按照上述思路，首先就是</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>R</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mi>R</mi><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>x</mi><mo stretchy="false">(</mo><msup><mi>R</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>&lt;</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R'=\frac{3}{4}R+\frac{1}{4}x(R'&lt;x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8019em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">x</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></span></p><p>接着就是</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mi>R</mi><mn>1</mn><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mo>=</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><msup><mi>R</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>y</mi><mo stretchy="false">(</mo><msubsup><mi>R</mi><mn>1</mn><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mo>&lt;</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>9</mn><mn>16</mn></mfrac><mi>R</mi><mo>+</mo><mfrac><mn>3</mn><mn>16</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>y</mi></mrow><annotation encoding="application/x-tex">R''_1=\frac{3}{4}R'+\frac{1}{4}y(R''_1&lt;y)=\frac{9}{16}R+\frac{3}{16}x+\frac{1}{4}y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0489em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-2.453em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-2.453em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">16</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">9</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">16</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span></span></p><p><strong>反之</strong>，首先选择一个较大的P=y，再选择一个较小的P=x，即首先就是</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msup><mi>R</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>=</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mi>R</mi><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>y</mi><mo stretchy="false">(</mo><msup><mi>R</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>&lt;</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R'=\frac{3}{4}R+\frac{1}{4}y(R'&lt;y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8019em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span></span></p><p>此时我们还不一点可以继续选择x，因为此时的R’可能已经超过了x的值，那么我们按照最优的情况计算，可以选择x，那么就是</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mi>R</mi><mn>2</mn><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mo>=</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><msup><mi>R</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>x</mi><mo stretchy="false">(</mo><msubsup><mi>R</mi><mn>2</mn><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mo>&lt;</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>9</mn><mn>16</mn></mfrac><mi>R</mi><mo>+</mo><mfrac><mn>3</mn><mn>16</mn></mfrac><mi>y</mi><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>x</mi></mrow><annotation encoding="application/x-tex">R''_2=\frac{3}{4}R'+\frac{1}{4}x(R''_2&lt;x)=\frac{9}{16}R+\frac{3}{16}y+\frac{1}{4}x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0489em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-2.453em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">x</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-2.453em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">16</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">9</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">16</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathnormal">x</span></span></span></span></span></p><p>可以发现</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msubsup><mi>R</mi><mn>2</mn><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mo>−</mo><msubsup><mi>R</mi><mn>1</mn><mrow><mo mathvariant="normal">′</mo><mo mathvariant="normal">′</mo></mrow></msubsup><mo>=</mo><mfrac><mn>1</mn><mn>16</mn></mfrac><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">)</mo><mo>&lt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">R''_2-R''_1=\frac{1}{16}(x-y)&lt;0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0489em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-2.453em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0489em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8019em;"><span style="top:-2.453em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">16</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></span></p><p>即会使得最终的答案变小。因此最优的策略就是按照增序进行叠加计算</p><p>==注意点：==</p><p>四舍五入的语句</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs cpp">cout &lt;&lt; (<span class="hljs-type">int</span>)(res + <span class="hljs-number">0.5</span>) &lt;&lt; <span class="hljs-string">"\n"</span>;<br></code></pre></td></tr></tbody></table></figure></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; k;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        cin &gt;&gt; a[i];<br>    }<br><br>    <span class="hljs-built_in">sort</span>(a, a + n);<br><br>    <span class="hljs-type">int</span> i = <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">for</span> (; i &lt; n; i++) {<br>        <span class="hljs-keyword">if</span> (a[i] &gt; k) {<br>            <span class="hljs-keyword">break</span>;<br>        }<br>    }<br><br>    <span class="hljs-type">double</span> res = k;<br><br>    <span class="hljs-keyword">for</span> (; i &lt; n; i++) {<br>        res = (res * <span class="hljs-number">3.0</span> + a[i] * <span class="hljs-number">1.0</span>) / <span class="hljs-number">4.0</span>;<br>    }<br><br>    cout &lt;&lt; (<span class="hljs-type">int</span>)(res + <span class="hljs-number">0.5</span>) &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="9-货运公司">9. 货运公司</h3><p><a href="https://www.acwing.com/problem/content/5380/">https://www.acwing.com/problem/content/5380/</a></p><blockquote><p>题意：给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个运货订单，每个订单需要运送一定量的货物，同时给予一定量的报酬。现在有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个卡车进行运送，每个卡车有运送容量。需要合理规划卡车的装配，使得最终获得的总报酬最大。有以下约束条件：</p><ul><li><strong>每一个订单</strong>只能用一辆货车运送</li><li><strong>每一辆货车</strong>只能运送不超过最大容量的货物</li><li><strong>每一辆货车</strong>只能运送一次</li><li><strong>每一辆货车</strong>在装得下的情况下只能装运一个订单</li></ul><p>思路：一开始尝试采用 01 背包的思路，但是题目中货物是否可以被运送是取决于卡车的装载的。思考无果后考虑贪心。<strong>考虑一般情况</strong></p><ul><li><p>为了让总收益最大，很显然报酬越多，就越优先选择该订单</p></li><li><p>在报酬一致时，货物量越少，就越优先选择该订单</p></li><li><p>按照上述规则对订单排序后，我们需要安排合理的货车进行运输。我们优先将运输量小的货车安排给合适的订单。因为运输量大的货车既可以满足当前订单，同样也满足后续的订单，那么为了最大化增加利润，就需要在运输量小的货车满足当前订单时，保留运输量大的货车用于后续的订单。我们可以画一个图进一步理解为什么这么安排货车进行运输：</p>  <img src="D:/BaiduSyncdisk/_images/typora-user-images/202402012332958.png" alt="image-20231223222627134" style="zoom: 50%;">  <div style="text-align: center">图例：我们按照上述红笔的需要进行货车的选择进行运输</div><p>如果红色序号顺序变掉了，则就不能将上述图例中的 6 个订单全部运输了</p></li></ul><p>时间复杂度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(nk)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">nk</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">1010</span>;<br><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">Item</span> {<br>    <span class="hljs-type">int</span> id, weight, money;<br>    <span class="hljs-type">bool</span> <span class="hljs-keyword">operator</span>&lt; (<span class="hljs-type">const</span> Item&amp; t) <span class="hljs-type">const</span> {<br>        <span class="hljs-keyword">if</span> (money != t.money) <span class="hljs-keyword">return</span> money &gt; t.money;<br>        <span class="hljs-keyword">else</span> <span class="hljs-keyword">return</span> weight &lt; t.weight;<br>    }<br>} item[N];<br><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">Car</span> {<br>    <span class="hljs-type">int</span> id, v;<br>    <span class="hljs-type">bool</span> <span class="hljs-keyword">operator</span>&lt; (<span class="hljs-type">const</span> Car&amp; t) <span class="hljs-type">const</span> {<br>        <span class="hljs-keyword">return</span> v &lt; t.v;<br>    }<br>} car[N];<br><br>pair&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt; res[N];<br><span class="hljs-type">bool</span> vis[N];<br><span class="hljs-type">int</span> cnt, sum;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-type">int</span> n;<br>    cin &gt;&gt; n;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        item[i].id = i;<br>        cin &gt;&gt; item[i].weight &gt;&gt; item[i].money;<br>    }<br><br>    <span class="hljs-type">int</span> k;<br>    cin &gt;&gt; k;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= k; i++) {<br>        car[i].id = i;<br>        cin &gt;&gt; car[i].v;<br>    }<br><br>    <span class="hljs-built_in">sort</span>(item + <span class="hljs-number">1</span>, item + n + <span class="hljs-number">1</span>);<br>    <span class="hljs-built_in">sort</span>(car + <span class="hljs-number">1</span>, car + k + <span class="hljs-number">1</span>);<br><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">1</span>; j &lt;= k; j++) {<br>            <span class="hljs-keyword">if</span> (!vis[j] &amp;&amp; car[j].v &gt;= item[i].weight) {<br>                sum += item[i].money;<br>                vis[j] = <span class="hljs-literal">true</span>;<br>                res[cnt++] = {item[i].id, car[j].id};<br>                <span class="hljs-keyword">break</span>;<br>            }<br>        }<br>    }<br><br>    cout &lt;&lt; cnt &lt;&lt; <span class="hljs-string">" "</span> &lt;&lt; sum &lt;&lt; <span class="hljs-string">"\n"</span>;<br><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; cnt; i++) {<br>        cout &lt;&lt; res[i].first &lt;&lt; <span class="hljs-string">" "</span> &lt;&lt; res[i].second &lt;&lt; <span class="hljs-string">"\n"</span>;<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="10-部分背包问题">10. 部分背包问题</h3><p><a href="https://www.luogu.com.cn/problem/P2240">https://www.luogu.com.cn/problem/P2240</a></p><blockquote><ul><li>题意：给定一组物品，每一个物品有一定的重量与价值，现在给定一个背包总承重，需要从中选择合适的物品进行装载，使得最终装载的物品的总价值最大。现在规定每一个物品可以进行随意分割，按照单位重量计算价值</li><li>思路：<strong>考虑一般情况</strong>。很显然，如果不说可以随意分割，那就是一个 01 背包，既然说了可以随意分割，那就直接贪心选择即可，按照单位重量的价值降序进行选择</li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">110</span>;<br><br><span class="hljs-type">int</span> n, W;<br><span class="hljs-type">double</span> res;<br><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">Item</span> {<br>    <span class="hljs-type">double</span> w;<br>    <span class="hljs-type">double</span> val;<br>    <br>    <span class="hljs-type">bool</span> <span class="hljs-keyword">operator</span>&lt; (<span class="hljs-type">const</span> Item&amp; t) <span class="hljs-type">const</span> {<br>        <span class="hljs-keyword">return</span> <span class="hljs-keyword">this</span>-&gt;val / <span class="hljs-keyword">this</span>-&gt;w &gt; t.val / t.w;<br>    }<br>} a[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; W;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) cin &gt;&gt; a[i].w &gt;&gt; a[i].val;    <br>    <br>    <span class="hljs-built_in">sort</span>(a, a + n);<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        <span class="hljs-keyword">if</span> (W &gt; a[i].w) {<br>            W -= a[i].w;<br>            res += a[i].val;<br>        } <span class="hljs-keyword">else</span> {<br>            res += W * a[i].val / a[i].w;<br>            <span class="hljs-keyword">break</span>;<br>        }<br>    }<br>    <br>    cout &lt;&lt; fixed &lt;&lt; <span class="hljs-built_in">setprecision</span>(<span class="hljs-number">2</span>) &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    cout &lt;&lt; fixed &lt;&lt; <span class="hljs-built_in">setprecision</span>(<span class="hljs-number">2</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="11-排队接水">11. 排队接水</h3><p><a href="https://www.luogu.com.cn/problem/P1223">https://www.luogu.com.cn/problem/P1223</a></p><blockquote><ul><li><p>题意：贪心经典，排队接水，每一个人都有一个打水时间，现在询问如何排队能够使得平均等待时间最短</p></li><li><p>思路：<strong>考虑一般情况</strong>。注意问的是平均等待时间，如果是打完水总花费时间那就是一个定值，即所有人打水的时间总和。现在为了让平均等待时间最短，我们的第一反应应该是，在人多的时候让打水时间短的先打，人少的时候让打水时间长的再打，那么很显然就是按照打水时间的升序进行排队。为了证明这个结论，我们采用数学归纳法，如下，对于一个排队序列来说，我们将其中的两个顺序进行颠倒，计算结果得到让打水时间短的排在前面更优，进而推广到一般情况，于是按照打水时间升序进行打水的策略是最优的。证毕</p>  <img src="D:/BaiduSyncdisk/_images/typora-user-images/202402012332959.png" alt="demo" style="zoom: 67%;"></li><li><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">1010</span>;<br><br><span class="hljs-type">int</span> n;<br><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">Item</span> {<br>    <span class="hljs-type">int</span> time;<br>    <span class="hljs-type">int</span> i;<br>    <br>    <span class="hljs-type">bool</span> <span class="hljs-keyword">operator</span>&lt; (<span class="hljs-type">const</span> Item&amp; t) <span class="hljs-type">const</span> {<br>        <span class="hljs-keyword">return</span> <span class="hljs-keyword">this</span>-&gt;time &lt; t.time;<br>    }<br>} a[N]; <br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        cin &gt;&gt; a[i].time;<br>        a[i].i = i;<br>    }<br>    <br>    <span class="hljs-built_in">sort</span>(a + <span class="hljs-number">1</span>, a + n + <span class="hljs-number">1</span>);<br>    <br>    <span class="hljs-type">double</span> sum = <span class="hljs-number">0</span>;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        sum += a[i].time * (n - i);<br>        cout &lt;&lt; a[i].i &lt;&lt; <span class="hljs-string">" \n"</span>[i == n];<br>    }<br>    <br>    cout &lt;&lt; fixed &lt;&lt; <span class="hljs-built_in">setprecision</span>(<span class="hljs-number">2</span>) &lt;&lt; sum / n &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="12-线段覆盖">12. 线段覆盖</h3><p><a href="https://www.luogu.com.cn/problem/P1803">https://www.luogu.com.cn/problem/P1803</a></p><blockquote><ul><li>题意：给定一个数轴和数轴上的 n 个线段，现在需要从中选出尽可能多的 k 个线段，使得线段之间两两不相交，问 k 最大是多少</li><li>思路：绞尽脑汁想着各种情况进行分类讨论，什么包含、相交、相离，但是都忽视了最重要的一点，那就是一开始应该选择线段。即<strong>考虑边界</strong>，对于最左侧我们应该选择哪一条线段呢？答案是最左侧的右端点最靠左的那一条线段，这样子后续的线段才能有更多的摆放空间</li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">1000010</span>;<br><br><span class="hljs-type">int</span> n;<br><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">Item</span> {<br>    <span class="hljs-type">int</span> l, r;<br>    <span class="hljs-type">bool</span> <span class="hljs-keyword">operator</span>&lt; (<span class="hljs-type">const</span> Item&amp; t) <span class="hljs-type">const</span> {<br>        <span class="hljs-keyword">return</span> <span class="hljs-keyword">this</span>-&gt;r &lt; t.r;<br>    }<br>} a[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) cin &gt;&gt; a[i].l &gt;&gt; a[i].r;<br>    <br>    <span class="hljs-built_in">sort</span>(a + <span class="hljs-number">1</span>, a + n + <span class="hljs-number">1</span>);<br>    <br>    <span class="hljs-type">int</span> right = <span class="hljs-number">-1</span>, res = <span class="hljs-number">0</span>;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-keyword">if</span> (a[i].l &gt;= right) {<br>            res++;<br>            right = a[i].r;<br>        }<br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="13-小A的糖果">13. 小A的糖果</h3><p><a href="https://www.luogu.com.cn/problem/P3817">https://www.luogu.com.cn/problem/P3817</a></p><blockquote><ul><li>题意：给定一个序列，现在每次可以选择序列中的某个数进行减一操作，问最少需要减多少可以使得序列中相邻两个元素之和不超过规定的数值 x</li><li>思路：同 T12 的思路展开，<strong>考虑边界</strong>，我们直接从任意的一个边界开始思考，假设从左边界开始考虑。为了让左边界满足，那么 a[1] 与 a[2] 之和就需要满足条件，为了后续可以尽可能少的进行减一操作，很显然我们最好从两个数的右边的数开始下手，于是贪心的结果就有了</li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">100010</span>;<br><br><span class="hljs-type">int</span> n, x;<br><span class="hljs-type">int</span> a[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; x;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) cin &gt;&gt; a[i];<br>    <br>    <span class="hljs-type">int</span> res = <span class="hljs-number">0</span>;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">2</span>; i &lt;= n; i++) {<br>        <span class="hljs-keyword">if</span> (a[i] + a[i - <span class="hljs-number">1</span>] &gt; x) {<br>            <span class="hljs-type">int</span> eat = a[i] + a[i - <span class="hljs-number">1</span>] - x;<br>            res += eat;<br>            a[i] = a[i] &gt;= eat ? a[i] - eat : <span class="hljs-number">0</span>;<br>        }<br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="15-陶陶摘苹果（升级版）">15. 陶陶摘苹果（升级版）</h3><p><a href="https://www.luogu.com.cn/problem/P1478">https://www.luogu.com.cn/problem/P1478</a></p><blockquote><ul><li>题意：现在有 n 个苹果，含有高度和消耗体力值两个属性，现在你有一个最大高度以及总体力值，每次只能摘高度值不超过最大高度的苹果且会消耗相应的体力值。问最多可以摘多少个苹果</li><li>思路：既然是想要摘的苹果尽可能多，那么高度都是浮云，体力值才是关键。为了摘得尽可能多的苹果，自然是消耗体力越少越优先摘。于是按照消耗体力值升序排序后，扫描一遍即可计数</li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">5010</span>;<br><br><span class="hljs-type">int</span> n, s;<br><span class="hljs-type">int</span> aa, bb;<br><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">Apple</span> {<br>    <span class="hljs-type">int</span> h, cost;<br>    <span class="hljs-type">bool</span> <span class="hljs-keyword">operator</span>&lt; (<span class="hljs-type">const</span> Apple&amp; t) <span class="hljs-type">const</span> {<br>        <span class="hljs-keyword">return</span> <span class="hljs-keyword">this</span>-&gt;cost &lt; t.cost;<br>    }<br>} a[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; s;<br>    cin &gt;&gt; aa &gt;&gt; bb;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) cin &gt;&gt; a[i].h &gt;&gt; a[i].cost;<br>    <br>    <span class="hljs-built_in">sort</span>(a + <span class="hljs-number">1</span>, a + n + <span class="hljs-number">1</span>);<br>    <br>    <span class="hljs-type">int</span> cnt = <span class="hljs-number">0</span>;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-keyword">if</span> (s &gt;= a[i].cost &amp;&amp; a[i].h &lt;= aa + bb) {<br>            cnt++;<br>            s -= a[i].cost;<br>        }<br>    }<br>    <br>    cout &lt;&lt; cnt &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="16-NOIP2018-提高组-铺设道路">16. [NOIP2018 提高组] 铺设道路</h3><p><a href="https://www.luogu.com.cn/problem/P5019">https://www.luogu.com.cn/problem/P5019</a></p><blockquote><ul><li><p>题意：给定一个序列，每次可以选择一个连续的区间使得其中的所有数 -1，前提是区间中的数都要 &gt; 1，问最少选择几次区间可以将所有的数全部都减到 0</p></li><li><p>思路：很显然的一个贪心思路就是，对于一个数，左右两侧均呈现递减的趋势，那么这个区间的选择数就是这个最大的数的值。但是这不符合代码逻辑，我们尝试<strong>从端点开始考虑</strong>。为了更好的理解，我们从递推的角度进行推演。假设填平前 i 个数花费的选择次数为 f[i]，那么对于第一个数 a[1]，我们一定需要花费 a[1] 次选择来将其减到零，于是 f[1] = a[1]；对于后续的数而言，可以发现</p><ul><li>如果 a[i] &gt; a[i-1]，那么之前的选择没有完全波及当前的数，还需要一定的次数来将其减到零，于是 <code>f[i] = f[i-1] + a[i] - a[i-1]</code></li><li>如果 a[i] <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≤</mo></mrow><annotation encoding="application/x-tex">\le</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≤</span></span></span></span> a[i-1]，那么当前的数值一定会被之前的选择“波及”而直接减到零，于是 <code>f[i] = f[i-1]</code></li></ul><p>从递推的角度理解之后我们直接滚动一遍数组记录増势即可</p></li><li><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">100010</span>;<br><br><span class="hljs-type">int</span> n, a[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) cin &gt;&gt; a[i];<br>    <br>    <span class="hljs-type">int</span> res = a[<span class="hljs-number">1</span>];<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">2</span>; i &lt;= n; i++)<br>        <span class="hljs-keyword">if</span> (a[i] &gt; a[i - <span class="hljs-number">1</span>])<br>            res += a[i] - a[i - <span class="hljs-number">1</span>];<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>} <br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="17-USACO1-3-混合牛奶-Mixing-Milk">17. [USACO1.3] 混合牛奶 Mixing Milk</h3><p><a href="https://www.luogu.com.cn/problem/P1208">https://www.luogu.com.cn/problem/P1208</a></p><blockquote><ul><li>题意：现在需要收集 need 容量的牛奶，同时有 n 个农户提供奶源，每一个商户的奶源有一定的数量与单价，问如何采购可以在满足需求 need 的情况下成本最低化</li><li>思路：很显然单价越低越优先采购</li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">2000010</span>;<br><br><span class="hljs-type">int</span> need, n;<br><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">Milk</span> {<br>    <span class="hljs-type">int</span> per, tot;<br>    <span class="hljs-type">bool</span> <span class="hljs-keyword">operator</span>&lt; (<span class="hljs-type">const</span> Milk&amp; t) <span class="hljs-type">const</span> {<br>        <span class="hljs-keyword">return</span> per &lt; t.per;<br>    }<br>} a[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; need &gt;&gt; n;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) cin &gt;&gt; a[i].per &gt;&gt; a[i].tot;<br>    <br>    <span class="hljs-built_in">sort</span>(a + <span class="hljs-number">1</span>, a + n + <span class="hljs-number">1</span>);<br>    <br>    <span class="hljs-type">int</span> res = <span class="hljs-number">0</span>;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-keyword">if</span> (need &gt;= a[i].tot) {<br>            need -= a[i].tot;<br>            res += a[i].tot * a[i].per;<br>        } <span class="hljs-keyword">else</span> {<br>            res += need * a[i].per;<br>            need = <span class="hljs-number">0</span>;<br>        }<br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="18-NOIP2007-普及组-纪念品分组">18. [NOIP2007 普及组] 纪念品分组</h3><p><a href="https://www.luogu.com.cn/problem/P1094">https://www.luogu.com.cn/problem/P1094</a></p><blockquote><ul><li>题意：给定 n 个数，问如何分组可以使得在每一组之和不超过定值 lim 且每一组最多两个数的情况下，分得的组数最少</li><li>思路：对序列没有要求，显然直接排序然后<strong>从一端开始考虑</strong>。首先有一个概念就是，为了尽可能的减少分得的组数，那么一定想要每一组分完后距离 lim 的差距越小越好。我们将 n 个数降序排序后，从最大值开始考虑，对于当前的数，为了找到一个数与当前数之和尽可能接近 lim，从贪心的角度来说一定是从当前数之后开始寻找那个配对的数从而使得两数之和与 lim 的差距最小，那么时间复杂度就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>。但是由于题目说了最多只有两个数，我们不妨换一种寻找配对数的策略，可以从原始的寻找方法得到启发，可以发现，原始的寻找配对数的方法中，如果找到的那个配对数可以与当前数之和不超过 lim 且之和尽可能的大，那么这个寻找到的配对数也一定可以和当前数的下一个数进行配对。由于我们一组只能有两个数，因此我们不用考虑两数组合之后比 lim 小的数尽可能的小，只需要考虑两个数能否组合即可，因此我们直接双指针进行寻找配对数操作即可</li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">30010</span>;<br><br><span class="hljs-type">int</span> lim, n;<br><span class="hljs-type">int</span> a[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; lim &gt;&gt; n;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) cin &gt;&gt; a[i];<br>    <br>    <span class="hljs-built_in">sort</span>(a + <span class="hljs-number">1</span>, a + n + <span class="hljs-number">1</span>, [&amp;](<span class="hljs-type">int</span> x, <span class="hljs-type">int</span> y) {<br>        <span class="hljs-keyword">return</span> x &gt; y;<br>    });<br>    <br>    <span class="hljs-type">int</span> res = <span class="hljs-number">0</span>, l = <span class="hljs-number">1</span>, r = n;<br>    <br>    <span class="hljs-keyword">while</span> (l &lt; r) {<br>        <span class="hljs-keyword">if</span> (a[l] + a[r] &lt;= lim) r--;<br>        l++, res++;<br>    }<br>    <br>    <span class="hljs-keyword">if</span> (l == r) res++;<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>} <br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="19-跳跳！">19. 跳跳！</h3><p><a href="https://www.luogu.com.cn/problem/P4995">https://www.luogu.com.cn/problem/P4995</a></p><blockquote><ul><li><p>题意：给定 n 个数，每一个数代表一个高度，现在有一只青蛙从地面（高度为 0）开始起跳，每一个数字只能被跳上去一次。问如何设定跳数顺序 h[]，可以使得下式尽可能的大</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mo stretchy="false">(</mo><msub><mi>h</mi><mi>i</mi></msub><mo>−</mo><msub><mi>h</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\sum_{i=1}^n (h_i-h_{i-1})^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.9291em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1141em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></p></li><li><p>思路：题目意思其实就是让每一次跳跃的两个高度之差尽可能的大。那么一个贪心的思路就是从地面跳到最高，再从最高跳到次低，接着从次低跳到次高，以此类推得到的结果就是最优解，为了证明我们可以用数学归纳法，假设当前只有两个高度可以选择，那么肯定是先跳高的再跳低的；假设当前有三个高度可以选择，那么可以证明先从地面跳到最高处，再跳到最低处，最后跳到第二高的方法得到的值最大</p></li><li><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">310</span>;<br><br><span class="hljs-type">int</span> n, a[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) cin &gt;&gt; a[i];<br><br>    <span class="hljs-built_in">sort</span>(a + <span class="hljs-number">1</span>, a + n + <span class="hljs-number">1</span>, [&amp;](<span class="hljs-type">int</span> x, <span class="hljs-type">int</span> y) {<br>        <span class="hljs-keyword">return</span> x &gt; y;<br>    });<br><br>    <span class="hljs-type">int</span> res = a[<span class="hljs-number">1</span>] * a[<span class="hljs-number">1</span>], l = <span class="hljs-number">1</span>, r = n;<br>    <br>    <span class="hljs-keyword">while</span> (l &lt; r) {<br>        res += (a[l] - a[r]) * (a[l] - a[r]);<br>        l++;<br>        <span class="hljs-keyword">if</span> (l &lt; r) {<br>            res += (a[l] - a[r]) * (a[l] - a[r]);<br>            r--;<br>        }<br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="20-安排时间">20. 安排时间</h3><p><a href="https://www.acwing.com/problem/content/5466/">https://www.acwing.com/problem/content/5466/</a></p><blockquote><ul><li>题意：给定 n 个时间段处理餐厅事务，m 个用餐预定信息，含有三个信息分别为订单发起时间，订单需要准备的时间，客人来用餐时间。问如何安排每一个时间段，可以让所有的用户得到正常的服务。如果不能全部满足则输出 -1</li><li>思路：玄学贪心，先到的用户先做它的菜</li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">110</span>;<br><br><span class="hljs-type">int</span> n, m;<br><span class="hljs-type">int</span> res[N];<br><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">Item</span> {<br>    <span class="hljs-type">int</span> id;<br>    <span class="hljs-type">int</span> begi, arri, time;<br>    <span class="hljs-type">bool</span> <span class="hljs-keyword">operator</span>&lt; (<span class="hljs-type">const</span> Item&amp; t) <span class="hljs-type">const</span> {<br>        <span class="hljs-keyword">return</span> arri &lt; t.arri;<br>    }<br>} a[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; m;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= m; i++) {<br>        cin &gt;&gt; a[i].begi &gt;&gt; a[i].arri &gt;&gt; a[i].time;<br>        a[i].id = i;<br>        res[a[i].arri] = m + <span class="hljs-number">1</span>; <span class="hljs-comment">// 接客 </span><br>    }<br>    <br>    <span class="hljs-built_in">sort</span>(a + <span class="hljs-number">1</span>, a + m + <span class="hljs-number">1</span>);<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= m; i++) {<br>        <span class="hljs-type">int</span> t = a[i].time;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = a[i].begi; j &lt;= a[i].arri - <span class="hljs-number">1</span>; j++) {<br>            <span class="hljs-keyword">if</span> (t &gt; <span class="hljs-number">0</span> &amp;&amp; !res[j]) {<br>                res[j] = a[i].id; <span class="hljs-comment">// 做菜 </span><br>                t--;<br>            }<br>        }<br>        <br>        <span class="hljs-keyword">if</span> (t) {<br>            cout &lt;&lt; <span class="hljs-number">-1</span> &lt;&lt; <span class="hljs-string">"\n"</span>;<br>            <span class="hljs-keyword">return</span>;<br>        }<br>    }<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) cout &lt;&lt; res[i] &lt;&lt; <span class="hljs-string">" \n"</span>[i == n];<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="21-NOIP2004-提高组-合并果子">21. [NOIP2004 提高组] 合并果子</h3><p><a href="https://www.luogu.com.cn/problem/P1090">https://www.luogu.com.cn/problem/P1090</a></p><blockquote><ul><li>题意：给定 n 个数，现在需要将其两两合并为最终的一个数，每次合并的代价是两个数之和，问如何对这些数进行合并可以使得总代价最小</li><li>思路：很显然一共需要合并 n-1 次。那么我们从三个物品开始考虑，然后使用数学归纳法推导到 n 个数。对于三个物品，很显然先将两个物品进行合并，再将这个结果与最大的数进行合并方案时最优的。那么推广到 n 个数，我们只需要每次合并当前局面中最小的两个数即可</li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">10010</span>;<br><br><span class="hljs-type">int</span> n, a[N];<br>priority_queue&lt;<span class="hljs-type">int</span>, vector&lt;<span class="hljs-type">int</span>&gt;, greater&lt;<span class="hljs-type">int</span>&gt;&gt; q;<br><span class="hljs-type">int</span> res;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-type">int</span> x;<br>        cin &gt;&gt; x;<br>        q.<span class="hljs-built_in">push</span>(x);<br>    }<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n - <span class="hljs-number">1</span>; i++) {<br>        <span class="hljs-type">int</span> a = q.<span class="hljs-built_in">top</span>(); q.<span class="hljs-built_in">pop</span>();<br>        <span class="hljs-type">int</span> b = q.<span class="hljs-built_in">top</span>(); q.<span class="hljs-built_in">pop</span>();<br>        <br>        res += a + b;<br>        <br>        q.<span class="hljs-built_in">push</span>(a + b);<br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>} <br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="22-删数问题">22. 删数问题</h3><p><a href="https://www.luogu.com.cn/problem/P1106">https://www.luogu.com.cn/problem/P1106</a></p><blockquote><ul><li>题意：给定一个数字字符串长度为 n，现在需要选择其中的 m 位数，使得这 m 位组成的数最小是多少 (m=n-k)</li><li>思路：<strong>我们从一端考虑问题</strong>。为了组成的数尽可能的小，我们希望从首位开始就尽可能的小，同时如果有重复的数字，就选择最左侧的，剩余的数字留给后续的位数继续选择。有一个需要注意的点就是选择每一位的数字时，范围的约束，我们假设需要从 10 位的数字串中选择 3 位，那么我们第一次选择时需要在 0-7位中选择一位，第二次选择在 上次选择的下一个位置到第8位中选择一位，以此类推进行贪心选择</li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br>string s, res;<br><span class="hljs-type">int</span> k;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; s &gt;&gt; k;<br>    <span class="hljs-type">int</span> n = s.<span class="hljs-built_in">size</span>();<br>    <br>    <span class="hljs-comment">// 特判保留的数字个数 n-k 为负数的情况 </span><br>    <span class="hljs-keyword">if</span> (n - k &lt;= <span class="hljs-number">0</span>) {<br>        cout &lt;&lt; <span class="hljs-number">0</span> &lt;&lt; <span class="hljs-string">"\n"</span>;<br>        <span class="hljs-keyword">return</span>;<br>    }<br>    <br>    <span class="hljs-comment">// 循环 n-k 次选择保留的数 </span><br>    <span class="hljs-type">int</span> l = <span class="hljs-number">0</span>, r = k;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n - k; i++) {<br>        <span class="hljs-type">int</span> idx = <span class="hljs-number">-1</span>;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = l; j &lt;= r; j++)<br>            <span class="hljs-keyword">if</span> (idx == <span class="hljs-number">-1</span> || s[j] &lt; s[idx])<br>                idx = j;<br>        <br>        res += s[idx];<br>        <br>        l = idx + <span class="hljs-number">1</span>, r++;<br>    }<br>    <br>    <span class="hljs-comment">// 删除前导零 </span><br>    <span class="hljs-type">int</span> i = <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">while</span> (res[i] == <span class="hljs-string">'0'</span>) i++;<br>    <br>    <span class="hljs-comment">// 输出结果 </span><br>    <span class="hljs-keyword">if</span> (i == res.<span class="hljs-built_in">size</span>()) <br>        cout &lt;&lt; <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">else</span><br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = i; j &lt; res.<span class="hljs-built_in">size</span>(); j++) <br>            cout &lt;&lt; res[j];<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="23-截断数组">23. 截断数组</h3><p><a href="https://www.acwing.com/problem/content/5483/">https://www.acwing.com/problem/content/5483/</a></p><blockquote><p>题意：定义平衡数组为奇偶数数量相同的数组，现在给定一个平衡数组，和一个总成本数，最多可以将原数组截成多少截子平衡数组，且截断代价总和不超过总成本，截断代价计算公式为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">∣</mi><msub><mi>a</mi><mi>i</mi></msub><mo>−</mo><msub><mi>a</mi><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">|a_i-a_{i+1}|</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mord">∣</span></span></span></span></p><p>思路：我们直接枚举所有的可以被截断的位置，统计所有的代价值，然后根据成本总数按降序枚举代价值即可</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;queue&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;cstring&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;vector&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;unordered_map&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;algorithm&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">110</span>;<br><br><span class="hljs-type">int</span> n, b;<br><span class="hljs-type">int</span> a[N], odd[N], even[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; b;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        cin &gt;&gt; a[i];<br>        <span class="hljs-keyword">if</span> (a[i] % <span class="hljs-number">2</span> == <span class="hljs-number">0</span>) {<br>            even[i] = even[i - <span class="hljs-number">1</span>] + <span class="hljs-number">1</span>;<br>            odd[i] = odd[i - <span class="hljs-number">1</span>];<br>        } <span class="hljs-keyword">else</span> {<br>            even[i] = even[i - <span class="hljs-number">1</span>];<br>            odd[i] = odd[i - <span class="hljs-number">1</span>] + <span class="hljs-number">1</span>;<br>        }<br>    }<br>    <br>    vector&lt;<span class="hljs-type">int</span>&gt; c;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">2</span>; i &lt;= n - <span class="hljs-number">2</span>; i += <span class="hljs-number">2</span>) {<br>        <span class="hljs-keyword">if</span> (odd[i] == even[i]) {<br>            c.<span class="hljs-built_in">push_back</span>(<span class="hljs-built_in">abs</span>(a[i] - a[i + <span class="hljs-number">1</span>]));<br>        }<br>    }<br>    <br>    <span class="hljs-type">int</span> res = <span class="hljs-number">0</span>;<br>    <span class="hljs-built_in">sort</span>(c.<span class="hljs-built_in">begin</span>(), c.<span class="hljs-built_in">end</span>());<br>    <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; x: c) {<br>        <span class="hljs-keyword">if</span> (b &gt;= x) {<br>            b -= x;<br>            res++;<br>        }<br>    }<br>    <br>    cout &lt;&lt; res;<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="24-修改后的最大二进制字符串">24. 修改后的最大二进制字符串</h3><p><a href="https://leetcode.cn/problems/maximum-binary-string-after-change/description/">https://leetcode.cn/problems/maximum-binary-string-after-change/description/</a></p><blockquote><p>题意：给定一个仅由01组成的字符串，现在可以执行下面两种操作任意次，使得最终的字符串对应的十进制数值最大，给出最终的字符串</p><ol><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>00</mn><mo>→</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">00 \to 10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">00</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span></span></span></span></li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10</mn><mo>→</mo><mn>01</mn></mrow><annotation encoding="application/x-tex">10 \to 01</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">10</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">01</span></span></span></span></li></ol><p>思路：</p><ul><li>有点像之前做的翻转转化为平移的问题，事实确实如此。首先需要确定一点就是：如果字符串起始有连续的1，则保留，因为两种操作都是针对0的。接着就是对从第一个0开始的尾串处理的思路</li><li>对于尾串而言，我们自然希望1越多、越靠前越好，但是第二种操作只能将1向后移，第一种操作又必须要连续的两个0，因此我们就产生了这样的贪心思路：将所有的1通过操作二移动到串尾，接着对尾串的开头连续的0串执行第二种操作，这样就可以得到最大数值的二进制串</li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span> {<br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-function">string <span class="hljs-title">maximumBinaryString</span><span class="hljs-params">(string binary)</span> </span>{<br>        <span class="hljs-type">int</span> n = binary.<span class="hljs-built_in">size</span>();<br><br>        string res;<br><br>        <span class="hljs-comment">// 前缀1全部加入答案</span><br>        <span class="hljs-type">int</span> i = <span class="hljs-number">0</span>;<br>        <span class="hljs-keyword">for</span> (i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>            <span class="hljs-keyword">if</span> (binary[i] == <span class="hljs-string">'0'</span>) {<br>                <span class="hljs-keyword">break</span>;<br>            }<br>            <span class="hljs-keyword">else</span> {<br>                res += <span class="hljs-string">'1'</span>;<br>            }<br>        }<br><br>        <span class="hljs-comment">// 非前缀1的部分一定可以操作为 00...011..1，进而转化为 11...11011...1</span><br>        <span class="hljs-type">int</span> zero = <span class="hljs-built_in">count</span>(binary.<span class="hljs-built_in">begin</span>() + i, binary.<span class="hljs-built_in">end</span>(), <span class="hljs-string">'0'</span>);<br>        <span class="hljs-keyword">if</span> (zero &gt; <span class="hljs-number">0</span>) {<br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">0</span>; j &lt; zero - <span class="hljs-number">1</span>; j++) res += <span class="hljs-string">'1'</span>;<br>            res += <span class="hljs-string">'0'</span>;<br>        }<br><br>        <span class="hljs-keyword">while</span> (res.<span class="hljs-built_in">size</span>() &lt; n) res += <span class="hljs-string">'1'</span>;<br><br>        <span class="hljs-keyword">return</span> res;<br>    }<br>};<br></code></pre></td></tr></tbody></table></figure><h3 id="25-子序列最大优雅度">25. 子序列最大优雅度</h3><p><a href="https://leetcode.cn/problems/maximum-elegance-of-a-k-length-subsequence/">https://leetcode.cn/problems/maximum-elegance-of-a-k-length-subsequence/</a></p><blockquote><p>标签：反悔贪心</p><p>题意：给定长度为 n 的物品序列，现在需要从中寻找一个长度为 k 的子序列使得子序列的总价值最大，给出最终的最大价值。一个序列的价值定义为每个物品的 <strong>本身价值之和</strong> + <strong>物品种类数的平方</strong></p><p>思路：我们采用反悔贪心的思路。首先按照物品本身价值进行降序排序并选择前 k 个物品作为集合 S，然后按顺序枚举剩下的物品进行替换决策。我们记每个物品有一个「本身价值」，集合 S 有一个「种类价值」。那么就会遇到下面两种情况：</p><ul><li>第 i 个物品对应的种类<strong>存在</strong>集合 S 中。此时该物品「一定不选」，因为无论当前物品替代集合中的 S 哪一个物品，本身价值的贡献会下降、种类价值的贡献保持不变。</li><li>第 i 个物品对应的种类<strong>不在</strong>集合 S 中。此时该物品一定选择吗？不一定。有两种情况：<ul><li>集合 S 没有重复种类的物品。此时该物品「一定不选」，同样的，无论当前物品替代集合中的 S 哪一个物品，本身价值的贡献会下降、种类价值的贡献保持不变。</li><li>集合 S 存在重复种类的物品。此时该物品「一定选」，因为一定可以提升种类价值的贡献，至于减去本身价值的损失后是否使得总贡献增加，并不具备可能的性质，只能每次替换物品时<strong>更新全局答案</strong>进行记录。</li></ul></li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><figcaption><span>[]</span></figcaption><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span> {<br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-function"><span class="hljs-type">long</span> <span class="hljs-type">long</span> <span class="hljs-title">findMaximumElegance</span><span class="hljs-params">(std::vector&lt;std::vector&lt;<span class="hljs-type">int</span>&gt;&gt;&amp; items, <span class="hljs-type">int</span> k)</span> </span>{<br>        <span class="hljs-keyword">using</span> ll = <span class="hljs-type">long</span> <span class="hljs-type">long</span>;<br><br>        std::<span class="hljs-built_in">sort</span>(items.<span class="hljs-built_in">rbegin</span>(), items.<span class="hljs-built_in">rend</span>());<br><br>        ll t = <span class="hljs-number">0</span>, cat = <span class="hljs-number">0</span>, res = <span class="hljs-number">0</span>;<br>        std::unordered_map&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt; cnt;<br>        std::stack&lt;ll&gt; stk;<br><br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; items.<span class="hljs-built_in">size</span>(); i++) {<br>            ll v = items[i][<span class="hljs-number">0</span>], c = items[i][<span class="hljs-number">1</span>];<br>            <span class="hljs-keyword">if</span> (i &lt; k) {<br>                <span class="hljs-comment">// 前 k 个物品</span><br>                t += v;<br>                cnt[c]++;<br>                cat += cnt[c] == <span class="hljs-number">1</span>;<br>                <span class="hljs-keyword">if</span> (cnt[c] &gt; <span class="hljs-number">1</span>) {<br>                    stk.<span class="hljs-built_in">push</span>(v);<br>                }<br>            } <span class="hljs-keyword">else</span> {<br>                <span class="hljs-comment">// 后 n-k 个物品</span><br>                <span class="hljs-keyword">if</span> (cnt[c]) {<br>                    <span class="hljs-comment">// 已经出现过的物品种类</span><br>                    <span class="hljs-keyword">continue</span>;<br>                } <span class="hljs-keyword">else</span> {<br>                    <span class="hljs-comment">// 没有出现过的物品种类</span><br>                    <span class="hljs-keyword">if</span> (stk.<span class="hljs-built_in">size</span>()) {<br>                        cat++;<br>                        t -= stk.<span class="hljs-built_in">top</span>();<br>                        t += v;<br><br>                        stk.<span class="hljs-built_in">pop</span>();<br>                        cnt[c]++;<br>                    }<br>                }<br>            }<br>            <span class="hljs-comment">// 更新全局答案</span><br>            res = <span class="hljs-built_in">max</span>(res, t + cat * cat);<br>        }<br><br>        <span class="hljs-keyword">return</span> res;<br>    }<br>};<br></code></pre></td></tr></tbody></table></figure><figure class="highlight python"><figcaption><span>[]</span></figcaption><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span>:<br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">findMaximumElegance</span>(<span class="hljs-params">self, items: <span class="hljs-type">List</span>[<span class="hljs-type">List</span>[<span class="hljs-built_in">int</span>]], k: <span class="hljs-built_in">int</span></span>) -&gt; <span class="hljs-built_in">int</span>:<br>        <span class="hljs-keyword">from</span> collections <span class="hljs-keyword">import</span> defaultdict<br>        items.sort(reverse=<span class="hljs-literal">True</span>)<br><br>        t, cat, res = <span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">0</span><br>        cnt, stk = defaultdict(<span class="hljs-built_in">int</span>), []<br><br>        <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-built_in">len</span>(items)):<br>            v, c = items[i][<span class="hljs-number">0</span>], items[i][<span class="hljs-number">1</span>]<br>            <span class="hljs-keyword">if</span> i &lt; k:<br>                t += v<br>                cnt[c] += <span class="hljs-number">1</span><br>                cat += cnt[c] == <span class="hljs-number">1</span><br>                <span class="hljs-keyword">if</span> cnt[c] &gt; <span class="hljs-number">1</span>:<br>                    stk.append(v)<br>            <span class="hljs-keyword">else</span>:<br>                <span class="hljs-keyword">if</span> cnt[c]:<br>                    <span class="hljs-keyword">continue</span><br>                <span class="hljs-keyword">else</span>:<br>                    <span class="hljs-keyword">if</span> <span class="hljs-built_in">len</span>(stk):<br>                        t -= stk[-<span class="hljs-number">1</span>]<br>                        t += v<br>                        cat += <span class="hljs-number">1</span><br>                        stk.pop()<br>                        cnt[c] += <span class="hljs-number">1</span><br>            res = <span class="hljs-built_in">max</span>(res, t + cat**<span class="hljs-number">2</span>)<br><br>        <span class="hljs-keyword">return</span> res<br></code></pre></td></tr></tbody></table></figure><h3 id="【按位贪心-分类讨论】">【按位贪心/分类讨论】</h3><p><a href="https://www.acwing.com/problem/content/1000/">https://www.acwing.com/problem/content/1000/</a></p><blockquote><p>题意：给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个运算数和对应的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个操作，仅有 <code>&amp;, |, ^</code> 三种。现在需要选择一个数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mi>m</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">num \in [0,m]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mclose">]</span></span></span></span> 使得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">num</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span></span></span></span> 经过这 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个二元运算后得到的结果 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mi>e</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">res</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">res</span></span></span></span> 最大。给出最终的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mi>e</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">res</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">res</span></span></span></span>。</p><p>思路：首先不难发现这些运算操作在二进制位上是相互独立的，没有进位或结尾的情况，因此我们可以「按二进制位」单独考虑。由于运算数和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 均 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≤</mo><mn>1</mn><msup><mn>0</mn><mn>9</mn></msup></mrow><annotation encoding="application/x-tex">\le 10^9</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span></span></span></span>，因此我们枚举 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mn>30</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0,30]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">30</span><span class="mclose">]</span></span></span></span> 对应的二进制位。接下来我们单独考虑第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 位的情况，显然的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">num</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span></span></span></span> 的第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 位可以取 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 也可以取 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>，对应的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mi>e</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">res</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">res</span></span></span></span> 的第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 位可以取 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 也可以取 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>。有以下 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">2^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span> 种运算情况：</p><table><thead><tr><th style="text-align:center"></th><th style="text-align:center">$\text{num 的第 i 位填 1 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> res 第 i 位的情况}$</th><th style="text-align:center">$\text{num 的第 i 位填 0 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">→</span></span></span></span> res 第 i 位的情况}$</th><th style="text-align:center">选择方案</th></tr></thead><tbody><tr><td style="text-align:center">情况一</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">1 \to 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0 \to 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></td><td style="text-align:center">num 填 0，res 填 0</td></tr><tr><td style="text-align:center">情况二</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">1 \to 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">0 \to 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></td><td style="text-align:center">num 填 0，res 填 1</td></tr><tr><td style="text-align:center">情况三</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">1 \to 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0 \to 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span></td><td style="text-align:center">待定</td></tr><tr><td style="text-align:center">情况四</td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">1 \to 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></td><td style="text-align:center"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">0 \to 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></td><td style="text-align:center">num 填 0，res 填 1</td></tr></tbody></table><p>选择方案时我们需要考虑两个约束，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">num</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span></span></span></span> 不能超过 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span>，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mi>e</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">res</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">res</span></span></span></span> 尽可能大，因此有了上述表格第四列的贪心选择结果。之所以情况三待定是因为其对两种约束的相互矛盾的，并且其余情况没有增加 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">num</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span></span></span></span>，即对 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">num</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span></span></span></span> 上限的约束仅存在于情况三。假设有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 位是上述情况三，那么显然的其余位枚举顺序是随意的，因为答案是固定的。对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个情况三，从贪心的角度考虑，我们希望 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mi>e</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">res</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">res</span></span></span></span> 尽可能大，那么就有「高位能出 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 就出 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>」的结论。因此我们需要从高位到低位逆序枚举这 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 个情况三，由于其他位枚举顺序随意，因此总体就是从高位到低位枚举。</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><br><span class="hljs-keyword">using</span> ll = <span class="hljs-type">long</span> <span class="hljs-type">long</span>;<br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-type">int</span> n, m;<br>    cin &gt;&gt; n &gt;&gt; m;<br>    <br>    vector&lt;pair&lt;string, <span class="hljs-type">int</span>&gt;&gt; <span class="hljs-built_in">a</span>(n);<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        cin &gt;&gt; a[i].first &gt;&gt; a[i].second;<br>    }<br>    <br>    <span class="hljs-type">int</span> res = <span class="hljs-number">0</span>, num = <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">30</span>; i &gt;= <span class="hljs-number">0</span>; i--) {<br>        <span class="hljs-type">int</span> x = <span class="hljs-number">1</span> &lt;&lt; i, y = <span class="hljs-number">0</span>;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">0</span>; j &lt; n; j++) {<br>            string op = a[j].first;<br>            <span class="hljs-type">int</span> t = a[j].second &amp; (<span class="hljs-number">1</span> &lt;&lt; i);<br>            <span class="hljs-keyword">if</span> (op == <span class="hljs-string">"AND"</span>) {<br>                x &amp;= t, y &amp;= t;<br>            } <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (op == <span class="hljs-string">"OR"</span>) {<br>                x |= t, y |= t;<br>            } <span class="hljs-keyword">else</span> {<br>                x ^= t, y ^= t;<br>            }<br>        }<br>        <span class="hljs-keyword">if</span> ((num | (<span class="hljs-number">1</span> &lt;&lt; i)) &lt;= m &amp;&amp; x &gt; y) {<br>            num |= <span class="hljs-number">1</span> &lt;&lt; i;<br>            res |= <span class="hljs-number">1</span> &lt;&lt; i;<br>        } <span class="hljs-keyword">else</span> {<br>            res += y;<br>        }<br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    std::ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    std::cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    std::cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure>]]></content>
+    <content type="html"><![CDATA[<h3 id="数论">数论</h3><p>整数问题。</p><h3 id="【质数】Divide-and-Equalize">【质数】Divide and Equalize</h3><blockquote><p>题意：给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个数，问能否找到一个数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">num</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord mathnormal">m</span></span></span></span>，使得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><msup><mi>m</mi><mi>n</mi></msup><mo>=</mo><msubsup><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>a</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">num^n = \prod_{i=1}^{n}a_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6644em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.104em;vertical-align:-0.2997em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∏</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8043em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p><p>原始思路：起初我的思路是二分，我们需要寻找一个数使得n个数相乘为原数组所有元素之积，那么我们预计算出所有数之积，并且在数组最大值和最小值之间进行二分，每次二分出来的数计算n次方进行笔比较即可。但是有一个很大的问题是，数据量是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mn>4</mn></msup></mrow><annotation encoding="application/x-tex">10^4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span>，而数字最大为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mn>6</mn></msup></mrow><annotation encoding="application/x-tex">10^6</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span></span></span></span>，最大之积为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mn>10</mn></msup></mrow><annotation encoding="application/x-tex">10^{10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span></span></span></span></span></span></span></span> 吗？不是！最大之和才是，最大之积是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mrow><mn>6</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mn>4</mn></msup></mrow></msup></mrow><annotation encoding="application/x-tex">10^{6\times10^4}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9869em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9869em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">6</span><span class="mbin mtight">×</span><span class="mord mtight">1</span><span class="mord mtight"><span class="mord mtight">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></p><p>最终思路：我们可以将选数看做多个水池匀水的过程。现在每一个水池的水高都不定，很显然我们一定可以一个值使得所有的水池的高度一致，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mfrac><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>a</mi><mi>i</mi></msub></mrow><mi>n</mi></mfrac></mrow><annotation encoding="application/x-tex">\frac{\sum_{i=1}^{n}a_i}{n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.405em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.06em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.535em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight"><span class="mop op-symbol small-op mtight" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7385em;"><span style="top:-2.1786em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3214em;"><span></span></span></span></span></span></span><span class="mspace mtight" style="margin-right:0.1952em;"></span><span class="mord mtight"><span class="mord mathnormal mtight">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>。但是我们最终的数字是一个整数，显然不可以直接求和然后除以n，那么应该如何分配呢？我们知道水池之所以可以直接除以n，是因为水的最小分配单位是无穷小，可以随意分割；而对于整数而言，最小分配单位是什么呢？答案是<strong>质因子</strong>！为了通过分配最小单位使得最终的“水池高度一致”，我们需要让每一个“水池”获得的数值相同的质因子数量相同。于是我们只需要统计一下数组中所有数的质因子数量即可。如果对于每一种质因子的数量都可以均匀分配每一个数（即数量是n的整数倍），那么就一定可以找到这个数使得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>u</mi><msup><mi>m</mi><mi>n</mi></msup><mo>=</mo><msubsup><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>a</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">num^n = \prod_{i=1}^{n}a_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6644em;"></span><span class="mord mathnormal">n</span><span class="mord mathnormal">u</span><span class="mord"><span class="mord mathnormal">m</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.104em;vertical-align:-0.2997em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∏</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8043em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br><span class="hljs-type">int</span> n;<br>cin &gt;&gt; n;<br><br><span class="hljs-comment">// 统计所有数字的所有质因数</span><br>unordered_map&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt; m;<br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br><span class="hljs-type">int</span> x;<br>cin &gt;&gt; x;<br><br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> k = <span class="hljs-number">2</span>; k &lt;= x / k; k++) {<br><span class="hljs-keyword">if</span> (x % k == <span class="hljs-number">0</span>) {<br><span class="hljs-keyword">while</span> (x % k == <span class="hljs-number">0</span>) {<br>m[k]++;<br>x /= k;<br>}<br>}<br>}<br><br><span class="hljs-keyword">if</span> (x &gt; <span class="hljs-number">1</span>) {<br>m[x]++;<br>}<br>}<br><br><span class="hljs-comment">// 查看每一种质因数是否是n的整数倍</span><br><span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; x: m) {<br><span class="hljs-keyword">if</span> (x.second % n) {<br>cout &lt;&lt; <span class="hljs-string">"No\n"</span>;<br><span class="hljs-keyword">return</span>;<br>}<br>}<br><br>cout &lt;&lt; <span class="hljs-string">"Yes\n"</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【整除】Deja-Vu">【整除】Deja Vu</h3><p><a href="https://codeforces.com/contest/1891/problem/B">https://codeforces.com/contest/1891/problem/B</a></p><blockquote><p>题意：给点序列 a 和 b，对于 b 中的每一个元素 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>b</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">b_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，如果 a 中的元素 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">a_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 能够整除 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><msub><mi>b</mi><mi>i</mi></msub></msup></mrow><annotation encoding="application/x-tex">2^{b_i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>，则将 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">a_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 加上 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mrow><msub><mi>b</mi><mi>i</mi></msub><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">2^{b_i - 1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span>。给出最后的 a 序列</p><p>思路一：暴力枚举。</p><ul><li><p>我们很容易就想到暴力的做法，即两层循环，第一层枚举 b 中的元素，第二层枚举 a 中的元素，如果 a 中的元素能够整除 2 的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>b</mi><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">b^i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8247em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span></span> 次方，就将 a 中相应的元素加上一个值即可。但是时间复杂度肯定过不了，考虑优化。</p></li><li><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(nm)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">nm</span><span class="mclose">)</span></span></span></span></p></li></ul><p>思路二：整除优化。</p><ul><li><p>现在我们假设a中有一个数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">a_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><msub><mi>b</mi><mi>i</mi></msub></msup></mrow><annotation encoding="application/x-tex">2^{b_i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span> 的整数倍（其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>b</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">b_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 是b序列中第一个枚举到的能够让 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">a_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 整除的数），那么就有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub><mo>=</mo><mi>k</mi><msup><mn>2</mn><msub><mi>b</mi><mi>i</mi></msub></msup><mo stretchy="false">(</mo><mi>k</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a_j = k2^{b_i}(k=1,2,...,)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0991em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">...</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mclose">)</span></span></span></span>，那么 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">a_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 就要加上 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mrow><msub><mi>b</mi><mi>i</mi></msub><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">2^{b_i-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span>，于是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">a_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 就变为了 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><msup><mn>2</mn><msub><mi>b</mi><mi>i</mi></msub></msup><mo>+</mo><msup><mn>2</mn><mrow><msub><mi>b</mi><mi>i</mi></msub><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mo stretchy="false">(</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mn>2</mn><mrow><msub><mi>b</mi><mi>i</mi></msub><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">k2^{b_i}+2^{b_i-1}=(2k+1)2^{b_i-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9324em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8491em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0991em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span>。此后 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">a_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span> 就一定是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>t</mi></msup><mo stretchy="false">(</mo><mi>t</mi><mo>∈</mo><mrow><mo fence="true">[</mo><mn>1</mn><mo separator="true">,</mo><msub><mi>b</mi><mi>i</mi></msub><mo>−</mo><mn>1</mn><mo fence="true">]</mo></mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">2^t(t\in \left[ 1,b_i-1 \right])</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0436em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">b</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose delimcenter" style="top:0em;">]</span></span><span class="mclose">)</span></span></span></span> 的倍数。因此我们需要做的就是首先找到b序列中第一个数x，能够在a中找到数是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>x</mi></msup></mrow><annotation encoding="application/x-tex">2^x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6644em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span></span> 的整数倍。这一步可以这样进行：对于 a中的每一个数，我们进行30次循环统计当前数是否是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>i</mi></msup></mrow><annotation encoding="application/x-tex">2^i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8247em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span></span></span></span> 的倍数，如果是就用哈希表记录当前的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>。最后我们在遍历寻找 x 时，只需要查看当前的 x 是否被哈希过即可。接着我们统计b序列中从x开始的严格降序序列c（由题意知，次序列的数量一定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≤</mo></mrow><annotation encoding="application/x-tex">\le</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≤</span></span></span></span> 30，因为b序列中数值的值域为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mn>1</mn><mtext>&nbsp;</mtext><mn>30</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[1~30]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">1</span><span class="mspace nobreak">&nbsp;</span><span class="mord">30</span><span class="mclose">]</span></span></span></span>）。最后我们再按照原来的思路，双重循环 a 序列和 c 序列即可。</p></li><li><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>30</mn><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(30n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">30</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br><span class="hljs-type">int</span> n, m;<br>cin &gt;&gt; n &gt;&gt; m;<br><br><span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">a</span><span class="hljs-params">(n + <span class="hljs-number">1</span>)</span>, <span class="hljs-title">b</span><span class="hljs-params">(m + <span class="hljs-number">1</span>)</span></span>;<br>unordered_map&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt; ha;<br><br>    <span class="hljs-comment">// 边读边哈希</span><br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>cin &gt;&gt; a[i];<br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">30</span>; j &gt;= <span class="hljs-number">1</span>; j--) {<br><span class="hljs-keyword">if</span> (a[i] % (<span class="hljs-number">1</span> &lt;&lt; j) == <span class="hljs-number">0</span>) {<br>ha[j]++;<br>}<br>}<br>}<br><br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= m; i++) {<br>cin &gt;&gt; b[i];<br>}<br><br><span class="hljs-comment">// 寻找b中第一个能够让a[j]整除的数b[flag]</span><br><span class="hljs-type">int</span> flag = <span class="hljs-number">-1</span>;<br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= m; i++) {<br><span class="hljs-keyword">if</span> (ha[b[i]]) {<br>flag = i;<br><span class="hljs-keyword">break</span>;<br>}<br>}<br><br>    <span class="hljs-comment">// 特判</span><br><span class="hljs-keyword">if</span> (flag == <span class="hljs-number">-1</span>) {<br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">1</span>; j &lt;= n; j++) {<br>cout &lt;&lt; a[j] &lt;&lt; <span class="hljs-string">" \n"</span>[j == n];<br>}<br><span class="hljs-keyword">return</span>;<br>}<br><br>    <span class="hljs-comment">// 寻找b中从flag开始的严格单调递减的序列c</span><br>vector&lt;<span class="hljs-type">int</span>&gt; c;<br>c.<span class="hljs-built_in">push_back</span>(b[flag]);<br><span class="hljs-keyword">for</span> (; flag &lt;= m; flag++) {<br><span class="hljs-keyword">if</span> (b[flag] &lt; c.<span class="hljs-built_in">back</span>()) {<br>c.<span class="hljs-built_in">push_back</span>(b[flag]);<br>}<br>}<br><br>    <span class="hljs-comment">// 暴力循环一遍即可</span><br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">1</span>; j &lt;= n; j++) {<br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> k = <span class="hljs-number">0</span>; k &lt; c.<span class="hljs-built_in">size</span>(); k++) {<br><span class="hljs-keyword">if</span> (a[j] % (<span class="hljs-number">1</span> &lt;&lt; c[k]) == <span class="hljs-number">0</span>) {<br>a[j] += <span class="hljs-number">1</span> &lt;&lt; (c[k] - <span class="hljs-number">1</span>);<br>}<br>}<br>}<br><br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">1</span>; j &lt;= n; j++) {<br>cout &lt;&lt; a[j] &lt;&lt; <span class="hljs-string">" \n"</span>[j == n];<br>}<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【组合数学】序列数量">【组合数学】序列数量</h3><p><a href="https://www.acwing.com/problem/content/5571/">https://www.acwing.com/problem/content/5571/</a></p><blockquote><p>题意：给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i(i=1,2,\cdots,n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">i</span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 个苹果，将其分给 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 个人，问一共有多少种分配方案，给出结果对 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mn>6</mn></msup><mo>+</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">10^6+3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">6</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">3</span></span></span></span> 取模的结果</p><p>思路：整数分配问题。我们采用隔板法，隔板法相关例题见这篇博客：<a href="https://www.acwing.com/solution/content/241669/%E3%80%82%E4%B8%8B%E9%9D%A2%E5%BC%80%E5%A7%8B%E8%AE%B2%E8%A7%A3">https://www.acwing.com/solution/content/241669/。下面开始讲解</a></p><ul><li><p>利用<strong>隔板法</strong>推导结果。首先我们考虑当前局面，即只有 i 个苹果的情况下的方案数。于是题目就是 i 个苹果分给 m 个人，允许分到 0 个。于是借鉴上述链接中“少分型”的思路，先借 m 个苹果，那么此时局面中就有 i+m 个苹果，现在就等价于将 i+m 个苹果分给 m 个人，每人至少分得 1 个苹果。（分完以后每个人都还回去就行了），此时的隔板操作就是”标准型”，即 i+m 个苹果产生 i+m-1 个间隔，在其中插 m-1 块板，从而将划分出来的 m 个部分分给 m 个人。此时的划分方案就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>C</mi><mrow><mi>i</mi><mo>+</mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">C_{i+m-1}^{m-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.1894em;vertical-align:-0.3352em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8542em;"><span style="top:-2.4231em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">+</span><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.1031em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3352em;"><span></span></span></span></span></span></span></span></span></span>，那么对于所有的 i，结果就是</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.25em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>C</mi><mrow><mi>i</mi><mo>+</mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msubsup></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msubsup><mi>C</mi><mi>m</mi><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mo>+</mo><msubsup><mi>C</mi><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mo>+</mo><mo>⋯</mo><mo>+</mo><msubsup><mi>C</mi><mrow><mi>n</mi><mo>+</mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msubsup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><msubsup><mi>C</mi><mi>m</mi><mn>1</mn></msubsup><mo>+</mo><msubsup><mi>C</mi><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mo>+</mo><mo>⋯</mo><mo>+</mo><msubsup><mi>C</mi><mrow><mi>n</mi><mo>+</mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow><mi>n</mi></msubsup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo stretchy="false">(</mo><mo>−</mo><msubsup><mi>C</mi><mi>m</mi><mn>0</mn></msubsup><mo>+</mo><msubsup><mi>C</mi><mi>m</mi><mn>0</mn></msubsup><mo stretchy="false">)</mo><mo>+</mo><msubsup><mi>C</mi><mi>m</mi><mn>1</mn></msubsup><mo>+</mo><msubsup><mi>C</mi><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mo>+</mo><mo>⋯</mo><mo>+</mo><msubsup><mi>C</mi><mrow><mi>n</mi><mo>+</mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow><mi>n</mi></msubsup></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>−</mo><msubsup><mi>C</mi><mi>m</mi><mn>0</mn></msubsup><mo>+</mo><mo stretchy="false">(</mo><msubsup><mi>C</mi><mi>m</mi><mn>0</mn></msubsup><mo>+</mo><msubsup><mi>C</mi><mi>m</mi><mn>1</mn></msubsup><mo>+</mo><msubsup><mi>C</mi><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mo>+</mo><mo>⋯</mo><mo>+</mo><msubsup><mi>C</mi><mrow><mi>n</mi><mo>+</mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>−</mo><mn>1</mn><mo>+</mo><mo stretchy="false">(</mo><msubsup><mi>C</mi><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow><mn>1</mn></msubsup><mo>+</mo><msubsup><mi>C</mi><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mo>+</mo><mo>⋯</mo><mo>+</mo><msubsup><mi>C</mi><mrow><mi>n</mi><mo>+</mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>−</mo><mn>1</mn><mo>+</mo><mo stretchy="false">(</mo><msubsup><mi>C</mi><mrow><mi>m</mi><mo>+</mo><mn>2</mn></mrow><mn>2</mn></msubsup><mo>+</mo><mo>⋯</mo><mo>+</mo><msubsup><mi>C</mi><mrow><mi>n</mi><mo>+</mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mo stretchy="false">)</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mo>−</mo><mn>1</mn><mo>+</mo><msubsup><mi>C</mi><mrow><mi>n</mi><mo>+</mo><mi>m</mi></mrow><mi>n</mi></msubsup></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}\sum_{i=1}^n C_{i+m-1}^{m-1} &amp;= C_{m}^{m-1} + C_{m+1}^{m-1} + \cdots + C_{n+m-1}^{m-1} \\&amp;= C_{m}^{1} + C_{m+1}^{2} + \cdots + C_{n+m-1}^{n} \\&amp;= (-C_{m}^{0}+C_{m}^{0})+C_{m}^{1} + C_{m+1}^{2} + \cdots + C_{n+m-1}^{n} \\&amp;= -C_{m}^{0}+(C_{m}^{0}+C_{m}^{1} + C_{m+1}^{2} + \cdots + C_{n+m-1}^{n}) \\&amp;= -1+(C_{m+1}^{1} + C_{m+1}^{2} + \cdots + C_{n+m-1}^{n}) \\&amp;= -1+(C_{m+2}^{2} + \cdots + C_{n+m-1}^{n}) \\&amp;= -1+C_{n+m}^n\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:12.3496em;vertical-align:-5.9248em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.4248em;"><span style="top:-8.4248em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.433em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">+</span><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3253em;"><span></span></span></span></span></span></span></span></span><span style="top:-5.983em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"></span></span><span style="top:-4.4589em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"></span></span><span style="top:-2.9348em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"></span></span><span style="top:-1.4107em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"></span></span><span style="top:0.1134em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"></span></span><span style="top:1.6134em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.9248em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:6.4248em;"><span style="top:-8.4248em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.4436em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3148em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.4436em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3148em;"><span></span></span></span></span></span></span></span></span><span style="top:-5.983em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.4589em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.9348em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">−</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-1.4107em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:0.1134em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">+</span><span class="mord mtight">2</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:1.6134em;"><span class="pstrut" style="height:3.6514em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mord">−</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">+</span><span class="mord mathnormal mtight">m</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3053em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:5.9248em;"><span></span></span></span></span></span></span></span></span></span></span></span></p></li><li><p>利用<strong>乘法逆元</strong>计算组合数。结果已经知道了，现在如何计算上述表达式呢？由于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>×</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">n\times m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 超过了常规递推的组合数计算方法内存空间，因此我们采用乘法逆元的思路计算。值得一提的是，本题在计算某个数关于 <code>1e6+3</code> 的乘法逆元时，不需要判断两者是否互质，因为 <code>1e6+3</code> 是一个质数，并且数据范围中的数均小于 <code>1e6+3</code>，因此两数一定互质，可以直接使用费马小定理计算乘法逆元</p></li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mn>1</mn><mi>e</mi><mn>6</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\log 1e6)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span><span class="mord mathnormal">e</span><span class="mord">6</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><span class="hljs-keyword">using</span> ll = <span class="hljs-type">long</span> <span class="hljs-type">long</span>;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">7e5</span> + <span class="hljs-number">10</span>;<br><span class="hljs-type">const</span> <span class="hljs-type">int</span> mod = <span class="hljs-number">1e6</span> + <span class="hljs-number">3</span>;<br><br><span class="hljs-type">int</span> fact[N], infact[N];<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">qmi</span><span class="hljs-params">(<span class="hljs-type">int</span> a, <span class="hljs-type">int</span> b, <span class="hljs-type">int</span> p)</span> </span>{<br>    <span class="hljs-type">int</span> res = <span class="hljs-number">1</span> % p;<br>    <span class="hljs-keyword">while</span> (b) {<br>        <span class="hljs-keyword">if</span> (b &amp; <span class="hljs-number">1</span>) res = (ll)res * a % p;<br>        a = (ll)a * a % p;<br>        b &gt;&gt;= <span class="hljs-number">1</span>;<br>    }<br>    <span class="hljs-keyword">return</span> res;<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">init</span><span class="hljs-params">()</span> </span>{<br>    fact[<span class="hljs-number">0</span>] = <span class="hljs-number">1</span>, infact[<span class="hljs-number">0</span>] = <span class="hljs-number">1</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt; N; i++) {<br>        fact[i] = (ll)fact[i - <span class="hljs-number">1</span>] * i % mod;<br>        infact[i] = (ll)infact[i - <span class="hljs-number">1</span>] * <span class="hljs-built_in">qmi</span>(i, mod - <span class="hljs-number">2</span>, mod) % mod;<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-built_in">init</span>();<br>    <br>    <span class="hljs-type">int</span> n, m;<br>    cin &gt;&gt; n &gt;&gt; m;<br>    <br>    cout &lt;&lt; (ll)fact[n + m] * infact[n] % mod * infact[m] % mod - <span class="hljs-number">1</span>;<br>    <br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure>]]></content>
     
     
     <categories>
@@ -1587,11 +1587,11 @@
   
   
   <entry>
-    <title>binary-search</title>
-    <link href="/Algorithm/binary-search/"/>
-    <url>/Algorithm/binary-search/</url>
+    <title>data-structure</title>
+    <link href="/Algorithm/data-structure/"/>
+    <url>/Algorithm/data-structure/</url>
     
-    <content type="html"><![CDATA[<h3 id="二分">二分</h3><p>二分本质上是一个线性的算法思维，只是比线性思维更进一步的是，二分思维需要提炼出题面中两个线性相关的变量，即单调变化的两个变量，从而采用二分加速检索。</p><h3 id="【二分答案】Building-an-Aquarium">【二分答案】Building an Aquarium</h3><p><a href="https://codeforces.com/contest/1873/problem/E">https://codeforces.com/contest/1873/problem/E</a></p><blockquote><p>题意：想象有一个二维平面，现在有一个数列，每一个数表示平面对应列的高度，现在要给这个平面在两边加上护栏，问护栏最高可以设置为多高，可以使得在完全填满的情况下，使用的水量不会超过给定的用水量。已知最大用水量为k</p><p>思路：对于一个护栏高度，水池高度低于护栏高度的地方都需要被水填满。为了便于分析，我们可以将水池高度进行排序。那么就会很显然的一个二分题目了，我们需要二分的就是护栏的高度（最小为1，最大需要考虑一下，就是只有一列的情况下，最大高度就是最高水池高度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>max</mi><mo>⁡</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>+</mo><mi>m</mi><mi>a</mi><mi>x</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\max(a_i)+max(k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">max</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">ma</span><span class="mord mathnormal">x</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span>），check的条件就是当前h的护栏高度时，消耗的水量与最大用水量之间的大小关系，如果超过了，那么高度就要下降，反之可以上升。由于是求最大高度，因此要使用的是求右边界的二分板子</p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-keyword">typedef</span> <span class="hljs-type">long</span> <span class="hljs-type">long</span> ll;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span></span><br><span class="hljs-function"></span>{<br>    <span class="hljs-type">int</span> n, w;<br>    cin &gt;&gt; n &gt;&gt; w;<br>    <span class="hljs-function">vector&lt;ll&gt; <span class="hljs-title">a</span><span class="hljs-params">(n)</span></span>;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i ++)<br>        cin &gt;&gt; a[i];<br>    <br>    <span class="hljs-built_in">sort</span>(a.<span class="hljs-built_in">begin</span>(), a.<span class="hljs-built_in">end</span>());<br>    <br>    ll l = <span class="hljs-number">0</span>, r = <span class="hljs-number">2e9</span> + <span class="hljs-number">1</span>;<br>    <span class="hljs-keyword">while</span> (l &lt; r)<br>    {<br>        ll h = (l + r + <span class="hljs-number">1</span>) &gt;&gt; <span class="hljs-number">1</span>;<br>        <br>        ll t = <span class="hljs-number">0</span>;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i ++)<br>            <span class="hljs-keyword">if</span> (a[i] &lt; h)<br>                t += h - a[i];<br>            <span class="hljs-keyword">else</span> <span class="hljs-keyword">break</span>;<br>        <br>        <span class="hljs-keyword">if</span> (t &lt;= w) l = h;<br>        <span class="hljs-keyword">else</span> r = h - <span class="hljs-number">1</span>;<br>    }<br>    cout &lt;&lt; r &lt;&lt; endl;<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span></span><br><span class="hljs-function"></span>{<br>    <span class="hljs-type">int</span> T; cin &gt;&gt; T;<br>    <span class="hljs-keyword">while</span> (T --) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【二分答案】分组">【二分答案】分组</h3><p><a href="https://www.lanqiao.cn/problems/5129/learning/">https://www.lanqiao.cn/problems/5129/learning/</a></p><blockquote><p>题意：给定一个序列，现在需要将这个数列分为k组，如何分组可以使得每一组的极差中，最大值最小</p><p>最开始想到的思路：很容易联想到的一种方法其实就是高中组合数学中学到的“隔板法”，现在有n个数，需要分成k组，则方案数就是在n-1个空档中插入k-1个隔板，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>C</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">C_{n-1}^{k-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2139em;vertical-align:-0.3246em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8892em;"><span style="top:-2.4337em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.1031em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3246em;"><span></span></span></span></span></span></span></span></span></span> 种方案</p><p>时间复杂度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></p><p>优化思路：上述思路是正向思维，即对于构思分组情况计算极差。我们不妨逆向思维，即枚举极差的情况，判断此时的分组情况。如果对于当前的极差lim，我们显然可以分成n组，即有一个最大分组值；我们也可以求一个最小分组值cnt，即如果再少分一组那么此时的极差就会超过当前约束的极差值lim。因此对于当前约束的极差值lim，我们可以求一个最小分组值cnt</p><ul><li>如果当前的最小分组值cnt &gt; k，那么 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">[</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo separator="true">,</mo><mi>n</mi><mo fence="true">]</mo></mrow><annotation encoding="application/x-tex">\left [ cnt,n \right ]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose delimcenter" style="top:0em;">]</span></span></span></span></span> 就无法包含k，也就是说当前约束的极差lim不符合条件，lim偏小</li><li>如果当前的最小分组值cnt &lt;= k，那么 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">[</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo separator="true">,</mo><mi>n</mi><mo fence="true">]</mo></mrow><annotation encoding="application/x-tex">\left [ cnt,n \right ]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose delimcenter" style="top:0em;">]</span></span></span></span></span> 就一定包含k，且当前分组的最小极差一定是 &lt;= 约束的极差值lim，lim偏大</li></ul><p>于是二分极差的思路就跃然纸上了。我们二分极差，然后根据可以分组的最小数量cnt判断二分的结果进行左右约束调整即可。</p><p>时间复杂度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><br><span class="hljs-function"><span class="hljs-type">bool</span> <span class="hljs-title">check</span><span class="hljs-params">(<span class="hljs-type">int</span> lim, vector&lt;<span class="hljs-type">int</span>&gt;&amp; a, <span class="hljs-type">int</span> n, <span class="hljs-type">int</span> k)</span> </span>{<br>    <span class="hljs-type">int</span> cnt = <span class="hljs-number">1</span>; <span class="hljs-comment">// 当前可以分的最小组数</span><br>    <span class="hljs-type">int</span> pre = a[<span class="hljs-number">0</span>];<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        <span class="hljs-keyword">if</span> (a[i] - pre &gt; lim) {<br>            pre = a[i];<br>            cnt++;<br>        }<br>    }<br>    <span class="hljs-keyword">return</span> cnt &lt;= k;<br>}<br><br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-type">int</span> n, k;<br>    cin &gt;&gt; n &gt;&gt; k;<br><br>    <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">a</span><span class="hljs-params">(n)</span></span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        cin &gt;&gt; a[i];<br>    }<br><br>    <span class="hljs-built_in">sort</span>(a.<span class="hljs-built_in">begin</span>(), a.<span class="hljs-built_in">begin</span>() + n);<br><br>    <span class="hljs-type">int</span> l = <span class="hljs-number">0</span>, r = a[n - <span class="hljs-number">1</span>] - a[<span class="hljs-number">0</span>];<br>    <span class="hljs-keyword">while</span> (l &lt; r) {<br>        <span class="hljs-type">int</span> mid = (l + r) &gt;&gt; <span class="hljs-number">1</span>;<br>        <span class="hljs-keyword">if</span> (<span class="hljs-built_in">check</span>(mid, a, n, k)) {<br>            <span class="hljs-comment">// 分的最小组数 &lt;= k，则当前极差大了</span><br>            r = mid;<br>        } <span class="hljs-keyword">else</span> {<br>            <span class="hljs-comment">// 分的最小组数 &gt;  k，则当前极差小了</span><br>            l = mid + <span class="hljs-number">1</span>;<br>        }<br>    }<br><br>    cout &lt;&lt; r &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-number">0</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-number">0</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br>    <span class="hljs-comment">// cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) {<br>        <span class="hljs-built_in">solve</span>();<br>    }    <br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【二分答案】木材加工">【二分答案】木材加工</h3><p><a href="https://www.luogu.com.cn/problem/P2440">https://www.luogu.com.cn/problem/P2440</a></p><blockquote><p>题意：给定一个序列，现在需要将这个序列切分为等长的 k 段，长度必须为整数且尽可能的长，如果无法切分可以将多余的长度丢弃，问最长的长度是多少</p><p>思路：可以发现切分长度与切分的段数具有单调性，即切分的长度越长，切出来的段数就越少，可以进行二分。二分的思路就是直接二分答案，根据长度判断可切得的段数，最终套右边界的模板找到最大的长度即可。需要注意的是，对于无法切割的情况，就是需要切出的段数 k 超过了序列之和</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mrow><mo stretchy="false">(</mo><mn>1</mn><mi>e</mi><mn>8</mn><mo stretchy="false">)</mo></mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\log {(1e8)})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">1</span><span class="mord mathnormal">e</span><span class="mord">8</span><span class="mclose">)</span></span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long </span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">1e5</span> + <span class="hljs-number">10</span>;<br><br><span class="hljs-type">int</span> n, k;<br><span class="hljs-type">int</span> a[N];<br><br><span class="hljs-function"><span class="hljs-type">bool</span> <span class="hljs-title">chk</span><span class="hljs-params">(<span class="hljs-type">int</span> x)</span> </span>{<br>    <span class="hljs-type">int</span> sum = <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++)<br>        sum += a[i] / x;<br>    <span class="hljs-keyword">return</span> sum &gt;= k;<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; k;<br>    <br>    <span class="hljs-type">int</span> sum = <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        cin &gt;&gt; a[i];<br>        sum += a[i];<br>    }<br>    <br>    <span class="hljs-type">int</span> l = <span class="hljs-number">1</span>, r = <span class="hljs-number">1e8</span>;<br>    <span class="hljs-keyword">while</span> (l &lt; r) {<br>        <span class="hljs-type">int</span> mid = (l + r + <span class="hljs-number">1</span>) &gt;&gt; <span class="hljs-number">1</span>;<br>        <span class="hljs-keyword">if</span> (<span class="hljs-built_in">chk</span>(mid)) l = mid;<br>        <span class="hljs-keyword">else</span> r = mid - <span class="hljs-number">1</span>;<br>    }<br>    <br>    <span class="hljs-keyword">if</span> (k &gt; sum) cout &lt;&lt; <span class="hljs-string">"0\n"</span>;<br>    <span class="hljs-keyword">else</span> cout &lt;&lt; r &lt;&lt; <span class="hljs-string">"\n"</span>;    <br>} <br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();    <br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【二分答案】跳石头">【二分答案】跳石头</h3><p><a href="https://www.luogu.com.cn/problem/P2678">https://www.luogu.com.cn/problem/P2678</a></p><blockquote><p>题意：给定 n 个递增的不重复数，现在可以从其中拿掉 k 个数，使得相邻数字之间的最小差值最大，问最大的最小差值是多少</p><p>思路：<strong>二分答案。将答案看做 y 轴，拿掉的数的数量看做 x 轴，就是二分 y 值。<strong>可以发现拿的数字越多，最小差值就越大，具有单调性，可以二分。我们直接二分答案，即直接二分最小差值的具体数值，通过判断当前的最小差值至少需要拿掉多少个数才能满足，进行 check 操作。至于如何计算至少要拿掉的数字，我们采用</strong>右贪心</strong>准则，即检查当前点与上一个点之间的距离是否满足最小差值的要求，如果不满足就需要记数，为了便于后续的计算，直接将当前的下标用上一个点的下标覆盖掉即可</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long </span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">5e4</span> + <span class="hljs-number">10</span>;<br><br><span class="hljs-type">int</span> lim, n, k;<br><span class="hljs-type">int</span> a[N], b[N];<br><span class="hljs-type">bool</span> del[N];<br><br><span class="hljs-function"><span class="hljs-type">bool</span> <span class="hljs-title">ok</span><span class="hljs-params">(<span class="hljs-type">int</span> x)</span> </span>{<br>    <span class="hljs-type">int</span> cnt = <span class="hljs-number">0</span>;<br>    <span class="hljs-built_in">memset</span>(del, <span class="hljs-literal">false</span>, <span class="hljs-keyword">sizeof</span> del);<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        b[i] = a[i];<br>    }<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-keyword">if</span> (b[i] - b[i - <span class="hljs-number">1</span>] &lt; x) {<br>            del[i] = <span class="hljs-literal">true</span>;<br>            b[i] = b[i - <span class="hljs-number">1</span>];<br>            cnt++;<br>        }<br>    }<br>    <br>    <span class="hljs-keyword">if</span> (lim - b[n] &lt; x) {<br>        cnt++;<br>    }<br>    <br>    <span class="hljs-keyword">return</span> cnt &lt;= k;<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; lim &gt;&gt; n &gt;&gt; k;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++)<br>        cin &gt;&gt; a[i];<br>    <br>    <span class="hljs-type">int</span> l = <span class="hljs-number">1</span>, r = lim;<br>    <span class="hljs-keyword">while</span> (l &lt; r) {<br>        <span class="hljs-type">int</span> mid = (l + r + <span class="hljs-number">1</span>) &gt;&gt; <span class="hljs-number">1</span>;<br>        <span class="hljs-keyword">if</span> (<span class="hljs-built_in">ok</span>(mid)) l = mid;<br>        <span class="hljs-keyword">else</span> r = mid - <span class="hljs-number">1</span>;<br>    }<br>    <br>    cout &lt;&lt; r &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();    <br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【二分答案】路标设置">【二分答案】路标设置</h3><p><a href="https://www.luogu.com.cn/problem/P3853">https://www.luogu.com.cn/problem/P3853</a></p><blockquote><p>题意：与第四题题面几乎一致，只是现在不是从序列中拿走 k 数个，而是往序列中插入 k 个数（插入数字后要保证序列仍然没有重复数且递增），问在插入一定数量数字的情况下，最小的最大差值是多少</p><p>思路：**二分答案。将答案看做 y 轴，拿掉的数的数量看做 x 轴，就是二分 y 值。**同样可以发现，插入的数字越多，最大差值就越小，具有单调性，可以二分。我们依然直接二分答案，即直接二分最大差值的具体数值，通过判断当前的最大差值需要插入多少个数来检查当前状态是否合理。需要插入的数字的个数为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo fence="true">⌈</mo><mfrac><mrow><mi>a</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo>−</mo><mi>a</mi><mo stretchy="false">[</mo><mi>i</mi><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><mrow><mi>d</mi><mi>i</mi><mi>s</mi><msub><mi>t</mi><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></mrow></mfrac><mo fence="true">⌉</mo></mrow><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\left \lceil \frac{a[i]-a[i-1]}{dist_{max}}\right \rceil - 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">⌈</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">i</span><span class="mord mathnormal">s</span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">ma</span><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose">]</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.836em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">⌉</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long </span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">1e6</span>;<br><br><span class="hljs-type">int</span> lim, n, k;<br><span class="hljs-type">int</span> a[N];<br><br><span class="hljs-function"><span class="hljs-type">bool</span> <span class="hljs-title">chk</span><span class="hljs-params">(<span class="hljs-type">int</span> x)</span> </span>{<br>    <span class="hljs-type">int</span> cnt = <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt; n; i++) {<br>        <span class="hljs-type">int</span> gap = a[i] - a[i - <span class="hljs-number">1</span>];<br>        <span class="hljs-keyword">if</span> (gap &gt; x) {<br>            cnt += (gap + x - <span class="hljs-number">1</span>) / x - <span class="hljs-number">1</span>;<br>        }<br>    }<br>    <span class="hljs-keyword">return</span> cnt &gt; k;<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; lim &gt;&gt; n &gt;&gt; k;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        cin &gt;&gt; a[i];<br>    }<br>    <br>    <span class="hljs-type">int</span> l = <span class="hljs-number">1</span>, r = lim;<br>    <span class="hljs-keyword">while</span> (l &lt; r) {<br>        <span class="hljs-type">int</span> mid = (l + r) &gt;&gt; <span class="hljs-number">1</span>;<br>        <span class="hljs-keyword">if</span> (<span class="hljs-built_in">chk</span>(mid)) l = mid + <span class="hljs-number">1</span>;<br>        <span class="hljs-keyword">else</span> r = mid;<br>    }<br>    <br>    cout &lt;&lt; r &lt;&lt; <span class="hljs-string">"\n"</span>;<br>} <br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();    <br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【二分答案】数列分段-Section-II">【二分答案】数列分段 Section II</h3><p><a href="https://www.luogu.com.cn/problem/P1182">https://www.luogu.com.cn/problem/P1182</a></p><blockquote><p>题意：给定一个无序的序列，现在需要将这个序列进行分段（连续的），分成指定的段数。问应该如何分段可以使得所有段的分段和的最大值最小，输出这个最小的最大值</p><p>思路：**二分答案，将答案看做 y 轴，拿掉的数的数量看做 x 轴，就是二分 y 值。**可以发现，分的段数越多，所有分段和的最大值就越小，具有单调性，可以二分。我们直接二分答案，即直接二分分段最大值，通过判断当前最大值的约束条件下可以分的组数进行判断。至于如何计算当前最大值条件下可分得的组数，直接线性扫描进行局部求和即可</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long </span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">1e5</span> + <span class="hljs-number">10</span>;<br><br><span class="hljs-type">int</span> n, k;<br><span class="hljs-type">int</span> a[N];<br><br><span class="hljs-comment">// 当前分组时最大子段和为 x </span><br><span class="hljs-function"><span class="hljs-type">bool</span> <span class="hljs-title">chk</span><span class="hljs-params">(<span class="hljs-type">int</span> x)</span> </span>{<br>    <span class="hljs-type">int</span> cnt = <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>, s = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        <span class="hljs-keyword">if</span> (a[i] &gt; x) <span class="hljs-keyword">return</span> <span class="hljs-literal">true</span>;<br>        <br>        <span class="hljs-keyword">if</span> (s + a[i] &lt;= x) s += a[i];<br>        <span class="hljs-keyword">else</span> {<br>            cnt++;<br>            s = a[i];<br>        }<br>    }<br>    cnt += <span class="hljs-number">1</span>;<br>    <br>    <span class="hljs-keyword">return</span> cnt &gt; k;<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; k;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        cin &gt;&gt; a[i];<br>    }<br>    <br>    <span class="hljs-type">int</span> l = <span class="hljs-number">0</span>, r = <span class="hljs-number">1e9</span>;<br>    <span class="hljs-keyword">while</span> (l &lt; r) {<br>        <span class="hljs-type">int</span> mid = (l + r) &gt;&gt; <span class="hljs-number">1</span>;<br>        <span class="hljs-keyword">if</span> (<span class="hljs-built_in">chk</span>(mid)) l = mid + <span class="hljs-number">1</span>;<br>        <span class="hljs-keyword">else</span> r = mid;<br>    }<br>    <br>    cout &lt;&lt; r &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();    <br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【二分答案】kotori的设备">【二分答案】kotori的设备</h3><p><a href="https://www.luogu.com.cn/problem/P3743">https://www.luogu.com.cn/problem/P3743</a></p><blockquote><p>题意：现在有一批电子设备，每一个电子设备有一个电量消耗速度与当前剩余电量，现在有一个充电器有一个确定的充电速度。问这批设备最久可以运作多久？当可以无限运作时输出 -1</p><p>思路：可以发现，想要运行的越久，需要补充的电量就越多，具有单调性，可以直接二分答案。很显然我们可以根据期望运行的时间进行 check 操作，通过比对当前期望时间可以充的电量与需要补充的电量进行比对来修改边界值。<strong>需要注意的是</strong>边界的选择，最长可运行时间为 1e10，具体 <a href="https://www.luogu.com.cn/discuss/468555">推导</a> 待定</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">(</mo><mi>log</mi><mo>⁡</mo><mrow><mn>1</mn><msup><mn>0</mn><mn>10</mn></msup><mo stretchy="false">)</mo></mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n(\log{10^{10})})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">100010</span>;<br><br><span class="hljs-type">int</span> n;<br><span class="hljs-type">double</span> p, v[N], s[N];<br><br><span class="hljs-function"><span class="hljs-type">bool</span> <span class="hljs-title">chk</span><span class="hljs-params">(<span class="hljs-type">double</span> x)</span> </span>{<br>    <span class="hljs-comment">// x 为当前状态期望使用的时间 </span><br>    <span class="hljs-type">double</span> need = <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++)<br>        <span class="hljs-keyword">if</span> (v[i] * x &gt; s[i])<br>            need += v[i] * x - s[i];<br>    <span class="hljs-keyword">return</span> need &lt;= x * p;<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; p;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        cin &gt;&gt; v[i] &gt;&gt; s[i];<br>    }<br>    <br>    <span class="hljs-type">double</span> l = <span class="hljs-number">0</span>, r = <span class="hljs-number">1e10</span> + <span class="hljs-number">10</span>;<br>    <span class="hljs-keyword">while</span> (r - l &gt; <span class="hljs-number">1e-6</span>) {<br>        <span class="hljs-type">double</span> mid = (l + r) / <span class="hljs-number">2</span>; <br>        <span class="hljs-keyword">if</span> (<span class="hljs-built_in">chk</span>(mid)) l = mid;<br>        <span class="hljs-keyword">else</span> r = mid;<br>    }<br>    <br>    <span class="hljs-keyword">if</span> (r &gt; <span class="hljs-number">1e10</span>) cout &lt;&lt; <span class="hljs-number">-1</span> &lt;&lt; <span class="hljs-string">"\n"</span>;<br>    <span class="hljs-keyword">else</span> cout &lt;&lt; r &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【二分答案-并查集】关押罪犯-fire">【二分答案/并查集】关押罪犯 <span class="github-emoji"><span>🔥</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/1f525.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span></h3><p><a href="https://www.luogu.com.cn/problem/P1525">https://www.luogu.com.cn/problem/P1525</a></p><blockquote><p>题意：给定一个无向图，没有重边和自环，边权为正。现在需要将图中所有的顶点分为两部分，使得两部分中最大的边权尽可能小，问该最小边权是多少</p><p>思路一：二分答案。</p><ul><li>思路：本题一眼二分图问题，但是有些变化的是左右部并非散点图，其中是有连边的。如何求出最小边权呢？我们可以这么想，首先答案一定出自左右部的连边中（除非左右部全是散点，那答案就是 0），之所以可以采用二分图将点集分为两部分，是因为我们在某种规则下忽略了其中奇数环（二分图定理）的一些边。当规则定义为 <strong>忽略边权不超过 x</strong> 的边时，该图若可以二分，那么忽略边权比 x 值更大的边时，该图同样一定也可以二分，反之则不一定可以二分（特性图如下）。具备单调性，于是我们可以通过 <strong>二分阈值、检查图是否可二分</strong> 的方法来计算出答案</li></ul><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202402230117928.png" alt="image-20240223011704197"></p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Θ</mi><mo stretchy="false">(</mo><mi>log</mi><mo>⁡</mo><msub><mi>C</mi><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>×</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta(\log C_{max} \times (n+e))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Θ</span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">ma</span><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">e</span><span class="mclose">))</span></span></span></span></p><p>思路二：并查集。</p><ul><li><p>思路：</p></li><li><p>时间复杂度：</p></li></ul></blockquote><p>二分判定二分图代码：</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br></pre></td><td class="code"><pre><code class="hljs cpp">```<br><br>并查集代码：<br><br>```cpp<br></code></pre></td></tr></tbody></table></figure><h3 id="【二分答案】摆放棋子">【二分答案】摆放棋子</h3><p><a href="https://www.acwing.com/problem/content/5562/">https://www.acwing.com/problem/content/5562/</a></p><blockquote><p>题意：给定一个 01 序列表示一个 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">1\times n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 的棋子序列，其中 1 表示有棋子，0 表示没有棋子。现在给定 k 个棋子，如何放置可以使得棋盘上连续的棋子长度尽可能长？给出一个合法的最长的序列</p><p>思路：可以发现，想要序列尽可能的长，那么需要放置的棋子就要尽可能的多，具备单调性，可以二分。我们二分答案，即最长序列长度。对于已知的 k 个可放置棋子，我们需要找到最大的序列长度，于是我们套用<strong>寻找右边界</strong>模板。检查方法就是判断对于当前的长度，通过前缀和与滑动窗口的形式计算当前窗口内需要放置多少颗棋子才能连为一体：</p><ul><li>若需要的棋子数 &lt; k，说明可以继续增大长度</li><li>若需要的棋子数 &gt; k，说明当前长度无法满足，要缩小长度</li><li>若需要的棋子数 = k，归属在 &lt; k 的类比中即可。</li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;cstring&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;vector&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;queue&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;stack&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;algorithm&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;unordered_map&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;set&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">300010</span>;<br><br><span class="hljs-type">int</span> n, k;<br><span class="hljs-type">int</span> a[N], s[N];<br><br><span class="hljs-function"><span class="hljs-type">bool</span> <span class="hljs-title">chk</span><span class="hljs-params">(<span class="hljs-type">int</span> x)</span> </span>{<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = x; i &lt;= n; i++) {<br>        <span class="hljs-keyword">if</span> (x - (s[i] - s[i - x]) &lt;= k) {<br>            <span class="hljs-keyword">return</span> <span class="hljs-literal">true</span>;<br>        }<br>    }<br>    <span class="hljs-keyword">return</span> <span class="hljs-literal">false</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; k;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        cin &gt;&gt; a[i];<br>        s[i] = s[i - <span class="hljs-number">1</span>] + a[i];<br>    }<br><br>    <span class="hljs-type">int</span> l = <span class="hljs-number">0</span>, r = n;<br>    <span class="hljs-keyword">while</span> (l &lt; r) {<br>        <span class="hljs-type">int</span> mid = (l + r + <span class="hljs-number">1</span>) &gt;&gt; <span class="hljs-number">1</span>;<br>        <span class="hljs-keyword">if</span> (<span class="hljs-built_in">chk</span>(mid)) l = mid;<br>        <span class="hljs-keyword">else</span> r = mid - <span class="hljs-number">1</span>;<br>    }<br><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = r; i &lt;= n; i++) {<br>        <span class="hljs-keyword">if</span> (r - (s[i] - s[i - r]) &lt;= k) {<br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = i - r + <span class="hljs-number">1</span>; j &lt;= i; j++) {<br>                a[j] = <span class="hljs-number">1</span>;<br>            }<br>            <span class="hljs-keyword">break</span>;<br>        }<br>    }<br><br>    cout &lt;&lt; r &lt;&lt; <span class="hljs-string">"\n"</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) cout &lt;&lt; a[i] &lt;&lt; <span class="hljs-string">" "</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【二分答案】盖楼">【二分答案】盖楼</h3><p><a href="https://www.acwing.com/problem/content/description/5569/">https://www.acwing.com/problem/content/description/5569/</a></p><blockquote><p>题意：给定 H 个正整数分别为 1 到 H。现在要将这 H 个正整数分给两个人，其中一个人希望获得可以不被 x 整除的数，另一个人希望可以获得不被 y 整除的数。其中 x 和 y 均为质数，问最小的 H 是多少？</p><p>思路：</p><ul><li><p>很显然的一个二分答案。因为 H 越大，越有可能满足这两个人的要求，具备单调性。</p></li><li><p>那么现在的问题就是检查函数的设计。对于当前的高度 h，即 h 个正整数，很显然可以划分出下面的集合关系</p>  <img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404091829325.png" alt="集合关系" style="zoom:50%;"><p>其中 h-p-q+a 的部分两个人都可以获得。最策略是，p-a 与 q-a 的都给另外的人，如果不够，从 h-p-q+a 中拿，如果还不够那就说明当前 h 无法满足，需要增大 h，反之说明 h 可以满足，继续寻找答案的左边界。</p></li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Θ</mi><mo stretchy="false">(</mo><mi>log</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>N</mi><mo>+</mo><mi>M</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta(\log (N+M))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Θ</span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mclose">))</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><span class="hljs-keyword">typedef</span> <span class="hljs-type">long</span> <span class="hljs-type">long</span> ll;<br><br>ll n, m, x, y;<br><br><span class="hljs-function"><span class="hljs-type">bool</span> <span class="hljs-title">chk</span><span class="hljs-params">(ll h)</span> </span>{<br>    ll p = h / x, q = h / y, a = h / (x * y);<br><br>    <span class="hljs-keyword">return</span> <span class="hljs-built_in">max</span>(<span class="hljs-number">0ll</span>, n - (q - a)) + <span class="hljs-built_in">max</span>(<span class="hljs-number">0ll</span>, m - (p - a)) &lt;= h - p - q + a;<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; m &gt;&gt; x &gt;&gt; y;<br><br>    ll l = <span class="hljs-number">1</span>, r = <span class="hljs-number">1e15</span>;<br>    <span class="hljs-keyword">while</span> (l &lt; r) {<br>        ll mid = (l + r) &gt;&gt; <span class="hljs-number">1</span>;<br>        <span class="hljs-keyword">if</span> (<span class="hljs-built_in">chk</span>(mid)) r = mid;<br>        <span class="hljs-keyword">else</span> l = mid + <span class="hljs-number">1</span>;<br>    }<br><br>    cout &lt;&lt; r &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><p>小插曲。一开始写的检查函数 code 始终只能通过 9/10，检查函数的代码如下：</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-function"><span class="hljs-type">bool</span> <span class="hljs-title">chk</span><span class="hljs-params">(ll h)</span> </span>{<br>    ll p = h / x, q = h / y, a = h / (x * y);<br><br>    <span class="hljs-keyword">if</span> (h - p &gt;= n &amp;&amp; h - q &gt;= m) {<br>        <span class="hljs-comment">// 如果满足第一人，检查第二人是否满足</span><br>        <span class="hljs-keyword">if</span> (q - a &gt;= n) <span class="hljs-keyword">return</span> <span class="hljs-literal">true</span>;<br>        <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (h - (n - (q - a)) - q &gt;= m) <span class="hljs-keyword">return</span> <span class="hljs-literal">true</span>;<br><br>        <span class="hljs-comment">// 如果满足第二人，检查第一人是否满足</span><br>        <span class="hljs-keyword">if</span> (p - a &gt;= m) <span class="hljs-keyword">return</span> <span class="hljs-literal">true</span>;<br>        <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (h - (n - (p - a)) - p &gt;= n) <span class="hljs-keyword">return</span> <span class="hljs-literal">true</span>;<br>    }<br>    <span class="hljs-keyword">return</span> <span class="hljs-literal">false</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><p>其实是因为有逻辑错误，即对于当前的 h，必须两个人都能满足才行。而上述代码可能会出现第一个人不满足，但是第二个人满足，却返回 true 的情况，而这是错误的。因此上述代码改成两个人同时满足就可以通过了，即如果有一个人不满足，则返回 false：</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-function"><span class="hljs-type">bool</span> <span class="hljs-title">chk</span><span class="hljs-params">(ll h)</span> </span>{<br>    ll p = h / x, q = h / y, a = h / (x * y);<br><br>    <span class="hljs-keyword">if</span> (h - p &gt;= n &amp;&amp; h - q &gt;= m) {<br>        <span class="hljs-comment">// 如果满足第一人，检查第二人是否满足</span><br>        <span class="hljs-keyword">if</span> (q - a &gt;= n) <span class="hljs-keyword">return</span> <span class="hljs-literal">true</span>;<br>        <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (h - (n - (q - a)) - q &gt;= m) <span class="hljs-keyword">return</span> <span class="hljs-literal">true</span>;<br><br>        <span class="hljs-keyword">return</span> <span class="hljs-literal">false</span>;<br>    }<br>    <span class="hljs-keyword">return</span> <span class="hljs-literal">false</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><p>小结：过多的分支语句不如一个 max 来的更加清晰，也可以避免一定的逻辑错误。</p><h3 id="【二分答案】找出叠涂元素">【二分答案】找出叠涂元素</h3><p><a href="https://leetcode.cn/problems/first-completely-painted-row-or-column/description/">https://leetcode.cn/problems/first-completely-painted-row-or-column/description/</a></p><blockquote><p>题意：给定一个 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">m\times n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 的矩阵，以及一个存储矩阵中所有元素值的数组，现在从左往右将数组中的元素在对应矩阵中涂色，问序列中最左边使得矩阵中某一行或列全部涂色的下标是什么</p><p>思路：显然，下标越往右越有可能出现矩阵中某一行或列全部涂色，具备单调性，可以二分答案。我们直接二分序列下标，对于 chk 函数我们直接 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>×</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\times m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span> 模拟即可</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>×</mo><mi>m</mi><mi>log</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>n</mi><mo>×</mo><mi>m</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\times m \log (n\times m))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">m</span><span class="mclose">))</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span> {<br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">firstCompleteIndex</span><span class="hljs-params">(vector&lt;<span class="hljs-type">int</span>&gt;&amp; arr, vector&lt;vector&lt;<span class="hljs-type">int</span>&gt;&gt;&amp; mat)</span> </span>{<br>        <span class="hljs-type">int</span> m = mat.<span class="hljs-built_in">size</span>(), n = mat[<span class="hljs-number">0</span>].<span class="hljs-built_in">size</span>();<br><br>        <span class="hljs-keyword">auto</span> chk = [&amp;](<span class="hljs-type">int</span> idx) {<br>            <span class="hljs-type">bool</span> vis[<span class="hljs-number">100010</span>] {};<br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt;= idx; i++) {<br>                vis[arr[i]] = <span class="hljs-literal">true</span>;<br>            }<br><br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; m; i++) {<br>                <span class="hljs-type">int</span> s = <span class="hljs-number">0</span>;<br>                <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">0</span>; j &lt; n; j++) {<br>                    s += vis[mat[i][j]];<br>                }<br>                <span class="hljs-keyword">if</span> (s == n) <span class="hljs-keyword">return</span> <span class="hljs-literal">true</span>;<br>            }<br><br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">0</span>; j &lt; n; j++) {<br>                <span class="hljs-type">int</span> s = <span class="hljs-number">0</span>;<br>                <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; m; i++) {<br>                    s += vis[mat[i][j]];<br>                }<br>                <span class="hljs-keyword">if</span> (s == m) <span class="hljs-keyword">return</span> <span class="hljs-literal">true</span>;<br>            }<br><br>            <span class="hljs-keyword">return</span> <span class="hljs-literal">false</span>;<br>        };<br><br>        <span class="hljs-type">int</span> l = <span class="hljs-number">0</span>, r = m * n - <span class="hljs-number">1</span>;<br>        <span class="hljs-keyword">while</span> (l &lt; r) {<br>            <span class="hljs-type">int</span> mid = (l + r) &gt;&gt; <span class="hljs-number">1</span>;<br>            <span class="hljs-keyword">if</span> (<span class="hljs-built_in">chk</span>(mid)) r = mid;<br>            <span class="hljs-keyword">else</span> l = mid + <span class="hljs-number">1</span>;<br>        }<br>        <span class="hljs-keyword">return</span> r;<br>    }<br>};<br></code></pre></td></tr></tbody></table></figure><h3 id="【二分答案-双指针】找出唯一性数组的中位数">【二分答案/双指针】找出唯一性数组的中位数</h3><p><a href="https://leetcode.cn/problems/find-the-median-of-the-uniqueness-array/">https://leetcode.cn/problems/find-the-median-of-the-uniqueness-array/</a></p><blockquote><p>题意：给定一个数组，返回其「唯一性数组」的中位数。如果唯一性数组有奇数个元素，则返回中间元素，如果有偶数个元素，则返回两个中间元素中较小的那个，即左边那个中间元素。唯一性数组定义为：原数组的所有非空子数组中不同元素的个数组成的序列。</p><p>思路：</p><ul><li>首先根据题意可以明确的计算出这个唯一性数组的元素个数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mi>o</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">tot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span><span class="mord mathnormal">o</span><span class="mord mathnormal">t</span></span></span></span>。假设原数组有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个元素，则其唯一性数组一定有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mi>t</mi><mi>o</mi><mi>t</mi><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>n</mi><mo stretchy="false">)</mo><mo>×</mo><mi>n</mi></mrow><mn>2</mn></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle tot=\frac{(1+n) \times n}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span><span class="mord mathnormal">o</span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 个元素，且最小值一定是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，最大值 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≤</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">\le n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>。最暴力的做法就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 的维护出这个唯一性数组然后返回其中位数，但这对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">10^5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span> 级别的数据显然是不合适的。观察其他性质。</li><li>不难发现，对于「不同元素的个数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>」而言，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 越大，满足条件的子数组数量就越多，反之 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 越小，则满足条件的子数组数量就越少，具备单调性，考虑二分。</li><li>问题就是在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mn>1</mn><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[1,n]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">]</span></span></span></span> 内找到一个尽可能小的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mi>p</mi><mi>p</mi><mi>e</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">upper</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">u</span><span class="mord mathnormal">pp</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span></span></span></span> 使得满足「不同元素的个数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≤</mo><mi>u</mi><mi>p</mi><mi>p</mi><mi>e</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">\le upper</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">u</span><span class="mord mathnormal">pp</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span></span></span></span> 的子数组」的数量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mo>≥</mo><mo stretchy="false">⌈</mo><mfrac><mrow><mi>t</mi><mi>o</mi><mi>t</mi></mrow><mn>2</mn></mfrac><mo stretchy="false">⌉</mo></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \ge \lceil \frac{tot}{2} \rceil</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.9781em;vertical-align:-0.686em;"></span><span class="mopen">⌈</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2921em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="mord mathnormal">o</span><span class="mord mathnormal">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌉</span></span></span></span>。</li><li>我们能否在指定不同元素个数上限为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mi>p</mi><mi>p</mi><mi>e</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">upper</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">u</span><span class="mord mathnormal">pp</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span></span></span></span> 的情况下，快速计算合法子数组的个数呢？答案是可以的，我们引入双指针算法进行 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi><mi>h</mi><mi>e</mi><mi>c</mi><mi>k</mi></mrow><annotation encoding="application/x-tex">check</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">h</span><span class="mord mathnormal">ec</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span>。我们枚举子数组的右边界并利用哈希表维护子数组的左边界进行统计即可。</li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><figcaption><span>[]</span></figcaption><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span> {<br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">medianOfUniquenessArray</span><span class="hljs-params">(vector&lt;<span class="hljs-type">int</span>&gt;&amp; nums)</span> </span>{<br>        <span class="hljs-keyword">using</span> ll = <span class="hljs-type">long</span> <span class="hljs-type">long</span>;<br>        <span class="hljs-type">int</span> n = nums.<span class="hljs-built_in">size</span>();<br>        <br>        ll tot = <span class="hljs-number">1ll</span> * (<span class="hljs-number">1</span> + n) * n / <span class="hljs-number">2</span>;<br>        <span class="hljs-type">int</span> ma = *<span class="hljs-built_in">max_element</span>(nums.<span class="hljs-built_in">begin</span>(), nums.<span class="hljs-built_in">end</span>());<br>        <span class="hljs-type">int</span> f[ma + <span class="hljs-number">1</span>];<br>        <span class="hljs-built_in">memset</span>(f, <span class="hljs-number">0</span>, <span class="hljs-keyword">sizeof</span> f);<br><br>        <span class="hljs-comment">// 双指针 check</span><br>        <span class="hljs-keyword">auto</span> chk = [&amp;](<span class="hljs-type">int</span> upper) -&gt; <span class="hljs-type">bool</span> {<br>            ll cnt = <span class="hljs-number">0</span>;<br>            <span class="hljs-type">int</span> size = <span class="hljs-number">0</span>; <span class="hljs-comment">// 哈希表大小</span><br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> l = <span class="hljs-number">0</span>, r = <span class="hljs-number">0</span>; r &lt; n; r++) {<br>                size += f[nums[r]] == <span class="hljs-number">0</span>;<br>                f[nums[r]]++;<br>                <span class="hljs-keyword">while</span> (size &gt; upper) {<br>                    --f[nums[l]];<br>                    size -= f[nums[l]] == <span class="hljs-number">0</span>;<br>                    l++;<br>                }<br>                <span class="hljs-comment">// 此时以 nums[r] 结尾且 l 为左边界的所有子数组的不同元素的数量均 &lt;= upper</span><br>                cnt += r - l + <span class="hljs-number">1</span>;<br>            }<br>            <span class="hljs-built_in">memset</span>(f, <span class="hljs-number">0</span>, <span class="hljs-keyword">sizeof</span> f);<br>            <span class="hljs-keyword">return</span> cnt &gt;= (tot + <span class="hljs-number">1</span>) / <span class="hljs-number">2</span>; <span class="hljs-comment">// 上取整写法</span><br>        };<br><br>        <span class="hljs-type">int</span> l = <span class="hljs-number">1</span>, r = n;<br>        <span class="hljs-keyword">while</span> (l &lt; r) {<br>            <span class="hljs-type">int</span> mid = (l + r) &gt;&gt; <span class="hljs-number">1</span>;<br>            <span class="hljs-keyword">if</span> (<span class="hljs-built_in">chk</span>(mid)) r = mid;<br>            <span class="hljs-keyword">else</span> l = mid + <span class="hljs-number">1</span>;<br>        }<br>        <span class="hljs-keyword">return</span> r;<br>    }<br>};<br></code></pre></td></tr></tbody></table></figure><figure class="highlight plaintext"><figcaption><span>[]</span></figcaption><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br></pre></td><td class="code"><pre><code class="hljs python3">class Solution:<br>    def medianOfUniquenessArray(self, nums: List[int]) -&gt; int:<br>        n = len(nums)<br>        ma = max(nums)<br>        tot = (1 + n) * n &gt;&gt; 1<br><br>        def chk(upper: int) -&gt; int:<br>            f = [0] * (ma + 1)<br>            cnt, l, size = 0, 0, 0<br>            for r in range(n):<br>                size += f[nums[r]] == 0<br>                f[nums[r]] += 1<br>                while size &gt; upper:<br>                    f[nums[l]] -= 1<br>                    size -= f[nums[l]] == 0<br>                    l += 1<br>                cnt += r - l + 1<br>            return cnt &gt;= (tot + 1) &gt;&gt; 1<br><br>        l, r = 1, n<br>        while l &lt; r:<br>            mid = (l + r) &gt;&gt; 1<br>            if chk(mid): r = mid<br>            else: l = mid + 1<br>        return r<br></code></pre></td></tr></tbody></table></figure><h3 id="【二分查找】Bomb">【二分查找】Bomb</h3><p><a href="https://codeforces.com/contest/1996/problem/F">https://codeforces.com/contest/1996/problem/F</a></p><blockquote><p>题意：给定两个序列 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span> 以及最大操作次数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span>，问在不超过最大操作次数的情况下，最多可以获得多少收益？收益的计算方法为：每次选择 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> 中一个数 <code>a[i]</code> 加入收益，并将该数减去 <code>b[i]</code> 直到为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>。</p><p>思路：</p><ul><li>暴力抬手：最暴力的做法是我们每次操作时选择 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> 中的最大值进行计数并对其修改，但是显然 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(nk)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">nk</span><span class="mclose">)</span></span></span></span> 会超时，我们考虑加速这个过程。</li><li>加速进程：由于最终结果中每次操作都是最优的，因此操作次数越多获得的收益就越大，具备单调性。如何利用这个单调性呢？关键在于全体数据的操作次数，我们记作 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>k</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">k'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span>。假设我们对 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> 中全体数据可取的范围设置一个下限 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">lower</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span></span></span></span>，那么显然的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">lower</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span></span></span></span> 越高，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>k</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">k'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> 越小，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">lower</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span></span></span></span> 越低，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>k</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">k'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> 越大。于是根据单调的传递性可知 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">lower</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span></span></span></span> 也和最终的收益具备单调性，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">lower</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span></span></span></span> 越高，收益越小，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">lower</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span></span></span></span> 越低，收益越大。因此我们二分 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">lower</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span></span></span></span>，使得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>k</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">k'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> 左逼近 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span>。</li><li>细节处理：二分的边界是什么？这需要从计算最终结果的角度来思考。对于二分出的下界 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">lower</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span></span></span></span>，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span>，此时扫描一遍计算出来的操作次数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>k</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">k'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> 一定是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>k</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>≤</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">k'\le k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8879em;vertical-align:-0.136em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span>。也就是说我们还需要操作 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo>−</mo><msup><mi>k</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">k-k'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> 次，显然我们可以将每个数操作后维护到最终的结果，然后执行上述的暴力思路，但是这样仍然会超时，因为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo>−</mo><msup><mi>k</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">k-k'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> 的计算结果最大是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>，此时进行暴力仍然会达到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>。正难则反，我们不妨将上述二分出的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">lower</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span></span></span></span> 减一进行最终答案的计算，这样对应的操作次数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>k</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">k'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> 一定会超过 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span>，并且对于超过 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 的操作，累加到答案的数值一定都是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">lower-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，这一步也是本题最精妙的一步。从上述分析不难发现，二分的下界需要设定为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，因为后续累加答案时会对 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">lower</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span></span></span></span> 进行减一操作；二分的上界需要设定为严格大于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mn>9</mn></msup></mrow><annotation encoding="application/x-tex">10^9</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span></span></span></span> 的数，比如 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mn>9</mn></msup><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">10^9+1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，因为我们需要保证 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">lower-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 后对应的操作次数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>k</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">k'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> 严格大于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span>。</li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\log k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><br><span class="hljs-keyword">using</span> ll = <span class="hljs-type">long</span> <span class="hljs-type">long</span>;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-type">int</span> n, k;<br>    std::cin &gt;&gt; n &gt;&gt; k;<br>    <br>    <span class="hljs-function">std::vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">a</span><span class="hljs-params">(n)</span>, <span class="hljs-title">b</span><span class="hljs-params">(n)</span></span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) std::cin &gt;&gt; a[i];<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) std::cin &gt;&gt; b[i];<br>    <br>    <span class="hljs-keyword">auto</span> chk = [&amp;](<span class="hljs-type">int</span> x) {<br>        ll cnt = <span class="hljs-number">0</span>;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>            <span class="hljs-keyword">if</span> (a[i] &lt; x) <span class="hljs-keyword">continue</span>;<br>            cnt += (a[i] - x) / b[i] + <span class="hljs-number">1</span>;<br>        }<br>        <span class="hljs-keyword">return</span> cnt;<br>    };<br>    <br>    <span class="hljs-type">int</span> l = <span class="hljs-number">1</span>, r = <span class="hljs-number">1e9</span> + <span class="hljs-number">10</span>;<br>    <span class="hljs-keyword">while</span> (l &lt; r) {<br>        <span class="hljs-type">int</span> mid = (l + r) &gt;&gt; <span class="hljs-number">1</span>;<br>        <span class="hljs-keyword">if</span> (<span class="hljs-built_in">chk</span>(mid) &lt;= k) r = mid;<br>        <span class="hljs-keyword">else</span> l = mid + <span class="hljs-number">1</span>;<br>    }<br>    <br>    <span class="hljs-type">int</span> best = r - <span class="hljs-number">1</span>;<br>    ll res = <span class="hljs-number">0</span>, cnt = <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        <span class="hljs-keyword">if</span> (a[i] &lt; best) <span class="hljs-keyword">continue</span>;<br>        ll t = (a[i] - best) / b[i] + <span class="hljs-number">1</span>;<br>        cnt += t;<br>        res += t * a[i] - t * (t - <span class="hljs-number">1</span>) / <span class="hljs-number">2</span> * b[i];<br>    }<br>    res -= (cnt - k) * best;<br>    <br>    std::cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    std::ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    std::cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br>    std::cin &gt;&gt; T;<br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure>]]></content>
+    <content type="html"><![CDATA[<h2 id="数据结构">数据结构</h2><p>数据结构由 <strong>数据</strong> 和 <strong>结构</strong> 两部分组成。我们主要讨论的是后者，即结构部分。</p><p>按照 <strong>逻辑结构</strong> 可以将其区分为 <strong>线性结构</strong> 和 <strong>非线性结构</strong>。</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202408301527018.png" alt="线性数据结构 与 非线性数据结构"></p><p>按照 <strong>物理结构</strong> 可以将其区分为 <strong>连续结构</strong> 和 <strong>分散结构</strong>。</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202408301528541.png" alt="连续空间存储 与 分散空间存储"></p><h2 id="【模板】双链表">【模板】双链表</h2><p><a href="https://www.acwing.com/problem/content/829/">https://www.acwing.com/problem/content/829/</a></p><blockquote><p>思路：用两个空结点作为起始状态的边界，避免所有边界讨论。</p><p>时间复杂度：插入、删除结点均为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span></span></span></span>，遍历为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br><span class="line">82</span><br><span class="line">83</span><br><span class="line">84</span><br><span class="line">85</span><br><span class="line">86</span><br><span class="line">87</span><br><span class="line">88</span><br><span class="line">89</span><br><span class="line">90</span><br><span class="line">91</span><br><span class="line">92</span><br><span class="line">93</span><br><span class="line">94</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><br><span class="hljs-keyword">using</span> ll = <span class="hljs-type">long</span> <span class="hljs-type">long</span>;<br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-keyword">template</span>&lt;<span class="hljs-keyword">class</span> <span class="hljs-title class_">T</span>&gt;<br><span class="hljs-keyword">class</span> <span class="hljs-title class_">myList</span> {<br><span class="hljs-keyword">private</span>:<br>    <span class="hljs-type">int</span> idx;<br>    std::vector&lt;T&gt; val;<br>    std::vector&lt;<span class="hljs-type">int</span>&gt; left, right;<br><br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-built_in">myList</span>(<span class="hljs-type">const</span> <span class="hljs-type">int</span> n) {<br>        idx = <span class="hljs-number">2</span>;<br>        val.<span class="hljs-built_in">resize</span>(n + <span class="hljs-number">10</span>);<br>        left.<span class="hljs-built_in">resize</span>(n + <span class="hljs-number">10</span>);<br>        right.<span class="hljs-built_in">resize</span>(n + <span class="hljs-number">10</span>);<br>        left[<span class="hljs-number">1</span>] = <span class="hljs-number">0</span>, right[<span class="hljs-number">0</span>] = <span class="hljs-number">1</span>;<br>    }<br><br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">push_back</span><span class="hljs-params">(T x)</span> </span>{<br>        <span class="hljs-built_in">insert_left</span>(<span class="hljs-number">1</span>, x);<br>    }<br><br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">push_front</span><span class="hljs-params">(T x)</span> </span>{<br>        <span class="hljs-built_in">insert_right</span>(<span class="hljs-number">0</span>, x);<br>    }<br><br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">insert_left</span><span class="hljs-params">(<span class="hljs-type">int</span> k, T x)</span> </span>{<br>        <span class="hljs-built_in">insert_right</span>(left[k], x);<br>    }<br><br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">insert_right</span><span class="hljs-params">(<span class="hljs-type">int</span> k, T x)</span> </span>{<br>        val[idx] = x;<br>        right[idx] = right[k];<br>        left[right[k]] = idx;<br>        left[idx] = k;<br>        right[k] = idx++;<br>    }<br><br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">erase</span><span class="hljs-params">(<span class="hljs-type">int</span> k)</span> </span>{<br>        right[left[k]] = right[k];<br>        left[right[k]] = left[k];<br>    }<br><br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">output</span><span class="hljs-params">()</span> </span>{<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = right[<span class="hljs-number">0</span>]; i != <span class="hljs-number">1</span>; i = right[i]) {<br>            cout &lt;&lt; val[i] &lt;&lt; <span class="hljs-string">" \n"</span>[i == <span class="hljs-number">1</span>];<br>        }<br>    }<br>};<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-type">int</span> n;<br>    cin &gt;&gt; n;<br><br>    <span class="hljs-function">myList&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">ls</span><span class="hljs-params">(n)</span></span>;<br><br>    <span class="hljs-keyword">while</span> (n--) {<br>        string op;<br>        cin &gt;&gt; op;<br><br>        <span class="hljs-type">int</span> k, x;<br><br>        <span class="hljs-keyword">if</span> (op == <span class="hljs-string">"L"</span>) {<br>            cin &gt;&gt; x;<br>            ls.<span class="hljs-built_in">push_front</span>(x);<br>        } <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (op == <span class="hljs-string">"R"</span>) {<br>            cin &gt;&gt; x;<br>            ls.<span class="hljs-built_in">push_back</span>(x);<br>        } <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (op == <span class="hljs-string">"D"</span>) {<br>            cin &gt;&gt; k;<br>            ls.<span class="hljs-built_in">erase</span>(k + <span class="hljs-number">1</span>);<br>        } <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (op == <span class="hljs-string">"IL"</span>) {<br>            cin &gt;&gt; k &gt;&gt; x;<br>            ls.<span class="hljs-built_in">insert_left</span>(k + <span class="hljs-number">1</span>, x);<br>        } <span class="hljs-keyword">else</span> {<br>            cin &gt;&gt; k &gt;&gt; x;<br>            ls.<span class="hljs-built_in">insert_right</span>(k + <span class="hljs-number">1</span>, x);<br>        }<br>    }<br><br>    ls.<span class="hljs-built_in">output</span>();<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    std::ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    std::cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    std::cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">import</span> heapq<br><span class="hljs-keyword">from</span> collections <span class="hljs-keyword">import</span> defaultdict<br><span class="hljs-keyword">from</span> typing <span class="hljs-keyword">import</span> <span class="hljs-type">List</span>, <span class="hljs-type">Tuple</span><br><span class="hljs-keyword">import</span> math<br><span class="hljs-keyword">from</span> itertools <span class="hljs-keyword">import</span> combinations<br><br>II = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">int</span>(<span class="hljs-built_in">input</span>())<br>FI = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">float</span>(<span class="hljs-built_in">input</span>())<br>MII = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">tuple</span>(<span class="hljs-built_in">map</span>(<span class="hljs-built_in">int</span>, <span class="hljs-built_in">input</span>().split()))<br>LII = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">list</span>(<span class="hljs-built_in">map</span>(<span class="hljs-built_in">int</span>, <span class="hljs-built_in">input</span>().split()))<br><br><br><span class="hljs-keyword">class</span> <span class="hljs-title class_">myList</span>:<br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">__init__</span>(<span class="hljs-params">self, n: <span class="hljs-built_in">int</span></span>) -&gt; <span class="hljs-literal">None</span>:<br>        <span class="hljs-variable language_">self</span>.val = [<span class="hljs-number">0</span>] * (n + <span class="hljs-number">10</span>)<br>        <span class="hljs-variable language_">self</span>.left = [<span class="hljs-number">0</span>] * (n + <span class="hljs-number">10</span>)<br>        <span class="hljs-variable language_">self</span>.right = [<span class="hljs-number">0</span>] * (n + <span class="hljs-number">10</span>)<br>        <span class="hljs-variable language_">self</span>.idx = <span class="hljs-number">2</span><br>        <span class="hljs-variable language_">self</span>.right[<span class="hljs-number">0</span>] = <span class="hljs-number">1</span><br>        <span class="hljs-variable language_">self</span>.left[<span class="hljs-number">1</span>] = <span class="hljs-number">0</span><br><br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">push_front</span>(<span class="hljs-params">self, x: <span class="hljs-built_in">int</span></span>):<br>        <span class="hljs-variable language_">self</span>.insert_right(<span class="hljs-number">0</span>, x)<br><br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">push_back</span>(<span class="hljs-params">self, x: <span class="hljs-built_in">int</span></span>):<br>        <span class="hljs-variable language_">self</span>.insert_left(<span class="hljs-number">1</span>, x)<br><br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">insert_left</span>(<span class="hljs-params">self, k: <span class="hljs-built_in">int</span>, x: <span class="hljs-built_in">int</span></span>):<br>        <span class="hljs-variable language_">self</span>.insert_right(<span class="hljs-variable language_">self</span>.left[k], x)<br><br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">insert_right</span>(<span class="hljs-params">self, k: <span class="hljs-built_in">int</span>, x: <span class="hljs-built_in">int</span></span>):<br>        <span class="hljs-variable language_">self</span>.val[<span class="hljs-variable language_">self</span>.idx] = x<br>        <span class="hljs-variable language_">self</span>.right[<span class="hljs-variable language_">self</span>.idx] = <span class="hljs-variable language_">self</span>.right[k]<br>        <span class="hljs-variable language_">self</span>.left[<span class="hljs-variable language_">self</span>.right[k]] = <span class="hljs-variable language_">self</span>.idx<br>        <span class="hljs-variable language_">self</span>.left[<span class="hljs-variable language_">self</span>.idx] = k<br>        <span class="hljs-variable language_">self</span>.right[k] = <span class="hljs-variable language_">self</span>.idx<br>        <span class="hljs-variable language_">self</span>.idx += <span class="hljs-number">1</span><br><br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">erase</span>(<span class="hljs-params">self, k: <span class="hljs-built_in">int</span></span>):<br>        <span class="hljs-variable language_">self</span>.left[<span class="hljs-variable language_">self</span>.right[k]] = <span class="hljs-variable language_">self</span>.left[k]<br>        <span class="hljs-variable language_">self</span>.right[<span class="hljs-variable language_">self</span>.left[k]] = <span class="hljs-variable language_">self</span>.right[k]<br><br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">output</span>(<span class="hljs-params">self</span>) -&gt; <span class="hljs-literal">None</span>:<br>        i = <span class="hljs-variable language_">self</span>.right[<span class="hljs-number">0</span>]<br>        <span class="hljs-keyword">while</span> i != <span class="hljs-number">1</span>:<br>            <span class="hljs-built_in">print</span>(<span class="hljs-variable language_">self</span>.val[i], end=<span class="hljs-string">' '</span>)<br>            i = <span class="hljs-variable language_">self</span>.right[i]<br><br><br><span class="hljs-keyword">def</span> <span class="hljs-title function_">solve</span>() -&gt; <span class="hljs-literal">None</span>:<br>    n = II()<br><br>    ls = myList(n)<br><br>    <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(n):<br>        op = <span class="hljs-built_in">input</span>().split()<br><br>        <span class="hljs-keyword">if</span> op[<span class="hljs-number">0</span>] == <span class="hljs-string">'L'</span>:<br>            ls.push_front(<span class="hljs-built_in">int</span>(op[-<span class="hljs-number">1</span>]))<br>        <span class="hljs-keyword">elif</span> op[<span class="hljs-number">0</span>] == <span class="hljs-string">'R'</span>:<br>            ls.push_back(<span class="hljs-built_in">int</span>(op[-<span class="hljs-number">1</span>]))<br>        <span class="hljs-keyword">elif</span> op[<span class="hljs-number">0</span>] == <span class="hljs-string">'D'</span>:<br>            ls.erase(<span class="hljs-built_in">int</span>(op[-<span class="hljs-number">1</span>]) + <span class="hljs-number">1</span>)<br>        <span class="hljs-keyword">elif</span> op[<span class="hljs-number">0</span>] == <span class="hljs-string">'IL'</span>:<br>            ls.insert_left(<span class="hljs-built_in">int</span>(op[<span class="hljs-number">1</span>]) + <span class="hljs-number">1</span>, <span class="hljs-built_in">int</span>(op[-<span class="hljs-number">1</span>]))<br>        <span class="hljs-keyword">else</span>:<br>            ls.insert_right(<span class="hljs-built_in">int</span>(op[<span class="hljs-number">1</span>]) + <span class="hljs-number">1</span>, <span class="hljs-built_in">int</span>(op[-<span class="hljs-number">1</span>]))<br><br>    ls.output()<br><br><br><span class="hljs-keyword">if</span> __name__ == <span class="hljs-string">'__main__'</span>:<br>    T = <span class="hljs-number">1</span><br>    <span class="hljs-comment"># T = II()</span><br>    <span class="hljs-keyword">while</span> T: solve(); T -= <span class="hljs-number">1</span><br></code></pre></td></tr></tbody></table></figure><h2 id="【模板】单调栈">【模板】单调栈</h2><p><a href="https://www.acwing.com/problem/content/832/">https://www.acwing.com/problem/content/832/</a></p><blockquote><p>题意：对于一个序列中的每一个元素，寻找每一个元素左侧最近的比其小的元素。</p><p>思路一：暴力枚举</p><ul><li>显然对于每一个元素 <code>nums[i]</code>，我们可以枚举倒序 <code>[0, i-1]</code> 直到找到第一个 <code>nums[j] &lt; nums[i]</code></li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></li></ul><p>思路二：单调栈</p><ul><li><p>可以发现时间开销主要在倒序枚举上，我们能少枚举一些元素吗？答案是可以的。我们定义「寻找 <code>[i-1, 0]</code> 中比当前元素小的第一个元素」的行为叫做「寻找合法对象」。显然我们在枚举每一个元素时都需要 <strong>查询</strong> 和 <strong>维护</strong> 这样的合法对象线性序列，可以理解为记忆化从而加速查询。那么如何高效查询和维护这样的线性序列呢？不妨考虑每一个合法对象对曾经的合法对象的影响：</p><ul><li>若当前元素 <code>nums[i]</code> 可以成为后续 <code>[i+1, n-1]</code> 元素的合法对象。则从 <code>i-1</code> 开始一直往左，只要比当前大的元素，都不可能成为 <code>[i+1, n-1]</code> 的合法对象，肯定都被 <code>nums[i]</code> “拦住了”。那么在「合法对象序列」中插入当前合法对象之前，需要不断尾弹出比当前大的元素</li><li>若当前元素 <code>nums[i]</code> 不能成为后续 <code>[i+1, n-1]</code> 元素的合法对象。表明当前元素过大，此时就不用将当前元素插入「合法对象序列」</li></ul></li><li><p>经过上述两个角度的讨论，很容易发现这样维护出来的的合法序列是严格单调递增的。于是，在查询操作时仅需要进行尾比较与尾弹出即可，在维护操作时，仅需要尾插入即可。满足这样的线性数据结构有很多，如栈、队列、数组、链表，我们就使用栈来演示，与标题遥相呼应</p></li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><br><span class="hljs-keyword">using</span> ll = <span class="hljs-type">long</span> <span class="hljs-type">long</span>;<br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-type">int</span> n;<br>    cin &gt;&gt; n;<br><br>    <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">a</span><span class="hljs-params">(n)</span></span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        cin &gt;&gt; a[i];<br>    }<br><br>    stack&lt;<span class="hljs-type">int</span>&gt; s;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        <span class="hljs-comment">// 查询</span><br>        <span class="hljs-keyword">while</span> (s.<span class="hljs-built_in">size</span>() &amp;&amp; s.<span class="hljs-built_in">top</span>() &gt;= a[i]) {<br>            s.<span class="hljs-built_in">pop</span>();<br>        }<br>        cout &lt;&lt; (s.<span class="hljs-built_in">size</span>() ? s.<span class="hljs-built_in">top</span>() : <span class="hljs-number">-1</span>) &lt;&lt; <span class="hljs-string">" "</span>;<br>        <br>        <span class="hljs-comment">// 维护</span><br>        s.<span class="hljs-built_in">push</span>(a[i]);<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    std::ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    std::cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    std::cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><p>相似题：</p><p><a href="https://leetcode.cn/problems/next-greater-element-ii/">下一个更大元素 II</a></p><h2 id="【模板】单调队列">【模板】单调队列</h2><blockquote><p>标签：双端队列、滑动窗口、单调队列</p><p>题意：给定一个含有 n 个元素的序列，求解其中每个长度为 k 的子数组中的最值。</p><p>思路：显然我们可以 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(nk)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">nk</span><span class="mclose">)</span></span></span></span> 暴力求解，有没有什么方法可以将「求解子数组中最值」的时间开销从 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span> 降低为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 呢？有的！我们重新定义一个队列就好了。为了做到线性时间复杂度的优化，我们对队列做以下自定义，以「求解子数组最小值」为例：</p><ol><li>插入元素到队尾：此时和单调栈的逻辑类似。如果当前元素可以作为当前子数组或后续子数组的最小值，则需要从当前队尾开始依次弹出比当前元素严格大的元素，最后再将当前元素入队。<strong>注意</strong>：当遇到和当前元素值相等的元素时不能出队，因为每一个元素都会经历入队和出队的操作，一旦此时出队了，后续进行出队判定时会提前弹出本不应该出队的与其等值的元素。</li><li>弹出队头元素：如果队头元素和子数组左端点 <code>nums[i-k]</code> 的元素值相等，则弹出。</li><li>获得队头元素：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 的获取队头元素，即队列中的最小值。</li></ol><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br><span class="line">82</span><br><span class="line">83</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><br><span class="hljs-keyword">using</span> ll = <span class="hljs-type">long</span> <span class="hljs-type">long</span>;<br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-keyword">template</span>&lt;<span class="hljs-keyword">class</span> <span class="hljs-title class_">T</span>&gt;<br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">minQueue</span> {<br>    std::deque&lt;T&gt; q;<br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">pushBack</span><span class="hljs-params">(T x)</span> </span>{<br>        <span class="hljs-keyword">while</span> (q.<span class="hljs-built_in">size</span>() &amp;&amp; x &lt; q.<span class="hljs-built_in">back</span>()) {<br>            q.<span class="hljs-built_in">pop_back</span>();<br>        }<br>        q.<span class="hljs-built_in">push_back</span>(x);<br>    }<br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">popFront</span><span class="hljs-params">(T x)</span> </span>{<br>        <span class="hljs-keyword">if</span> (q.<span class="hljs-built_in">size</span>() &amp;&amp; q.<span class="hljs-built_in">front</span>() == x) {<br>            q.<span class="hljs-built_in">pop_front</span>();<br>        }<br>    }<br>    <span class="hljs-function">T <span class="hljs-title">getMinValue</span><span class="hljs-params">()</span> </span>{<br>        <span class="hljs-keyword">return</span> q.<span class="hljs-built_in">front</span>();<br>    }<br>};<br><br><span class="hljs-keyword">template</span>&lt;<span class="hljs-keyword">class</span> <span class="hljs-title class_">T</span>&gt;<br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">maxQueue</span> {<br>    std::deque&lt;T&gt; q;<br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">pushBack</span><span class="hljs-params">(T x)</span> </span>{<br>        <span class="hljs-keyword">while</span> (q.<span class="hljs-built_in">size</span>() &amp;&amp; x &gt; q.<span class="hljs-built_in">back</span>()) {<br>            q.<span class="hljs-built_in">pop_back</span>();<br>        }<br>        q.<span class="hljs-built_in">push_back</span>(x);<br>    }<br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">popFront</span><span class="hljs-params">(T x)</span> </span>{<br>        <span class="hljs-keyword">if</span> (q.<span class="hljs-built_in">size</span>() &amp;&amp; q.<span class="hljs-built_in">front</span>() == x) {<br>            q.<span class="hljs-built_in">pop_front</span>();<br>        }<br>    }<br>    <span class="hljs-function">T <span class="hljs-title">getMaxValue</span><span class="hljs-params">()</span> </span>{<br>        <span class="hljs-keyword">return</span> q.<span class="hljs-built_in">front</span>();<br>    }<br>};<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-type">int</span> n, k;<br>    cin &gt;&gt; n &gt;&gt; k;<br>    <br>    <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">nums</span><span class="hljs-params">(n)</span></span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        cin &gt;&gt; nums[i];<br>    }<br>    <br>    minQueue&lt;<span class="hljs-type">int</span>&gt; minq;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        minq.<span class="hljs-built_in">pushBack</span>(nums[i]);<br>        <span class="hljs-keyword">if</span> (i &gt;= k) {<br>            minq.<span class="hljs-built_in">popFront</span>(nums[i - k]);<br>        }<br>        <span class="hljs-keyword">if</span> (i &gt;= k - <span class="hljs-number">1</span>) {<br>            cout &lt;&lt; minq.<span class="hljs-built_in">getMinValue</span>() &lt;&lt; <span class="hljs-string">" \n"</span>[i == n - <span class="hljs-number">1</span>];<br>        }<br>    }<br>    <br>    maxQueue&lt;<span class="hljs-type">int</span>&gt; maxq;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        maxq.<span class="hljs-built_in">pushBack</span>(nums[i]);<br>        <span class="hljs-keyword">if</span> (i &gt;= k) {<br>            maxq.<span class="hljs-built_in">popFront</span>(nums[i - k]);<br>        }<br>        <span class="hljs-keyword">if</span> (i &gt;= k - <span class="hljs-number">1</span>) {<br>            cout &lt;&lt; maxq.<span class="hljs-built_in">getMaxValue</span>() &lt;&lt; <span class="hljs-string">" \n"</span>[i == n - <span class="hljs-number">1</span>];<br>        }<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    std::ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    std::cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    std::cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">from</span> collections <span class="hljs-keyword">import</span> defaultdict, deque<br><span class="hljs-keyword">from</span> typing <span class="hljs-keyword">import</span> <span class="hljs-type">List</span>, <span class="hljs-type">Tuple</span><br><span class="hljs-keyword">from</span> itertools <span class="hljs-keyword">import</span> combinations, permutations<br><span class="hljs-keyword">import</span> math, heapq, queue<br><br>II = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">int</span>(<span class="hljs-built_in">input</span>())<br>FI = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">float</span>(<span class="hljs-built_in">input</span>())<br>MII = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">tuple</span>(<span class="hljs-built_in">map</span>(<span class="hljs-built_in">int</span>, <span class="hljs-built_in">input</span>().split()))<br>LII = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">list</span>(<span class="hljs-built_in">map</span>(<span class="hljs-built_in">int</span>, <span class="hljs-built_in">input</span>().split()))<br><br><br><span class="hljs-keyword">def</span> <span class="hljs-title function_">solve</span>() -&gt; <span class="hljs-literal">None</span>:<br>    n, k = MII()<br>    nums = LII()<br>    <br>    qa, qb = deque(), deque()<br>    ra, rb = [], []<br>    <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(n):<br>        <span class="hljs-comment"># push back</span><br>        <span class="hljs-keyword">while</span> <span class="hljs-built_in">len</span>(qa) <span class="hljs-keyword">and</span> nums[i] &lt; qa[-<span class="hljs-number">1</span>]:<br>            qa.pop()<br>        qa.append(nums[i])<br>        <span class="hljs-keyword">while</span> <span class="hljs-built_in">len</span>(qb) <span class="hljs-keyword">and</span> nums[i] &gt; qb[-<span class="hljs-number">1</span>]:<br>            qb.pop()<br>        qb.append(nums[i])<br>        <span class="hljs-keyword">if</span> i &gt;= k:<br>            <span class="hljs-comment"># pop front</span><br>            <span class="hljs-keyword">if</span> <span class="hljs-built_in">len</span>(qa) <span class="hljs-keyword">and</span> qa[<span class="hljs-number">0</span>] == nums[i - k]:<br>                qa.popleft()<br>            <span class="hljs-keyword">if</span> <span class="hljs-built_in">len</span>(qb) <span class="hljs-keyword">and</span> qb[<span class="hljs-number">0</span>] == nums[i - k]:<br>                qb.popleft()<br>        <span class="hljs-keyword">if</span> i &gt;= k - <span class="hljs-number">1</span>:<br>            <span class="hljs-comment"># get ans</span><br>            ra.append(qa[<span class="hljs-number">0</span>])<br>            rb.append(qb[<span class="hljs-number">0</span>])<br>    <br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">' '</span>.join(<span class="hljs-built_in">map</span>(<span class="hljs-built_in">str</span>, ra)))<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">' '</span>.join(<span class="hljs-built_in">map</span>(<span class="hljs-built_in">str</span>, rb)))<br><br><br><span class="hljs-keyword">if</span> __name__ == <span class="hljs-string">'__main__'</span>:<br>    T = <span class="hljs-number">1</span><br>    <span class="hljs-comment"># T = II()</span><br>    <span class="hljs-keyword">while</span> T: solve(); T -= <span class="hljs-number">1</span><br></code></pre></td></tr></tbody></table></figure><h2 id="【模板】最近公共祖先">【模板】最近公共祖先</h2><p><a href="https://www.luogu.com.cn/problem/P3379">https://www.luogu.com.cn/problem/P3379</a></p><blockquote><p>题意：寻找树中指定两个结点的最近公共祖先 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>(Lowest&nbsp;Common&nbsp;Ancestor,&nbsp;简称&nbsp;LCA)</mtext></mrow><annotation encoding="application/x-tex">\text{(Lowest Common Ancestor, 简称 LCA)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">(Lowest&nbsp;Common&nbsp;Ancestor,&nbsp;</span><span class="mord cjk_fallback">简称</span><span class="mord">&nbsp;LCA)</span></span></span></span></span>。</p><p>思路：对于每次查询，我们可以从指定的两个结点开始往上跳，第一个公共结点就是目标的 LCA，每一次询问的时间复杂度均为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>，为了加速查询，我们可以采用倍增法，预处理出往上跳的结果，即 <code>fa[i][j]</code> 数组，表示 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 号点向上跳 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>j</mi></msup></mrow><annotation encoding="application/x-tex">2^j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8247em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span></span></span></span></span></span></span> 步后到达的结点。接下来在往上跳跃的过程中，利用二进制拼凑的思路，即可在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 的时间内查询到 LCA。</p><p>预处理：可以发现，对于 <code>fa[i][j]</code>，我们可以通过递推的方式获得，即 <code>fa[i][j] = fa[fa[i][j-1]][j-1]</code>，当前结点向上跳跃 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>j</mi></msup></mrow><annotation encoding="application/x-tex">2^j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8247em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span></span></span></span></span></span></span> 步可以拆分为先向上 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">2^{j-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8247em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span> 步, 在此基础之上再向上 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">2^{j-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8247em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span> 步.于是我们可以采用宽搜 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>o</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">or</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">or</span></span></span></span> 深搜的顺序维护 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mi>a</mi></mrow><annotation encoding="application/x-tex">fa</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mord mathnormal">a</span></span></span></span> 数组。</p><p>跳跃：我们首先需要将两个结点按照倍增的思路向上跳到同一个深度，接下来两个结点同时按照倍增的思路向上跳跃，为了确保求出最近的，我们需要确保在跳跃的步调一致的情况下，两者的祖先始终不相同，那么倍增结束后，两者的父结点就是最近公共祖先，即 <code>fa[x][k]</code> 或 <code>fa[y][k]</code></p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Θ</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>+</mo><mi>m</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta(n \log n + m \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Θ</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">n \log n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span></span></span></span> 为预处理每一个结点向上跳跃抵达的情况</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mi>log</mi><mo>⁡</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">m \log n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span></span></span></span> 为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 次询问的情况</li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">5e5</span> + <span class="hljs-number">10</span>;<br><br><span class="hljs-type">int</span> n, Q, root;<br>vector&lt;<span class="hljs-type">int</span>&gt; G[N];<br><span class="hljs-type">int</span> fa[N][<span class="hljs-number">20</span>], dep[N];<br>queue&lt;<span class="hljs-type">int</span>&gt; q;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">init</span><span class="hljs-params">()</span> </span>{<br>    dep[root] = <span class="hljs-number">1</span>;<br>    q.<span class="hljs-built_in">push</span>(root);<br><br>    <span class="hljs-keyword">while</span> (q.<span class="hljs-built_in">size</span>()) {<br>        <span class="hljs-type">int</span> now = q.<span class="hljs-built_in">front</span>();<br>        q.<span class="hljs-built_in">pop</span>();<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> ch: G[now]) {<br>            <span class="hljs-keyword">if</span> (!dep[ch]) {<br>                dep[ch] = dep[now] + <span class="hljs-number">1</span>;<br>                fa[ch][<span class="hljs-number">0</span>] = now;<br>                <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> k = <span class="hljs-number">1</span>; k &lt;= <span class="hljs-number">19</span>; k++) {<br>                    fa[ch][k] = fa[ fa[ch][k<span class="hljs-number">-1</span>] ][k<span class="hljs-number">-1</span>];<br>                }<br>                q.<span class="hljs-built_in">push</span>(ch);<br>            }<br>        }<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">lca</span><span class="hljs-params">(<span class="hljs-type">int</span> a, <span class="hljs-type">int</span> b)</span> </span>{<br>    <span class="hljs-keyword">if</span> (dep[a] &lt; dep[b]) <span class="hljs-built_in">swap</span>(a, b);<br><br>    <span class="hljs-comment">// 二进制拼凑从而跳到一样高</span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> k = <span class="hljs-number">19</span>; k &gt;= <span class="hljs-number">0</span>; k--)<br>        <span class="hljs-keyword">if</span> (dep[fa[a][k]] &gt;= dep[b])<br>            a = fa[a][k];<br><br>    <span class="hljs-keyword">if</span> (a == b) <span class="hljs-keyword">return</span> a;<br><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> k = <span class="hljs-number">19</span>; k &gt;= <span class="hljs-number">0</span>; k--)<br>        <span class="hljs-keyword">if</span> (fa[a][k] != fa[b][k])<br>            a = fa[a][k], b = fa[b][k];<br><br>    <span class="hljs-keyword">return</span> fa[a][<span class="hljs-number">0</span>];<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; Q &gt;&gt; root;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n - <span class="hljs-number">1</span>; ++i) {<br>        <span class="hljs-type">int</span> a, b;<br>        cin &gt;&gt; a &gt;&gt; b;<br>        G[a].<span class="hljs-built_in">push_back</span>(b);<br>        G[b].<span class="hljs-built_in">push_back</span>(a);<br>    }<br><br>    <span class="hljs-built_in">init</span>();<br><br>    <span class="hljs-keyword">while</span> (Q--) {<br>        <span class="hljs-type">int</span> a, b;<br>        cin &gt;&gt; a &gt;&gt; b;<br>        cout &lt;&lt; <span class="hljs-built_in">lca</span>(a, b) &lt;&lt; <span class="hljs-string">"\n"</span>;<br>    }<br>}    <br></code></pre></td></tr></tbody></table></figure><h2 id="【栈】验证栈序列">【栈】验证栈序列</h2><p><a href="https://www.luogu.com.cn/problem/P4387">https://www.luogu.com.cn/problem/P4387</a></p><blockquote><ul><li><p>题意：给定入栈序列与出栈序列，问出栈序列是否合法</p></li><li><p>思路：思路很简单，就是对于当前出栈的数，和入栈序列中最后已出栈的数之间，如果还有数没有出，那么就是不合法的出栈序列，反之合法。这是从入栈的结果来看的，如果这么判断就需要扫描入栈序列 n 次，时间复杂度为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>。我们按照入栈的顺序来看，对于当前待入栈的数，若与出栈序列的队头不等，则成功入栈等待后续出栈；若与出栈序列相等，则匹配成功直接出栈无需入栈，同时对已入栈的数与出栈序列队头不断匹配直到不相等。最后判断待入栈的数与出栈序列是否全部匹配掉了，如果全部匹配掉了说明该出栈序列合法，反之不合法</p><p>抽象总结上述思路：为了判断出栈序列是否合法，我们不妨思考：对于每一个出栈的数，出栈的时机是什么？可以发现出栈的时机无非两种：</p><ul><li>一入栈就出栈（对应于枚举待入栈序列时发现待入栈的数与出栈序列队头相等）</li><li>紧跟着刚出栈的数继续出栈（对应于枚举待入栈序列时发现待入栈的数与出栈序列队头相等之后，继续判断出栈序列队头与已入栈的数是否相等，若相等则不断判断并出栈）</li></ul></li><li><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-comment">// #include &lt;bits/stdc++.h&gt;</span><br><span class="hljs-comment">// #define int long long</span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;unordered_map&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;stack&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;queue&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-type">int</span> n;<br>    cin &gt;&gt; n;<br>    <br>    <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">a</span><span class="hljs-params">(n)</span>, <span class="hljs-title">b</span><span class="hljs-params">(n)</span></span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) cin &gt;&gt; a[i];<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) cin &gt;&gt; b[i];<br>    <br>    stack&lt;<span class="hljs-type">int</span>&gt; stk;<br>    <span class="hljs-type">int</span> i = <span class="hljs-number">0</span>, j = <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">while</span> (i &lt; n) {<br>        <span class="hljs-keyword">if</span> (a[i] != b[j]) stk.<span class="hljs-built_in">push</span>(a[i++]);<br>        <span class="hljs-keyword">else</span> {<br>            i++, j++;<br>            <span class="hljs-keyword">while</span> (!stk.<span class="hljs-built_in">empty</span>() &amp;&amp; b[j] == stk.<span class="hljs-built_in">top</span>()) {<br>                stk.<span class="hljs-built_in">pop</span>();<br>                j++;<br>            }<br>        }<br>    }<br>    <br>    cout &lt;&lt; (stk.<span class="hljs-built_in">empty</span>() ? <span class="hljs-string">"Yes"</span> : <span class="hljs-string">"No"</span>) &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br>    cin &gt;&gt; T;<br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h2 id="二叉搜索树-平衡树">二叉搜索树 &amp; 平衡树</h2><p>如果我们想要在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 时间复杂度内对数据进行增删查改的操作，就可以引入 <strong>二叉搜索树 (Binary Search Tree)</strong> 这一数据结构。然而，在某些极端的情况下，例如当插入的数据是单调不减或不增时，这棵树就会退化为一条链从而导致所有的增删查改操作退化到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>，这是我们不愿意看到的。因此我们引入 <strong>平衡二叉搜索树 (Balanced Binary Search Tree)</strong> 这一数据结构。</p><p>关于平衡二叉搜索树，有非常多的变种与实现，不同的应用场景会选择不同的平衡树类型。<strong>Treap</strong> 更灵活，通过随机化优先级实现预期的平衡，但在最坏情况下可能退化。<strong>AVL 树</strong> 严格保持平衡，保证了 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 的性能，但在频繁插入和删除的场景下可能有较大的旋转开销。<strong>红黑树</strong> 通过较宽松的平衡条件实现了较好的插入和删除性能，通常被广泛用于需要高效插入删除操作的系统（如 STL 中的 <code>map</code> 和 <code>set</code>）。一般来说，红黑树是一个较为通用的选择，而在需要严格平衡性时，AVL 树可能是更好的选择。</p><h3 id="Treap">Treap</h3><blockquote><p>Treap 是二叉搜索树和堆的结合体。它通过维护两种性质来保持平衡：</p><ul><li><strong>二叉搜索树性质</strong>：每个节点的左子树的所有节点值小于该节点的值，右子树的所有节点值大于该节点的值。</li><li><strong>堆性质</strong>：每个节点的优先级（通常随机生成）要大于或等于其子节点的优先级。</li></ul><p><strong>平衡机制</strong>：</p><ul><li>Treap 使用随机化优先级使得树的形状接近于理想的平衡树（期望树高为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>）。</li><li>通过旋转操作（左旋和右旋）在插入和删除时保持堆的性质。</li></ul><p><strong>优点</strong>：</p><ul><li>实现相对简单。</li><li>由于随机化的优先级，在期望情况下，树的高度是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>。</li><li>灵活性高，可以根据需要调整优先级函数。</li></ul><p><strong>缺点</strong>：</p><ul><li>最坏情况下，树的高度可能退化为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>（例如所有优先级相同或顺序生成的优先级），尽管发生概率很低。</li></ul></blockquote><h3 id="AVL-树">AVL 树</h3><blockquote><p>AVL 树是最早发明的自平衡二叉搜索树之一，1962 年由 Adelson-Velsky 和 Landis 发明。</p><ul><li><strong>平衡因子</strong>：每个节点的左右子树高度差不能超过 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，且需要记录每个节点的高度。</li></ul><p><strong>平衡机制</strong>：</p><ul><li>插入或删除节点后，如果某个节点的平衡因子不再为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 或 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，就需要通过旋转（单旋转或双旋转）来恢复平衡。</li><li>旋转操作包括：左旋转、右旋转、左右双旋转和右左双旋转。</li></ul><p><strong>优点</strong>：</p><ul><li>严格的平衡条件保证了树的高度始终为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>，因此搜索、插入和删除操作的时间复杂度为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>。</li></ul><p><strong>缺点</strong>：</p><ul><li>由于平衡条件严格，每次插入和删除后可能需要较多的旋转操作，从而导致实现较复杂，插入和删除操作的常数时间开销较大。</li></ul></blockquote><h3 id="红黑树">红黑树</h3><blockquote><p>红黑树是一种较为宽松的自平衡二叉搜索树，由 Rudolf Bayer 于 1972 年发明。</p><ul><li><strong>颜色属性</strong>：每个节点都有红色或黑色两种颜色，通过这些颜色约束树的平衡性。</li></ul><p><strong>平衡机制</strong>：</p><ul><li>通过遵循红黑树的五个性质来保持平衡：<ol><li>每个节点要么是红色，要么是黑色。</li><li>根节点是黑色。</li><li>叶子节点（NIL 节点）是黑色。</li><li>如果一个节点是红色的，那么它的子节点必须是黑色（红节点不能连续出现）。</li><li>从任一节点到其每个叶子节点的所有路径都包含相同数量的黑色节点。</li></ol></li><li>插入和删除操作可能破坏红黑树的性质，需要通过重新着色和旋转来恢复平衡。</li></ul><p><strong>优点</strong>：</p><ul><li>红黑树的高度最多是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Θ</mi><mo stretchy="false">(</mo><mn>2</mn><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta (2  \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Θ</span><span class="mopen">(</span><span class="mord">2</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>，因此搜索、插入和删除操作的时间复杂度仍为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>。</li><li>由于平衡条件较为宽松，插入和删除操作需要的旋转操作通常比 AVL 树少，效率更高。</li></ul><p><strong>缺点</strong>：</p><ul><li>实现较复杂，特别是插入和删除的平衡修复过程。</li><li>虽然红黑树的搜索效率与 AVL 树相似，但由于平衡条件较宽松，实际应用中的树高度通常略高于 AVL 树，因此搜索操作的效率稍低。</li></ul></blockquote><h2 id="【二叉树】美国血统">【二叉树】美国血统</h2><p><a href="https://www.luogu.com.cn/problem/P1827">https://www.luogu.com.cn/problem/P1827</a></p><blockquote><p>题意：给定二叉树的中序和先序序列，输出后序序列</p><p>思路：经典二叉树的题目，主要用于巩固加强对于递归的理解。指针其实是没有必要的，为了得到后序序列，我们只需要有一个 dfs 序即可，为了得到 dfs 序，我们只需要根据给出的中序和前序序列即可得到 dfs 序</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><p>指针做法</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">30</span>;<br><br>string mid, pre;<br><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">Node</span> {<br>    <span class="hljs-type">char</span> data;<br>    Node* le, * ri;<br>    <span class="hljs-built_in">Node</span>(<span class="hljs-type">char</span> _data) : <span class="hljs-built_in">data</span>(_data), <span class="hljs-built_in">le</span>(<span class="hljs-literal">nullptr</span>), <span class="hljs-built_in">ri</span>(<span class="hljs-literal">nullptr</span>) {}<br>};<br><br><span class="hljs-function">Node* <span class="hljs-title">build</span><span class="hljs-params">(<span class="hljs-type">int</span> i, <span class="hljs-type">int</span> j, <span class="hljs-type">int</span> p, <span class="hljs-type">int</span> q)</span> </span>{<br>    <span class="hljs-keyword">if</span> (i &gt; j) <span class="hljs-keyword">return</span> <span class="hljs-literal">nullptr</span>;<br>    <br>    Node* root = <span class="hljs-keyword">new</span> <span class="hljs-built_in">Node</span>(pre[i]);<br>    <br>    <span class="hljs-type">int</span> k; <span class="hljs-comment">// 根结点在中序序列的下标 </span><br>    <span class="hljs-keyword">for</span> (k = p; k &lt;= q; k++)<br>        <span class="hljs-keyword">if</span> (mid[k] == root-&gt;data)<br>            <span class="hljs-keyword">break</span>;<br>    <br>    root-&gt;le = <span class="hljs-built_in">build</span>(i + <span class="hljs-number">1</span>, k - p + i, p, k - <span class="hljs-number">1</span>);<br>    root-&gt;ri = <span class="hljs-built_in">build</span>(k - p + i + <span class="hljs-number">1</span>, j, k + <span class="hljs-number">1</span>, q);<br>    <br>    cout &lt;&lt; root-&gt;data; <br>    <br>    <span class="hljs-keyword">return</span> root;<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; mid &gt;&gt; pre;<br>    <br>    <span class="hljs-type">int</span> i = <span class="hljs-number">0</span>, j = pre.<span class="hljs-built_in">size</span>() - <span class="hljs-number">1</span>;<br>    <span class="hljs-type">int</span> p = <span class="hljs-number">0</span>, q = mid.<span class="hljs-built_in">size</span>() - <span class="hljs-number">1</span>;<br>    <br>    <span class="hljs-built_in">build</span>(i, j, p, q);<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><p>构造出 dfs 序直接输出</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">30</span>;<br><br>string mid, pre;<br><br><span class="hljs-comment">// 前序起始 i，前序末尾 j，中序起始 p，中序末尾 q </span><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">build</span><span class="hljs-params">(<span class="hljs-type">int</span> i, <span class="hljs-type">int</span> j, <span class="hljs-type">int</span> p, <span class="hljs-type">int</span> q)</span> </span>{<br>    <span class="hljs-keyword">if</span> (i &gt; j) <span class="hljs-keyword">return</span>;<br>    <br>    <span class="hljs-type">char</span> root = pre[i];<br>    <br>    <span class="hljs-type">int</span> k;<br>    <span class="hljs-keyword">for</span> (k = p; k &lt;= q; k++)<br>        <span class="hljs-keyword">if</span> (mid[k] == root)<br>            <span class="hljs-keyword">break</span>;<br>            <br>    <span class="hljs-built_in">build</span>(i + <span class="hljs-number">1</span>, k - p + i, p, k - <span class="hljs-number">1</span>);<br>    <span class="hljs-built_in">build</span>(k - p + i + <span class="hljs-number">1</span>, j, k + <span class="hljs-number">1</span>, q);<br>    <br>    cout &lt;&lt; root;<br>} <br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; mid &gt;&gt; pre;<br>    <br>    <span class="hljs-type">int</span> i = <span class="hljs-number">0</span>, j = pre.<span class="hljs-built_in">size</span>() - <span class="hljs-number">1</span>;<br>    <span class="hljs-type">int</span> p = <span class="hljs-number">0</span>, q = mid.<span class="hljs-built_in">size</span>() - <span class="hljs-number">1</span>;<br>    <br>    <span class="hljs-built_in">build</span>(i, j, p, q);<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h2 id="【二叉树】新二叉树">【二叉树】新二叉树</h2><p><a href="https://www.luogu.com.cn/problem/P1305">https://www.luogu.com.cn/problem/P1305</a></p><blockquote><p>题意：给定一棵二叉树的 n 个结点信息，分别为当前结点的数据信息、左孩子结点信息和右结点信息，输出这棵二叉树的前序序列</p><p>思路：我们首先将这棵二叉树构建出来，接着遍历输出前序序列即可。关键在于如何构建二叉树？我们使用数组存储二叉树，对于每一个树上结点，我们将数组中元素的索引存储为树上结点信息，每一个结点再存储左孩子与右孩子的信息</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">int</span> n;<br>string s;<br><span class="hljs-type">char</span> root;<br><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">Node</span> {<br>    <span class="hljs-type">char</span> l, r;<br>} tree[<span class="hljs-number">200</span>];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">pre</span><span class="hljs-params">(<span class="hljs-type">char</span> now)</span> </span>{<br>    <span class="hljs-keyword">if</span> (now == <span class="hljs-string">'*'</span>) <span class="hljs-keyword">return</span>;<br>    cout &lt;&lt; now;<br>    <span class="hljs-built_in">pre</span>(tree[now].l);<br>    <span class="hljs-built_in">pre</span>(tree[now].r);<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        cin &gt;&gt; s;<br>        <span class="hljs-keyword">if</span> (i == <span class="hljs-number">1</span>) root = s[<span class="hljs-number">0</span>];<br>        tree[s[<span class="hljs-number">0</span>]].l = s[<span class="hljs-number">1</span>];<br>        tree[s[<span class="hljs-number">0</span>]].r = s[<span class="hljs-number">2</span>];<br>    }<br>    <br>    <span class="hljs-built_in">pre</span>(root);<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h2 id="【二叉树】遍历问题">【二叉树】遍历问题</h2><p><a href="https://www.luogu.com.cn/problem/P1229">https://www.luogu.com.cn/problem/P1229</a></p><blockquote><p>题意：给定一棵二叉树的前序序列与后序序列，问中序序列的可能情况有多少种</p><p>思路：我们采用从最小结构单元的思路进行考虑，即假设当前二叉树只有一个根结点与两个叶子结点，而非两棵子树。然后将题意进行等价变换，即问对于已经固定的前序和后序二叉树，该二叉树有多少种不同的形状？对于当前的最小结构二叉树，形状就是 <strong>左右根</strong> or <strong>根左右</strong>，现在的根可以直接确定，那么就只能从左右孩子进行变形，很显然只能进行左右交换的变形，但是问题是一旦左右变换，前序 or 后序都会变掉，说明这种左右孩子都存在的前后序固定的二叉树是唯一的，那么如何才是不唯一的呢？我们考虑减少孩子数量。假设没有孩子，那么很显然也只有一个形状，就是一个根结点，故排除。于是答案就呼之欲出了，就是当根结点只有一个孩子时，这个孩子无论是在左边还是右边，前后序都是相同的，但是中序序列就不同了，于是就产生了两种中序序列。于是最终的结论是：对于前后序确定的二叉树来说，中序序列的情况是就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mtext>单分支结点数</mtext></msup></mrow><annotation encoding="application/x-tex">2^{\text{单分支结点数}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8413em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">单分支结点数</span></span></span></span></span></span></span></span></span></span></span></span></span> 个。现在的问题就转变为了在给定前后序的二叉树中求解单分支结点个数的问题。</p><p>如何寻找单分支结点呢？根据下面的递归图可以发现，无论是左单分支还是右单分支，如果 pre 的连续两个结点与 post 的连续两个结点对称相同，那么就一定有一个单分支结点，故只需要寻找前后序序列中连续两个字符对称相同的情况数 cnt 即可。最终的答案数就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mrow><mi>c</mi><mi>n</mi><mi>t</mi></mrow></msup></mrow><annotation encoding="application/x-tex">2^{cnt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7936em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">c</span><span class="mord mathnormal mtight">n</span><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span></span></span></span></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406061327025.png" alt="图例"></p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(nm)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">nm</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br>string pre, post;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; pre &gt;&gt; post;<br>    <br>    <span class="hljs-type">int</span> cnt = <span class="hljs-number">0</span>;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; pre.<span class="hljs-built_in">size</span>() - <span class="hljs-number">1</span>; i++)<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">0</span>; j &lt; post.<span class="hljs-built_in">size</span>(); j++)<br>            <span class="hljs-keyword">if</span> (pre[i] == post[j + <span class="hljs-number">1</span>] &amp;&amp; pre[i + <span class="hljs-number">1</span>] == post[j])<br>                cnt++;<br>    <br>    cout &lt;&lt; (<span class="hljs-number">1</span> &lt;&lt; cnt) &lt;&lt; <span class="hljs-string">"\n"</span>;<br>} <br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h2 id="【二叉树】医院设置-fire">【二叉树】医院设置 <span class="github-emoji"><span>🔥</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/1f525.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span></h2><p><a href="https://www.luogu.com.cn/problem/P1364">https://www.luogu.com.cn/problem/P1364</a></p><blockquote><p>题意：给定一棵二叉树，树中每一个结点存储了一个数值表示一个医院的人数，现在需要在所有的结点中将一个结点设置为医院使得其余结点中的所有人到达该医院走的路总和最小。路程为结点到医院的最短路，边权均为 1。给出最终的最短路径总和</p><p>思路一：暴力</p><ul><li><p>显然的对于已经设置好医院的局面，需要求解的路径总和就直接将树遍历一边即可。每一个结点都可以作为医院进行枚举，每次遍历是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 的</p></li><li><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></p></li></ul><p>思路二：带权树的重心</p><ul><li>TODO</li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li></ul></blockquote><p>暴力代码</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">110</span>;<br><br><span class="hljs-type">int</span> n;<br><br>vector&lt;<span class="hljs-type">int</span>&gt; G[N];<br><span class="hljs-type">int</span> cnt[N];<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">bfs</span><span class="hljs-params">(<span class="hljs-type">int</span> v)</span> </span>{<br>    <span class="hljs-type">int</span> res = <span class="hljs-number">0</span>;<br>    <span class="hljs-function">vector&lt;<span class="hljs-type">bool</span>&gt; <span class="hljs-title">vis</span><span class="hljs-params">(n + <span class="hljs-number">1</span>, <span class="hljs-literal">false</span>)</span></span>;<br>    <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">d</span><span class="hljs-params">(n + <span class="hljs-number">1</span>, <span class="hljs-number">0</span>)</span></span>; <span class="hljs-comment">// d[i] 表示点 i 到点 v 的距离</span><br>        <br>    queue&lt;<span class="hljs-type">int</span>&gt; q;<br>    vis[v] = <span class="hljs-literal">true</span>;<br>    d[v] = <span class="hljs-number">0</span>;<br>    q.<span class="hljs-built_in">push</span>(v);<br>    <br>    <span class="hljs-keyword">while</span> (q.<span class="hljs-built_in">size</span>()) {<br>        <span class="hljs-type">int</span> now = q.<span class="hljs-built_in">front</span>();<br>        q.<span class="hljs-built_in">pop</span>();<br>        <br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; ch: G[now]) {<br>            <span class="hljs-keyword">if</span> (!vis[ch]) {<br>                vis[ch] = <span class="hljs-literal">true</span>;<br>                d[ch] = d[now] + <span class="hljs-number">1</span>;<br>                q.<span class="hljs-built_in">push</span>(ch);<br>                <br>                res += cnt[ch] * d[ch]; <br>            }<br>        }<br>    }<br>    <br>    <span class="hljs-keyword">return</span> res;<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-type">int</span> count, l, r;<br>        cin &gt;&gt; count &gt;&gt; l &gt;&gt; r;<br>        cnt[i] = count;<br>        <br>        <span class="hljs-keyword">if</span> (l) {<br>            G[i].<span class="hljs-built_in">push_back</span>(l);<br>            G[l].<span class="hljs-built_in">push_back</span>(i);<br>        }<br>        <br>        <span class="hljs-keyword">if</span> (r) {<br>            G[i].<span class="hljs-built_in">push_back</span>(r);<br>            G[r].<span class="hljs-built_in">push_back</span>(i);<br>        }<br>    }<br>    <br>    <span class="hljs-type">int</span> res = <span class="hljs-number">1e7</span> + <span class="hljs-number">10</span>;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        res = <span class="hljs-built_in">min</span>(res, <span class="hljs-built_in">bfs</span>(i));<br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>} <br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><p>优化代码</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><br></code></pre></td></tr></tbody></table></figure><h2 id="【二叉树】二叉树深度">【二叉树】二叉树深度</h2><p><a href="https://www.luogu.com.cn/problem/P4913">https://www.luogu.com.cn/problem/P4913</a></p><blockquote><p>题意：给定一棵二叉树，求解这棵二叉树的深度</p><p>思路：有两个考点，一个是如何根据给定的信息（从根结点开始依次给出已存在树上结点的左右孩子的编号）构建二叉树，一个是如何求解已经构建好的二叉树的深度。对于构建二叉树，我们沿用 T5 数组模拟构建的思路，直接定义结点类型即可；对于求解深度，很显然的一个递归求解，即左右子树深度值 +1 即可</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">1000010</span>;<br><br><span class="hljs-type">int</span> n;<br><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">Node</span> {<br>    <span class="hljs-type">int</span> l, r;<br>} t[N];<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">dep</span><span class="hljs-params">(<span class="hljs-type">int</span> now)</span> </span>{<br>    <span class="hljs-keyword">if</span> (!now) <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">return</span> <span class="hljs-built_in">max</span>(<span class="hljs-built_in">dep</span>(t[now].l), <span class="hljs-built_in">dep</span>(t[now].r)) + <span class="hljs-number">1</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-type">int</span> x, y;<br>        cin &gt;&gt; x &gt;&gt; y;<br>        t[i].l = x, t[i].r = y;<br>    }<br>    <br>    cout &lt;&lt; <span class="hljs-built_in">dep</span>(<span class="hljs-number">1</span>);<br>} <br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h2 id="【完全二叉树】淘汰赛">【完全二叉树】淘汰赛</h2><p><a href="https://www.luogu.com.cn/problem/P1364">https://www.luogu.com.cn/problem/P1364</a></p><blockquote><p>题意：给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">2^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6644em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> 支球队的编号与能力值，进行淘汰赛，能力值者晋级下一轮直到赛出冠军。输出亚军编号</p><p>思路：很显然的一个完全二叉树的题目。我们都不需要进行递归操作，直接利用完全二叉树的下标性质利用数组模拟循环计算即可。给出的信息就是完全二叉树的全部叶子结点的信息，分别为球队编号 id 与球队能力值 val，我们从第 n-1 个结点开始循环枚举到第 1 个结点计算每一轮的胜者信息，最终输出最后一场的能力值较小者球队编号即可</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Θ</mi><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta(2n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Θ</span><span class="mopen">(</span><span class="mord">2</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">1</span> &lt;&lt; <span class="hljs-number">8</span>;<br><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">Node</span> {<br>    <span class="hljs-type">int</span> id, val;<br>} a[N];<br><br><span class="hljs-type">int</span> n;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n;<br>    <br>    n = <span class="hljs-number">1</span> &lt;&lt; n;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = n; i &lt;= <span class="hljs-number">2</span> * n - <span class="hljs-number">1</span>; i++) {<br>        a[i].id = i - n + <span class="hljs-number">1</span>;<br>        cin &gt;&gt; a[i].val;<br>    }<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = n - <span class="hljs-number">1</span>; i &gt;= <span class="hljs-number">1</span>; i--)<br>        <span class="hljs-keyword">if</span> (a[i * <span class="hljs-number">2</span>].val &gt; a[i * <span class="hljs-number">2</span> + <span class="hljs-number">1</span>].val) a[i] = a[i * <span class="hljs-number">2</span>];<br>        <span class="hljs-keyword">else</span> a[i] = a[i * <span class="hljs-number">2</span> + <span class="hljs-number">1</span>];<br>            <br>    <span class="hljs-keyword">if</span> (a[<span class="hljs-number">2</span>].val &gt; a[<span class="hljs-number">3</span>].val) cout &lt;&lt; a[<span class="hljs-number">3</span>].id;<br>    <span class="hljs-keyword">else</span> cout &lt;&lt; a[<span class="hljs-number">2</span>].id;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h2 id="【二叉树-LCA】二叉树问题">【二叉树/LCA】二叉树问题</h2><p><a href="https://www.luogu.com.cn/problem/P3884">https://www.luogu.com.cn/problem/P3884</a></p><blockquote><p>题意：给定一棵二叉树的结点关系信息，求出这棵二叉树的深度、宽度和两个指定结点之间的最短路径长度</p><p>思路：二叉树的构建直接采用有向图的构造方法。深度直接 dfs 即可，宽度直接在 dfs 遍历时哈希深度值即可。问题的关键在于如何求解两个给定结点之间的路径长度，很显然需要求解两个结点的 LCA，由于结点数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≤</mo><mn>100</mn></mrow><annotation encoding="application/x-tex">\le 100</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">100</span></span></span></span> 故直接采用暴力的方法，可以重定义结点，增加父结点域。也可以通过比对根结点到两个指定结点的路径信息得到 LCA 即最后一个相同的结点编号（本题采用），通过在 dfs 遍历树时存储路径即可得到根结点到两个指定结点的路径信息。之后直接根据题中新定义的路径长度输出即可，即</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>length</mtext><mo>=</mo><mn>2</mn><mo>×</mo><mo stretchy="false">(</mo><msub><mi>d</mi><mi>x</mi></msub><mo>−</mo><msub><mi>d</mi><mrow><mi>l</mi><mi>c</mi><mi>a</mi></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><msub><mi>d</mi><mi>y</mi></msub><mo>−</mo><msub><mi>d</mi><mrow><mi>l</mi><mi>c</mi><mi>a</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{length} = 2 \times (d_x - d_{lca}) + (d_y - d_{lca})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">length</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.01968em;">l</span><span class="mord mathnormal mtight">c</span><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.01968em;">l</span><span class="mord mathnormal mtight">c</span><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p><p>其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">d_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 表示：根结点到第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 号点之间的路径长度，在 dfs 时通过传递深度值维护得到</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">110</span>;<br><br><span class="hljs-type">int</span> n, x, y;<br>vector&lt;<span class="hljs-type">int</span>&gt; G[N];<br><span class="hljs-type">int</span> depth, width;<br>unordered_map&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt; ha; <span class="hljs-comment">// 将所有的深度值进行哈希</span><br><span class="hljs-type">int</span> d[N];                   <span class="hljs-comment">// d[i] 表示第 i 个点到根结点的边数</span><br>vector&lt;<span class="hljs-type">int</span>&gt; temp, rx, ry;   <span class="hljs-comment">// 根结点到 x 号点与 y 号点直接的路径结点编号</span><br><br><span class="hljs-comment">// 当前结点编号 now，当前深度 level</span><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">dfs</span><span class="hljs-params">(<span class="hljs-type">int</span> now, <span class="hljs-type">int</span> level)</span> </span>{<br>    depth = <span class="hljs-built_in">max</span>(depth, level);<br>    <br>    temp.<span class="hljs-built_in">push_back</span>(now);<br>    <span class="hljs-keyword">if</span> (now == x) rx = temp;<br>    <span class="hljs-keyword">if</span> (now == y) ry = temp;<br>    <br>    ha[level]++;<br>    d[now] = level - <span class="hljs-number">1</span>;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; ch: G[now]) {<br>        <span class="hljs-built_in">dfs</span>(ch, level + <span class="hljs-number">1</span>);<br>        temp.<span class="hljs-built_in">pop_back</span>();<br>    }<br>}<br><br><span class="hljs-comment">// 暴力 lca + 计算路径长度</span><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">len</span><span class="hljs-params">(<span class="hljs-type">int</span> x, <span class="hljs-type">int</span> y)</span> </span>{<br>    <span class="hljs-type">int</span> i = <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">while</span> (i &lt; rx.<span class="hljs-built_in">size</span>() &amp;&amp; i &lt; ry.<span class="hljs-built_in">size</span>() &amp;&amp; rx[i] == ry[i]) i++;<br>    <br>    <span class="hljs-type">int</span> lca = rx[--i];<br>    <br>    <span class="hljs-keyword">return</span> <span class="hljs-number">2</span> * (d[x] - d[lca]) + (d[y] - d[lca]);<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n - <span class="hljs-number">1</span>; i++) {<br>        <span class="hljs-type">int</span> a, b;<br>        cin &gt;&gt; a &gt;&gt; b;<br>        G[a].<span class="hljs-built_in">push_back</span>(b);<br>    }<br>    <br>    cin &gt;&gt; x &gt;&gt; y;<br>    <br>    <span class="hljs-comment">// 二叉树的深度 depth</span><br>    <span class="hljs-built_in">dfs</span>(<span class="hljs-number">1</span>, <span class="hljs-number">1</span>);<br>    cout &lt;&lt; depth &lt;&lt; <span class="hljs-string">"\n"</span>;<br>    <br>    <span class="hljs-comment">// 二叉树的宽度 width</span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; item: ha) width = <span class="hljs-built_in">max</span>(width, item.second);<br>    cout &lt;&lt; width &lt;&lt; <span class="hljs-string">"\n"</span>;<br>    <br>    <span class="hljs-comment">// 两个结点之间的路径长度</span><br>    cout &lt;&lt; <span class="hljs-built_in">len</span>(x, y) &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h2 id="【set】营业额统计">【set】营业额统计</h2><p><a href="https://www.luogu.com.cn/problem/P2234">https://www.luogu.com.cn/problem/P2234</a></p><blockquote><p>题意：给定一个序列 a，需要计算 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mstyle scriptlevel="0" displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>2</mn><mo separator="true">,</mo><mn>1</mn><mo>≤</mo><mi>j</mi><mo>&lt;</mo><mi>i</mi></mrow><mi>n</mi></munderover><mi>min</mi><mo>⁡</mo><mrow><mi mathvariant="normal">∣</mi><msub><mi>a</mi><mi>i</mi></msub><mo>−</mo><msub><mi>a</mi><mi>j</mi></msub><mi mathvariant="normal">∣</mi></mrow></mstyle></mrow><annotation encoding="application/x-tex">a_1 + \displaystyle \sum_{i=2,1 \le j &lt;i}^{n} \min {|a_i - a_j|}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:3.0652em;vertical-align:-1.4138em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">2</span><span class="mpunct mtight">,</span><span class="mord mtight">1</span><span class="mrel mtight">≤</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">&lt;</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4138em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">min</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord">∣</span></span></span></span></span> ，即计算每一个数与序列中当前数之前的数的最小差值之和</p><p>思路：很显然的思路，对于每一个数，我们需要对之前的序列在短时间内找到一个数值最接近当前数的数。</p><ul><li>TLE：一开始的思路是每次对之前的序列进行排序，然后二分查找与当前值匹配的数，为了确保所有的情况都找到，就直接判断二分查到的数，查到的数之前的一个数，之后的一个数，但是时间复杂度极高（我居然没想到），是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2 \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li><li>AC：后来看了题解才知道 <code>set</code> 的正确用法，就是一个 <strong>平衡树的 STL</strong>。我们对于之前的序列不断的插入平衡树中（默认升序排序），每次利用 <code>s.lower_bound(x)</code> 返回「集合 <code>s</code> 中第一个 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≥</mo></mrow><annotation encoding="application/x-tex">\ge</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≥</span></span></span></span> 当前数的迭代器」，然后进行判断即可。<code>lower_bound()</code> 的时间复杂度为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 。需要注意的是边界的判断，一开始的思路虽然会超时，但是二分后边界的判断很简单，使用 STL 后同样需要考虑边界的情况。分为三种（详情见代码）<ul><li>当前数比集合中所有的数都大，那么 <code>lower_bound</code> 就会返回 <code>s.end()</code> 答案就是当前数与集合中最后一个数的差值</li><li>当前数比集合中所有的数都小，那么 <code>lower_bound</code> 就会返回 <code>s.bigin()</code> 答案就是集合中第一个数与当前数的差值</li><li>当前数存在于集合中 or 集合中既有比当前数大的又有比当前数小的，那么就比较查到的数与查到的数前一个数和当前数的差值，取最小的差值即可</li></ul></li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><p>TLE 但逻辑清晰代码</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;algorithm&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">1</span> &lt;&lt; <span class="hljs-number">16</span>;<br><br><span class="hljs-type">int</span> n, a[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n;<br>    <br>    <span class="hljs-type">int</span> res = <span class="hljs-number">0</span>;<br>    cin &gt;&gt; a[<span class="hljs-number">1</span>];<br>    res += a[<span class="hljs-number">1</span>];<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">2</span>; i &lt;= n; i++) {<br>        <span class="hljs-comment">// 维护之前序列有序</span><br>        <span class="hljs-built_in">sort</span>(a + <span class="hljs-number">1</span>, a + i);<br>        cin &gt;&gt; a[i];<br><br>        <span class="hljs-comment">// 二分查找目标数</span><br>        <span class="hljs-type">int</span> l = <span class="hljs-number">1</span>, r = i - <span class="hljs-number">1</span>;<br>        <span class="hljs-keyword">while</span> (l &lt; r) {<br>            <span class="hljs-type">int</span> mid = (l + r) &gt;&gt; <span class="hljs-number">1</span>;<br>            <span class="hljs-keyword">if</span> (a[mid] &lt; a[i]) l = mid + <span class="hljs-number">1</span>;<br>            <span class="hljs-keyword">else</span> r = mid;<br>        }<br>        <br>        <span class="hljs-comment">// 边界判断</span><br>        <span class="hljs-type">int</span> ans = <span class="hljs-built_in">abs</span>(a[i] - a[r]);<br>        <span class="hljs-keyword">if</span> (r + <span class="hljs-number">1</span> &gt;= <span class="hljs-number">1</span> &amp;&amp; r + <span class="hljs-number">1</span> &lt;= i - <span class="hljs-number">1</span>) ans = <span class="hljs-built_in">min</span>(ans, <span class="hljs-built_in">abs</span>(a[i] - a[r + <span class="hljs-number">1</span>]));<br>        <span class="hljs-keyword">if</span> (r - <span class="hljs-number">1</span> &gt;= <span class="hljs-number">1</span> &amp;&amp; r - <span class="hljs-number">1</span> &lt;= i - <span class="hljs-number">1</span>) ans = <span class="hljs-built_in">min</span>(ans, <span class="hljs-built_in">abs</span>(a[i] - a[r - <span class="hljs-number">1</span>]));<br>        <br>        res += ans;<br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><p>AC 的 set 代码</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;algorithm&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;set&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">int</span> n, res;<br>set&lt;<span class="hljs-type">int</span>&gt; s;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n;<br>    <br>    <span class="hljs-type">int</span> x;<br>    cin &gt;&gt; x;<br>    res += x;<br>    s.<span class="hljs-built_in">insert</span>(x);<br>    <br>    <span class="hljs-keyword">while</span> (--n) {<br>        cin &gt;&gt; x;<br><br>        <span class="hljs-keyword">auto</span> it = s.<span class="hljs-built_in">lower_bound</span>(x);<br><br>        <span class="hljs-keyword">if</span> (it == s.<span class="hljs-built_in">end</span>()) {<br>            <span class="hljs-comment">// 没有比当前数大的</span><br>            res += x - *s.<span class="hljs-built_in">rbegin</span>();<br>        } <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (it == s.<span class="hljs-built_in">begin</span>()) {<br>            <span class="hljs-comment">// 没有比当前数小的</span><br>            res += *s.<span class="hljs-built_in">begin</span>() - x;<br>        } <span class="hljs-keyword">else</span> {<br>            <span class="hljs-comment">// 当前数已存在于集合中 or 既有比当前数大的也有比当前数小的</span><br>            <span class="hljs-keyword">auto</span> pre = it;<br>            pre--;<br>            res += <span class="hljs-built_in">min</span>(<span class="hljs-built_in">abs</span>(x - *it), <span class="hljs-built_in">abs</span>(x - *pre));<br>        }<br>        <br>        s.<span class="hljs-built_in">insert</span>(x);<br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h2 id="【multiset】切蛋糕">【multiset】切蛋糕</h2><p><a href="https://www.acwing.com/problem/content/description/5581/">https://www.acwing.com/problem/content/description/5581/</a></p><blockquote><p>题意：给定一个矩形，由左下角和右上角的坐标确定。现在对这个矩形进行切割操作，要么竖着切，要么横着切。现在需要给出每次切割后最大子矩形的面积</p><p>思路：STL set。很容易想到，横纵坐标是相互独立的，最大子矩形面积一定产生于最大长度与最大宽度的乘积。因此我们只需要维护一个序列的最大值即可。对于二维可以直接当做一维来看，于是问题就变成了，需要在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>log</mi><mo>⁡</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">\log n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span></span></span></span> 的时间复杂度内，对一个序列完成下面三个操作：</p><ol><li>删除一个数</li><li>增加一个数（执行两次）</li><li>获取最大值</li></ol><p>如何实现呢？我们需要有序记录每一个子线段的长度，并且子线段的长度可能重复，因此我们用 <code>std::multiset</code> 来存储所有子线段的长度</p><ol><li>使用 <code>M.erase(M.find(value))</code> 实现：删除一个子线段长度值</li><li>使用 <code>M.insert(value)</code> 实现：增加子线段一个长度值</li><li>使用 <code>*M.rbegin()</code> 实现：获取当前所有子线段长度的最大值</li></ol><p>由于给的是切割的位置坐标 <code>x</code>，因此上述操作 1 不能直接实现，我们需要利用给定的切割坐标 <code>x</code> 计算出当前切割位置对应子线段的长度。如何实现呢？我们知道，对于当前切割的坐标 <code>x</code>，对应的子线段的长度取决于当前切割坐标左右两个切割的位置 <code>rp, lp</code>，因此我们只需要存储每一个切割的坐标即可。由于切割位置不会重复，并且需要在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>log</mi><mo>⁡</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">\log n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span></span></span></span> 的时间复杂度内查询到，因此我们还是可以使用 <code>std::set</code> 来存储切割位置</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;set&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><span class="hljs-keyword">using</span> ll = <span class="hljs-type">long</span> <span class="hljs-type">long</span>;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">work</span><span class="hljs-params">(<span class="hljs-type">int</span> x, set&lt;<span class="hljs-type">int</span>&gt;&amp; S, multiset&lt;<span class="hljs-type">int</span>&gt;&amp; M)</span> </span>{<br>    set&lt;<span class="hljs-type">int</span>&gt;::iterator rp = S.<span class="hljs-built_in">upper_bound</span>(x), lp = rp;<br>    lp--;<br>    S.<span class="hljs-built_in">insert</span>(x);<br><br>    M.<span class="hljs-built_in">erase</span>(M.<span class="hljs-built_in">find</span>(*rp - *lp));<br>    M.<span class="hljs-built_in">insert</span>(*rp - x);<br>    M.<span class="hljs-built_in">insert</span>(x - *lp);<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-type">int</span> w, h, n;<br>    cin &gt;&gt; w &gt;&gt; h &gt;&gt; n;<br><br>    set&lt;<span class="hljs-type">int</span>&gt; S1, S2;<br>    multiset&lt;<span class="hljs-type">int</span>&gt; M1, M2;<br>    S<span class="hljs-number">1.</span><span class="hljs-built_in">insert</span>(<span class="hljs-number">0</span>), S<span class="hljs-number">1.</span><span class="hljs-built_in">insert</span>(w), M<span class="hljs-number">1.</span><span class="hljs-built_in">insert</span>(w);<br>    S<span class="hljs-number">2.</span><span class="hljs-built_in">insert</span>(<span class="hljs-number">0</span>), S<span class="hljs-number">2.</span><span class="hljs-built_in">insert</span>(h), M<span class="hljs-number">2.</span><span class="hljs-built_in">insert</span>(h);<br><br>    <span class="hljs-keyword">while</span> (n--) {<br>        <span class="hljs-type">char</span> op;<br>        <span class="hljs-type">int</span> x;<br>        cin &gt;&gt; op &gt;&gt; x;<br>        <span class="hljs-keyword">if</span> (op == <span class="hljs-string">'X'</span>) <span class="hljs-built_in">work</span>(x, S1, M1);<br>        <span class="hljs-keyword">else</span> <span class="hljs-built_in">work</span>(x, S2, M2);<br><br>        cout &lt;&lt; (ll)*M<span class="hljs-number">1.</span><span class="hljs-built_in">rbegin</span>() * *M<span class="hljs-number">2.</span><span class="hljs-built_in">rbegin</span>() &lt;&lt; <span class="hljs-string">"\n"</span>;<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h2 id="【并查集】Milk-Visits-S">【并查集】Milk Visits S</h2><p><a href="https://www.luogu.com.cn/problem/P5836">https://www.luogu.com.cn/problem/P5836</a></p><blockquote><p>题意：给定一棵树，结点被标记成两种，一种是 H，一种是 G，在每一次查询中，需要知道指定的两个结点之间是否含有某一种标记</p><p>思路：对于树上标记，我们可以将相同颜色的分支连成一个连通块</p><ul><li>如果查询的两个结点在同一个连通块，则查询两个结点所在的颜色与所需的颜色是否匹配即可</li><li>如果查询的两个结点不在同一个连通块，两个结点之间的路径一定是覆盖了两种颜色的标记，则答案一定是 1</li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Θ</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta(n+m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Θ</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">100010</span>;<br><br><span class="hljs-type">int</span> n, m, p[N];<br><span class="hljs-type">char</span> col[N];<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">find</span><span class="hljs-params">(<span class="hljs-type">int</span> x)</span> </span>{<br>    <span class="hljs-keyword">if</span> (p[x] != x) {<br>        p[x] = <span class="hljs-built_in">find</span>(p[x]);<br>    }<br>    <span class="hljs-keyword">return</span> p[x];<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; m;<br>    cin &gt;&gt; (col + <span class="hljs-number">1</span>);<br><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        p[i] = i;<br>    }<br><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n - <span class="hljs-number">1</span>; i++) {<br>        <span class="hljs-type">int</span> a, b;<br>        cin &gt;&gt; a &gt;&gt; b;<br>        <span class="hljs-keyword">if</span> (col[a] == col[b]) {<br>            p[<span class="hljs-built_in">find</span>(a)] = <span class="hljs-built_in">find</span>(b);<br>        }<br>    }<br><br>    string res;<br><br>    <span class="hljs-keyword">while</span> (m--) {<br>        <span class="hljs-type">int</span> u, v;<br>        cin &gt;&gt; u &gt;&gt; v;<br><br>        <span class="hljs-type">char</span> cow;<br>        cin &gt;&gt; cow;<br><br>        <span class="hljs-keyword">if</span> (<span class="hljs-built_in">find</span>(u) == <span class="hljs-built_in">find</span>(v)) {<br>            res += <span class="hljs-built_in">to_string</span>(col[u] == cow);<br>        } <span class="hljs-keyword">else</span> {<br>            res += <span class="hljs-string">'1'</span>;<br>        }<br>    }<br><br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h2 id="【并查集】尽量减少恶意软件的传播">【并查集】尽量减少恶意软件的传播</h2><p><a href="https://leetcode.cn/problems/minimize-malware-spread/description/">https://leetcode.cn/problems/minimize-malware-spread/description/</a></p><blockquote><p>题意：给定一个由邻接矩阵存储的无向图，其中某些结点具备感染能力，可以感染相连的所有结点，问消除哪一个结点的感染能力可以使得最终不被感染的结点数量尽可能少，给出消除的有感染能力的最小结点编号</p><p>思路：很显然我们可以将当前无向图的多个连通分量，共三种感染情况：</p><ol><li><p>如果一个连通分量中含有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\ge 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span> 个感染结点，则当前连通分量一定会被全部感染；</p></li><li><p>如果一个连通块中含有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 个感染结点，则无需任何操作；</p></li><li><p>如果一个连通块中含有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 个感染结点，则最佳实践就是移除该连通块中唯一的那个感染结点。</p></li></ol><p>当然了，由于只能移走一个感染结点，我们需要从所有只含有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 个感染结点的连通块中移走连通块结点最多的那个感染结点。因此我们需要统计每一个连通分量中感染结点的数量以及总结点数，采用并查集进行统计。需要注意的是题目中的“索引最小”指的是结点编号最小而非结点在序列 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mi>n</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>a</mi><mi>l</mi></mrow><annotation encoding="application/x-tex">initial</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">ini</span><span class="mord mathnormal">t</span><span class="mord mathnormal">ia</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span></span></span> 中的下标最小。算法流程见代码。</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span> {<br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-type">int</span> p[<span class="hljs-number">310</span>];<br><br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">Find</span><span class="hljs-params">(<span class="hljs-type">int</span> x)</span> </span>{<br>        <span class="hljs-keyword">if</span> (x != p[x]) p[x] = <span class="hljs-built_in">Find</span>(p[x]);<br>        <span class="hljs-keyword">return</span> p[x];<br>    }<br><br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">minMalwareSpread</span><span class="hljs-params">(vector&lt;vector&lt;<span class="hljs-type">int</span>&gt;&gt;&amp; graph, vector&lt;<span class="hljs-type">int</span>&gt;&amp; initial)</span> </span>{<br>        <span class="hljs-comment">// 1. 维护并查集数组：p[]</span><br>        <span class="hljs-type">int</span> n = graph.<span class="hljs-built_in">size</span>();<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) p[i] = i;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++)<br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">0</span>; j &lt; n; j++)<br>                <span class="hljs-keyword">if</span> (graph[i][j])<br>                    p[<span class="hljs-built_in">Find</span>(i)] = <span class="hljs-built_in">Find</span>(j);<br><br>        <span class="hljs-comment">// 2. 维护哈希表：每一个连通块中的感染结点数、总结点数</span><br>        unordered_map&lt;<span class="hljs-type">int</span>, pair&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt;&gt; ha;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; x: initial) ha[<span class="hljs-built_in">Find</span>(x)].first++;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) ha[<span class="hljs-built_in">Find</span>(i)].second++;<br><br>        <span class="hljs-comment">// 3. 排序：按照感染结点数升序，总结点数降序</span><br>        vector&lt;pair&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt;&gt; v;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; it: ha) v.<span class="hljs-built_in">push_back</span>(it.second);<br>        <span class="hljs-built_in">sort</span>(v.<span class="hljs-built_in">begin</span>(), v.<span class="hljs-built_in">end</span>(), [&amp;](pair&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt;&amp; x, pair&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt;&amp; y){<br>            <span class="hljs-keyword">if</span> (x.first == y.first) <span class="hljs-keyword">return</span> x.second &gt; y.second;<br>            <span class="hljs-keyword">return</span> x.first &lt; y.first;<br>        });<br><br>        <span class="hljs-comment">// 4. 寻找符合条件的连通块属性：找到序列中第一个含有 1 个感染结点的连通块祖宗结点编号 idx</span><br>        <span class="hljs-type">int</span> idx = <span class="hljs-number">-1</span>;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; v.<span class="hljs-built_in">size</span>(); i++) {<br>            <span class="hljs-keyword">if</span> (v[i].first == <span class="hljs-number">1</span>) {<br>                idx = i;<br>                <span class="hljs-keyword">break</span>;<br>            }<br>        }<br><br>        <span class="hljs-comment">// 5. 返回答案：比对感染结点所在的连通块属性与目标连通块属性</span><br>        <span class="hljs-keyword">if</span> (idx == <span class="hljs-number">-1</span>) {<br>            <span class="hljs-comment">// 特判没有连通块只含有 1 个感染结点的情况</span><br>            <span class="hljs-keyword">return</span> *<span class="hljs-built_in">min_element</span>(initial.<span class="hljs-built_in">begin</span>(), initial.<span class="hljs-built_in">end</span>());<br>        }<br>        <br>        <span class="hljs-type">int</span> res = n + <span class="hljs-number">10</span>;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; x: initial) {<br>            <span class="hljs-type">int</span> px = <span class="hljs-built_in">Find</span>(x);<br>            <span class="hljs-keyword">if</span> (ha[px].first == v[idx].first &amp;&amp; ha[px].second == v[idx].second) {<br>                res = <span class="hljs-built_in">min</span>(res, x);<br>            }<br>        }<br><br>        <span class="hljs-keyword">return</span> res;<br>    }<br>};<br></code></pre></td></tr></tbody></table></figure><h2 id="【并查集】账户合并">【并查集】账户合并</h2><p><a href="https://leetcode.cn/problems/accounts-merge/">https://leetcode.cn/problems/accounts-merge/</a></p><blockquote><p>题意：给定 n 个账户，每一个账户含有一个用户名和最多 m 个绑定的邮箱。由于一个用户可能注册多个账户，因此我们需要对所有的账户进行合并使得一个用户对应一个账户。合并的规则是将所有「含有相同邮箱的账户」视作同一个用户注册的账户。返回合并后的账户列表。</p><p>思路：这道题的需求很显然，我们需要合并含有相同邮箱的账户。显然有一个暴力的做法，我们直接枚举每一个账户中所有的邮箱，接着枚举剩余账户中的邮箱进行匹配，匹配上就进行合并，但这样做显然会造成大量的冗余匹配和冗余合并，我们不妨将这两个过程进行拆分。我们需要解决两个问题：</p><ul><li>哪些账户需要合并？很容易想到并查集这样的数据结构。我们使用哈希表存储每一个邮箱的账户编号，最后进行集合合并即可维护好每一个账号归属的集合编号。<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(nm)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">nm</span><span class="mclose">)</span></span></span></span></li><li>如何合并指定账户？对于上述维护好的集合编号，我们需要合并所有含有相同“祖先”的账户。排序去重或使用有序列表均可实现。<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><figcaption><span>[]</span></figcaption><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-keyword">struct</span> <span class="hljs-title class_">dsu</span> {<br>    <span class="hljs-type">int</span> n;<br>    std::vector&lt;<span class="hljs-type">int</span>&gt; p;<br>    <span class="hljs-built_in">dsu</span>(<span class="hljs-type">int</span> _n) { n = _n; p.<span class="hljs-built_in">resize</span>(n + <span class="hljs-number">1</span>); <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) p[i] = i; }<br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">find</span><span class="hljs-params">(<span class="hljs-type">int</span> x)</span> </span>{ <span class="hljs-keyword">return</span> (p[x] == x ? p[x] : p[x] = <span class="hljs-built_in">find</span>(p[x])); }<br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">merge</span><span class="hljs-params">(<span class="hljs-type">int</span> a, <span class="hljs-type">int</span> b)</span> </span>{ p[<span class="hljs-built_in">find</span>(a)] = <span class="hljs-built_in">find</span>(b); }<br>    <span class="hljs-function"><span class="hljs-type">bool</span> <span class="hljs-title">query</span><span class="hljs-params">(<span class="hljs-type">int</span> a, <span class="hljs-type">int</span> b)</span> </span>{ <span class="hljs-keyword">return</span> <span class="hljs-built_in">find</span>(a) == <span class="hljs-built_in">find</span>(b); }<br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">block</span><span class="hljs-params">()</span> </span>{ <span class="hljs-type">int</span> ret = <span class="hljs-number">0</span>; <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) ret += p[i] == i; <span class="hljs-keyword">return</span> ret; }<br>};<br><br><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span> {<br><span class="hljs-keyword">public</span>:<br>    vector&lt;vector&lt;string&gt;&gt; <span class="hljs-built_in">accountsMerge</span>(vector&lt;vector&lt;string&gt;&gt;&amp; accounts) {<br>        <span class="hljs-comment">// 维护每一个子账户归属的集合</span><br>        <span class="hljs-type">int</span> n = accounts.<span class="hljs-built_in">size</span>();<br>        unordered_map&lt;string, vector&lt;<span class="hljs-type">int</span>&gt;&gt; hash;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">1</span>; j &lt; accounts[i - <span class="hljs-number">1</span>].<span class="hljs-built_in">size</span>(); j++) {<br>                hash[accounts[i - <span class="hljs-number">1</span>][j]].<span class="hljs-built_in">push_back</span>(i);<br>            }<br>        }        <br>        <span class="hljs-function">dsu <span class="hljs-title">d</span><span class="hljs-params">(n)</span></span>;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; it: hash) {<br>            vector&lt;<span class="hljs-type">int</span>&gt; v = it.second;<br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt; v.<span class="hljs-built_in">size</span>(); i++) {<br>                d.<span class="hljs-built_in">merge</span>(v[i - <span class="hljs-number">1</span>], v[i]);<br>            }<br>        }<br><br>        <span class="hljs-comment">// 按照子账户归属的集合合并出最终的账户</span><br>        unordered_set&lt;<span class="hljs-type">int</span>&gt; fa;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>            fa.<span class="hljs-built_in">insert</span>(d.<span class="hljs-built_in">find</span>(i));<br>        }<br>        vector&lt;vector&lt;string&gt;&gt; res;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span> p: fa) {<br>            set&lt;string&gt; se;<br>            vector&lt;string&gt; ans;<br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>                <span class="hljs-keyword">if</span> (d.<span class="hljs-built_in">find</span>(i) == p) {<br>                    <span class="hljs-keyword">if</span> (ans.<span class="hljs-built_in">empty</span>()) {<br>                        ans.<span class="hljs-built_in">push_back</span>(accounts[i - <span class="hljs-number">1</span>][<span class="hljs-number">0</span>]);<br>                    }<br>                    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">1</span>; j &lt; accounts[i - <span class="hljs-number">1</span>].<span class="hljs-built_in">size</span>(); j++) {<br>                        se.<span class="hljs-built_in">insert</span>(accounts[i - <span class="hljs-number">1</span>][j]);<br>                    }<br>                }<br>            }<br>            <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span> mail: se) {<br>                ans.<span class="hljs-built_in">push_back</span>(mail);<br>            }<br>            res.<span class="hljs-built_in">push_back</span>(ans);<br>        }<br><br>        <span class="hljs-keyword">return</span> res;<br>    }<br>};<br></code></pre></td></tr></tbody></table></figure><figure class="highlight python"><figcaption><span>[]</span></figcaption><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">class</span> <span class="hljs-title class_">dsu</span>:<br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">__init__</span>(<span class="hljs-params">self, n: <span class="hljs-built_in">int</span></span>) -&gt; <span class="hljs-literal">None</span>:<br>        <span class="hljs-variable language_">self</span>.n = n<br>        <span class="hljs-variable language_">self</span>.p = [i <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(n + <span class="hljs-number">1</span>)]<br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">find</span>(<span class="hljs-params">self, x: <span class="hljs-built_in">int</span></span>) -&gt; <span class="hljs-built_in">int</span>:<br>        <span class="hljs-keyword">if</span> <span class="hljs-variable language_">self</span>.p[x] != x: <span class="hljs-variable language_">self</span>.p[x] = <span class="hljs-variable language_">self</span>.find(<span class="hljs-variable language_">self</span>.p[x])<br>        <span class="hljs-keyword">return</span> <span class="hljs-variable language_">self</span>.p[x]<br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">merge</span>(<span class="hljs-params">self, a: <span class="hljs-built_in">int</span>, b: <span class="hljs-built_in">int</span></span>) -&gt; <span class="hljs-literal">None</span>:<br>        <span class="hljs-variable language_">self</span>.p[<span class="hljs-variable language_">self</span>.find(a)] = <span class="hljs-variable language_">self</span>.find(b)<br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">query</span>(<span class="hljs-params">self, a: <span class="hljs-built_in">int</span>, b: <span class="hljs-built_in">int</span></span>) -&gt; <span class="hljs-built_in">bool</span>:<br>        <span class="hljs-keyword">return</span> <span class="hljs-variable language_">self</span>.find(a) == <span class="hljs-variable language_">self</span>.find(b)<br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">block</span>(<span class="hljs-params">self</span>) -&gt; <span class="hljs-built_in">int</span>:<br>        <span class="hljs-keyword">return</span> <span class="hljs-built_in">sum</span>([<span class="hljs-number">1</span> <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">1</span>, <span class="hljs-variable language_">self</span>.n + <span class="hljs-number">1</span>) <span class="hljs-keyword">if</span> <span class="hljs-variable language_">self</span>.p[i] == i])<br><br><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span>:<br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">accountsMerge</span>(<span class="hljs-params">self, accounts: <span class="hljs-type">List</span>[<span class="hljs-type">List</span>[<span class="hljs-built_in">str</span>]]</span>) -&gt; <span class="hljs-type">List</span>[<span class="hljs-type">List</span>[<span class="hljs-built_in">str</span>]]:<br>        <span class="hljs-keyword">from</span> collections <span class="hljs-keyword">import</span> defaultdict<br>        <br>        n = <span class="hljs-built_in">len</span>(accounts)<br>        <span class="hljs-built_in">hash</span> = defaultdict(<span class="hljs-built_in">list</span>)<br>        <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">1</span>, n + <span class="hljs-number">1</span>):<br>            <span class="hljs-keyword">for</span> j <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">1</span>, <span class="hljs-built_in">len</span>(accounts[i - <span class="hljs-number">1</span>])):<br>                <span class="hljs-built_in">hash</span>[accounts[i - <span class="hljs-number">1</span>][j]].append(i)<br>        <br>        d = dsu(n)<br>        <span class="hljs-keyword">for</span> _, ids <span class="hljs-keyword">in</span> <span class="hljs-built_in">hash</span>.items():<br>            <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">1</span>, <span class="hljs-built_in">len</span>(ids)):<br>                d.merge(ids[i - <span class="hljs-number">1</span>], ids[i])<br>        <br>        fa = <span class="hljs-built_in">set</span>()<br>        <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">1</span>, n + <span class="hljs-number">1</span>):<br>            fa.add(d.find(i))<br>        <br>        res = []<br>        <span class="hljs-keyword">for</span> p <span class="hljs-keyword">in</span> fa:<br>            ans = []<br>            se = <span class="hljs-built_in">set</span>()<br>            <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">1</span>, n + <span class="hljs-number">1</span>):<br>                <span class="hljs-keyword">if</span> d.find(i) == p:<br>                    <span class="hljs-keyword">if</span> <span class="hljs-built_in">len</span>(ans) == <span class="hljs-number">0</span>:<br>                        ans.append(accounts[i - <span class="hljs-number">1</span>][<span class="hljs-number">0</span>])<br>                    <span class="hljs-keyword">for</span> j <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">1</span>, <span class="hljs-built_in">len</span>(accounts[i - <span class="hljs-number">1</span>])):<br>                        se.add(accounts[i - <span class="hljs-number">1</span>][j])<br>            ans += <span class="hljs-built_in">sorted</span>(se)<br>            res.append(ans)<br>        <br>        <span class="hljs-keyword">return</span> res<br></code></pre></td></tr></tbody></table></figure><h2 id="【树状数组】将元素分配到两个数组中-II">【树状数组】将元素分配到两个数组中 II</h2><p><a href="https://leetcode.cn/problems/distribute-elements-into-two-arrays-ii/description/">https://leetcode.cn/problems/distribute-elements-into-two-arrays-ii/description/</a></p><blockquote><p>题意：给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个数，现在需要将这些数按照某种规则分配到两个数组 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> 中。初始化分配 <code>nums[0]</code> 到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 中，<code>nums[1]</code> 到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> 中，接下来对于剩余的每个元素 <code>nums[i]</code>，分配取决于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> 中比当前元素 <code>nums[i]</code> 大的个数，最终返回两个分配好的数组</p><p>思路：首先每一个元素都需要进行枚举，那么本题需要考虑的就是如何在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>log</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\log</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span></span></span></span> 时间复杂度内统计出两数组中比当前元素大的元素个数。针对 C++ 和 Python 分别讨论</p><ul><li>C++<ul><li>法一：<code>std::multiset&lt;int&gt;</code>。可惜不行，因为统计比当前元素大的个数时，<code>s.rbegin() - s.upper_bound(nums[i])</code> 是不合法的，因为 <code>std::multiset&lt;int&gt;</code> 的迭代器不是基于指针的，因此无法直接进行加减来计算地址差，遂作罢</li><li>法二：树状数组。很容易想到利用前缀和统计比当前数大的数字个数，但是由于此处需要对前缀和进行单点修改，因此时间复杂度肯定会寄。有什么数据结构支持「单点修改，区间更新」呢？我们引入树状数组。我们将数组元素哈希到 <code>[1, len(set(nums))]</code> 区间，定义哈希后的当前元素 <code>nums[i]</code> 为 <code>x</code>，对于当前哈希后的 <code>x</code> 而言想要知道两个数组中有多少数比当前数严格大，只需要计算前缀和数组 <code>arr</code> 中 <code>arr[n] - arr[x]</code> 的结果即可</li></ul></li><li>Python<ul><li><code>SortedList</code>。python 有一个 <code>sortedcontainers</code> 包其中有 <code>SortedList</code> 模块，可以实现 <code>std::multiset&lt;int&gt;</code> 所有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>log</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\log</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span></span></span></span> 操作并且可以进行随机下标访问，于是就可以进行下标访问 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 计算比当前数大的元素个数</li><li>题外话。LeetCode 可以进行第三方库导入的操作，某些比赛不允许，需要手搓 <code>SortedList</code> 模块，<s>当然可以用树状数组 or 线段树解决</s>，板子链接：<a href="https://blog.dwj601.cn/Algorithm/a_template/#SortedList">https://blog.dwj601.cn/Algorithm/a_template/#SortedList</a></li></ul></li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><figcaption><span>[]</span></figcaption><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-keyword">template</span>&lt;<span class="hljs-keyword">class</span> <span class="hljs-title class_">T</span>&gt;<br><span class="hljs-keyword">class</span> <span class="hljs-title class_">BinaryIndexedTree</span> {<br><span class="hljs-keyword">private</span>:<br>    std::vector&lt;T&gt; _arr;<br>    <span class="hljs-type">int</span> _n;<br><br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">lowbit</span><span class="hljs-params">(<span class="hljs-type">int</span> x)</span> </span>{ <span class="hljs-keyword">return</span> x &amp; (-x); }<br><br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-built_in">BinaryIndexedTree</span>(<span class="hljs-type">int</span> n) :_n(n) {<br>        _arr.<span class="hljs-built_in">resize</span>(_n + <span class="hljs-number">1</span>, <span class="hljs-number">0</span>);<br>    }<br><br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">add</span><span class="hljs-params">(<span class="hljs-type">int</span> pos, T x)</span> </span>{<br>        <span class="hljs-keyword">while</span> (pos &lt;= _n) {<br>            _arr[pos] += x;<br>            pos += <span class="hljs-built_in">lowbit</span>(pos);<br>        }<br>    }<br><br>    <span class="hljs-function">T <span class="hljs-title">sum</span><span class="hljs-params">(<span class="hljs-type">int</span> pos)</span> </span>{<br>        T ret = <span class="hljs-number">0</span>;<br>        <span class="hljs-keyword">while</span> (pos) {<br>            ret += _arr[pos];<br>            pos -= <span class="hljs-built_in">lowbit</span>(pos);<br>        }<br>        <span class="hljs-keyword">return</span> ret;<br>    }<br>};<br><br><br><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span> {<br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">resultArray</span><span class="hljs-params">(vector&lt;<span class="hljs-type">int</span>&gt;&amp; nums)</span> </span>{<br>        vector&lt;<span class="hljs-type">int</span>&gt; copy = nums;<br>        <span class="hljs-built_in">sort</span>(copy.<span class="hljs-built_in">begin</span>(), copy.<span class="hljs-built_in">end</span>());<br>        copy.<span class="hljs-built_in">erase</span>(<span class="hljs-built_in">unique</span>(copy.<span class="hljs-built_in">begin</span>(), copy.<span class="hljs-built_in">end</span>()), copy.<span class="hljs-built_in">end</span>());<br><br>        <span class="hljs-type">int</span> n = copy.<span class="hljs-built_in">size</span>(), cnt = <span class="hljs-number">1</span>;<br>        unordered_map&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt; a;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>            a[copy[i]] = cnt++;<br>        }<br><br>        vector&lt;<span class="hljs-type">int</span>&gt; v1, v2;<br>        v<span class="hljs-number">1.</span><span class="hljs-built_in">push_back</span>(nums[<span class="hljs-number">0</span>]);<br>        v<span class="hljs-number">2.</span><span class="hljs-built_in">push_back</span>(nums[<span class="hljs-number">1</span>]);<br><br>        <span class="hljs-function">BinaryIndexedTree&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">t1</span><span class="hljs-params">(n)</span>, <span class="hljs-title">t2</span><span class="hljs-params">(n)</span></span>;<br>        t<span class="hljs-number">1.</span><span class="hljs-built_in">add</span>(a[nums[<span class="hljs-number">0</span>]], <span class="hljs-number">1</span>);<br>        t<span class="hljs-number">2.</span><span class="hljs-built_in">add</span>(a[nums[<span class="hljs-number">1</span>]], <span class="hljs-number">1</span>);<br><br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">2</span>; i &lt; nums.<span class="hljs-built_in">size</span>(); i++) {<br>            <span class="hljs-type">int</span> d1 = t<span class="hljs-number">1.</span><span class="hljs-built_in">sum</span>(n) - t<span class="hljs-number">1.</span><span class="hljs-built_in">sum</span>(a[nums[i]]);<br>            <span class="hljs-type">int</span> d2 = t<span class="hljs-number">2.</span><span class="hljs-built_in">sum</span>(n) - t<span class="hljs-number">2.</span><span class="hljs-built_in">sum</span>(a[nums[i]]);<br><br>            <span class="hljs-keyword">if</span> (d1 &gt; d2) {<br>                v<span class="hljs-number">1.</span><span class="hljs-built_in">push_back</span>(nums[i]);<br>                t<span class="hljs-number">1.</span><span class="hljs-built_in">add</span>(a[nums[i]], <span class="hljs-number">1</span>);                <br>            } <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (d1 &lt; d2) {<br>                v<span class="hljs-number">2.</span><span class="hljs-built_in">push_back</span>(nums[i]);<br>                t<span class="hljs-number">2.</span><span class="hljs-built_in">add</span>(a[nums[i]], <span class="hljs-number">1</span>);<br>            } <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (d1 == d2 &amp;&amp; v<span class="hljs-number">1.</span><span class="hljs-built_in">size</span>() &lt; v<span class="hljs-number">2.</span><span class="hljs-built_in">size</span>()) {<br>                v<span class="hljs-number">1.</span><span class="hljs-built_in">push_back</span>(nums[i]);<br>                t<span class="hljs-number">1.</span><span class="hljs-built_in">add</span>(a[nums[i]], <span class="hljs-number">1</span>);<br>            } <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (d1 == d2 &amp;&amp; v<span class="hljs-number">1.</span><span class="hljs-built_in">size</span>() &gt; v<span class="hljs-number">2.</span><span class="hljs-built_in">size</span>()) {<br>                v<span class="hljs-number">2.</span><span class="hljs-built_in">push_back</span>(nums[i]);<br>                t<span class="hljs-number">2.</span><span class="hljs-built_in">add</span>(a[nums[i]], <span class="hljs-number">1</span>);<br>            } <span class="hljs-keyword">else</span> {<br>                v<span class="hljs-number">1.</span><span class="hljs-built_in">push_back</span>(nums[i]);<br>                t<span class="hljs-number">1.</span><span class="hljs-built_in">add</span>(a[nums[i]], <span class="hljs-number">1</span>);<br>            }<br>        }<br><br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> x: v2) {<br>            v<span class="hljs-number">1.</span><span class="hljs-built_in">push_back</span>(x);<br>        }<br><br>        <span class="hljs-keyword">return</span> v1;<br>    }<br>};<br></code></pre></td></tr></tbody></table></figure><figure class="highlight python"><figcaption><span>[]</span></figcaption><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">class</span> <span class="hljs-title class_">BinaryIndexedTree</span>:<br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">__init__</span>(<span class="hljs-params">self, n: <span class="hljs-built_in">int</span></span>):<br>        <span class="hljs-variable language_">self</span>._n = n<br>        <span class="hljs-variable language_">self</span>._arr = [<span class="hljs-number">0</span>] * (n + <span class="hljs-number">1</span>)<br><br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">_lowbit</span>(<span class="hljs-params">self, x: <span class="hljs-built_in">int</span></span>) -&gt; <span class="hljs-built_in">int</span>:<br>        <span class="hljs-keyword">return</span> x &amp; (-x)<br><br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">add</span>(<span class="hljs-params">self, pos: <span class="hljs-built_in">int</span>, x: <span class="hljs-built_in">int</span></span>) -&gt; <span class="hljs-literal">None</span>:<br>        <span class="hljs-keyword">while</span> pos &lt;= <span class="hljs-variable language_">self</span>._n:<br>            <span class="hljs-variable language_">self</span>._arr[pos] += x<br>            pos += <span class="hljs-variable language_">self</span>._lowbit(pos)<br><br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">sum</span>(<span class="hljs-params">self, pos: <span class="hljs-built_in">int</span></span>) -&gt; <span class="hljs-built_in">int</span>:<br>        ret = <span class="hljs-number">0</span><br>        <span class="hljs-keyword">while</span> pos:<br>            ret += <span class="hljs-variable language_">self</span>._arr[pos]<br>            pos -= <span class="hljs-variable language_">self</span>._lowbit(pos)<br>        <span class="hljs-keyword">return</span> ret<br><br><br><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span>:<br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">resultArray</span>(<span class="hljs-params">self, nums: <span class="hljs-type">List</span>[<span class="hljs-built_in">int</span>]</span>) -&gt; <span class="hljs-type">List</span>[<span class="hljs-built_in">int</span>]:<br>        copy = <span class="hljs-built_in">sorted</span>(<span class="hljs-built_in">set</span>(nums))<br>        <br>        n, cnt, a = <span class="hljs-built_in">len</span>(copy), <span class="hljs-number">1</span>, {}<br>        <span class="hljs-keyword">for</span> x <span class="hljs-keyword">in</span> copy:<br>            a[x] = cnt<br>            cnt += <span class="hljs-number">1</span><br><br>        v1, v2 = [nums[<span class="hljs-number">0</span>]], [nums[<span class="hljs-number">1</span>]]<br>        t1, t2 = BinaryIndexedTree(n), BinaryIndexedTree(n)<br>        t1.add(a[nums[<span class="hljs-number">0</span>]], <span class="hljs-number">1</span>)<br>        t2.add(a[nums[<span class="hljs-number">1</span>]], <span class="hljs-number">1</span>)<br>        <br>        <span class="hljs-keyword">for</span> x <span class="hljs-keyword">in</span> nums[<span class="hljs-number">2</span>:]:<br>            d1, d2 = t1.<span class="hljs-built_in">sum</span>(n) - t1.<span class="hljs-built_in">sum</span>(a[x]), t2.<span class="hljs-built_in">sum</span>(n) - t2.<span class="hljs-built_in">sum</span>(a[x])<br>            <br>            <span class="hljs-keyword">if</span> d1 &gt; d2:<br>                v1.append(x)<br>                t1.add(a[x], <span class="hljs-number">1</span>)<br>            <span class="hljs-keyword">elif</span> d1 &lt; d2:<br>                v2.append(x)<br>                t2.add(a[x], <span class="hljs-number">1</span>)<br>            <span class="hljs-keyword">elif</span> d1 == d2 <span class="hljs-keyword">and</span> <span class="hljs-built_in">len</span>(v1) &lt; <span class="hljs-built_in">len</span>(v2):<br>                v1.append(x)<br>                t1.add(a[x], <span class="hljs-number">1</span>)<br>            <span class="hljs-keyword">elif</span> d1 == d2 <span class="hljs-keyword">and</span> <span class="hljs-built_in">len</span>(v1) &gt; <span class="hljs-built_in">len</span>(v2):<br>                v2.append(x)<br>                t2.add(a[x], <span class="hljs-number">1</span>)<br>            <span class="hljs-keyword">else</span>:<br>                v1.append(x)<br>                t1.add(a[x], <span class="hljs-number">1</span>)<br>        <br>        <span class="hljs-keyword">return</span> v1 + v2<br></code></pre></td></tr></tbody></table></figure><figure class="highlight python"><figcaption><span>[]</span></figcaption><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span>:<br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">resultArray</span>(<span class="hljs-params">self, nums: <span class="hljs-type">List</span>[<span class="hljs-built_in">int</span>]</span>) -&gt; <span class="hljs-type">List</span>[<span class="hljs-built_in">int</span>]:<br>        <span class="hljs-keyword">from</span> sortedcontainers <span class="hljs-keyword">import</span> SortedList<br>        <br>        v1, v2 = copy.deepcopy(nums[:<span class="hljs-number">1</span>]), copy.deepcopy(nums[<span class="hljs-number">1</span>:<span class="hljs-number">2</span>])<br>        s1, s2 = SortedList(v1), SortedList(v2)<br><br>        <span class="hljs-keyword">for</span> x <span class="hljs-keyword">in</span> nums[<span class="hljs-number">2</span>:]:<br>            d1, d2 = <span class="hljs-built_in">len</span>(v1) - s1.bisect_right(x), <span class="hljs-built_in">len</span>(v2) - s2.bisect_right(x)<br>            <br>            <span class="hljs-keyword">if</span> d1 &gt; d2:<br>                v1.append(x)<br>                s1.add(x)<br>            <span class="hljs-keyword">elif</span> d1 &lt; d2:<br>                v2.append(x)<br>                s2.add(x)<br>            <span class="hljs-keyword">elif</span> d1 == d2 <span class="hljs-keyword">and</span> <span class="hljs-built_in">len</span>(v1) &lt; <span class="hljs-built_in">len</span>(v2):<br>                v1.append(x)<br>                s1.add(x)<br>            <span class="hljs-keyword">elif</span> d1 == d2 <span class="hljs-keyword">and</span> <span class="hljs-built_in">len</span>(v1) &gt; <span class="hljs-built_in">len</span>(v2):<br>                v2.append(x)<br>                s2.add(x)<br>            <span class="hljs-keyword">else</span>:<br>                v1.append(x)<br>                s1.add(x)<br>        <br>        <span class="hljs-keyword">return</span> v1 + v2<br></code></pre></td></tr></tbody></table></figure><h2 id="【线段树-二分】以组为单位订音乐会的门票-fire">【线段树/二分】以组为单位订音乐会的门票 <span class="github-emoji"><span>🔥</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/1f525.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span></h2><p><a href="https://leetcode.cn/problems/booking-concert-tickets-in-groups/">https://leetcode.cn/problems/booking-concert-tickets-in-groups/</a></p><blockquote><p>题意：给定一个长为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>≤</mo><mn>5</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mn>4</mn></msup></mrow><annotation encoding="application/x-tex">n\le 5 \times 10^4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">5</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span> 且初始值均为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 的数组 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span>，数组中的每个元素最多增加到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span>。现在需要以这个数组为基础进行 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo>≤</mo><mn>5</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mn>4</mn></msup></mrow><annotation encoding="application/x-tex">q\le 5 \times 10^4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">5</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span> 次询问，每次询问是以下两者之一：</p><ol><li>给定一个 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>i</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">lim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">im</span></span></span></span>，找到最小的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mi>l</mi><mi>i</mi><mi>m</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">i \in [0,lim]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6986em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">im</span><span class="mclose">]</span></span></span></span> 使得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>−</mo><msub><mi>a</mi><mi>i</mi></msub><mo>≥</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">m - a_i \ge k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span></li><li>给定一个 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>i</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">lim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">im</span></span></span></span>，找到最小的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mi>l</mi><mi>i</mi><mi>m</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">i \in [0,lim]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6986em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">im</span><span class="mclose">]</span></span></span></span> 使得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mi>m</mi><mo>×</mo><mo stretchy="false">(</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>0</mn></mrow><mi>i</mi></munderover><msub><mi>a</mi><mi>j</mi></msub><mo>≥</mo><mi>k</mi></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle m\times (i+1) - \sum_{j=0}^i a_j \ge k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:3.2254em;vertical-align:-1.4138em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8117em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4138em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span></li></ol><p>思路一：<strong>暴力</strong>。对于询问 1，我们直接顺序遍历 a 数组直到找到第一个符合条件的即可；对于询问 2，同样直接顺序遍历 a 数组直到找到第一个符合条件的即可。时间复杂度为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>q</mi><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(qn)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>。</p><p>思路二：<strong>线段树上二分</strong>。</p></blockquote><p>暴力代码：</p><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">class</span> <span class="hljs-title class_">BookMyShow</span>:<br><br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">__init__</span>(<span class="hljs-params">self, n: <span class="hljs-built_in">int</span>, m: <span class="hljs-built_in">int</span></span>):<br>        <span class="hljs-variable language_">self</span>.a = [<span class="hljs-number">0</span>] * n  <span class="hljs-comment"># a[i] 表示第 i 行已入座的人数</span><br>        <span class="hljs-variable language_">self</span>.n = n<br>        <span class="hljs-variable language_">self</span>.m = m<br><br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">gather</span>(<span class="hljs-params">self, k: <span class="hljs-built_in">int</span>, lim: <span class="hljs-built_in">int</span></span>) -&gt; <span class="hljs-type">List</span>[<span class="hljs-built_in">int</span>]:<br>        <span class="hljs-comment"># 在 [0, lim] 行中找到第一个可以容纳 k 人的行</span><br>        <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(lim + <span class="hljs-number">1</span>):<br>            <span class="hljs-keyword">if</span> <span class="hljs-variable language_">self</span>.m - <span class="hljs-variable language_">self</span>.a[i] &gt;= k:<br>                l, r = i, <span class="hljs-variable language_">self</span>.a[i]<br>                <span class="hljs-variable language_">self</span>.a[i] += k<br>                <span class="hljs-keyword">return</span> [l, r]<br>        <span class="hljs-keyword">return</span> []<br><br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">scatter</span>(<span class="hljs-params">self, k: <span class="hljs-built_in">int</span>, lim: <span class="hljs-built_in">int</span></span>) -&gt; <span class="hljs-built_in">bool</span>:<br>        <span class="hljs-comment"># 在 [0, lim] 行中找到最小的 i 使得 [0, i] 行可以容纳 k 人</span><br>        <span class="hljs-keyword">if</span> <span class="hljs-variable language_">self</span>.m * (lim + <span class="hljs-number">1</span>) - <span class="hljs-built_in">sum</span>(<span class="hljs-variable language_">self</span>.a[:lim+<span class="hljs-number">1</span>]) &lt; k:<br>            <span class="hljs-keyword">return</span> <span class="hljs-literal">False</span><br><br>        i = <span class="hljs-number">0</span><br>        <span class="hljs-keyword">while</span> k &gt; <span class="hljs-number">0</span>:<br>            <span class="hljs-keyword">if</span> <span class="hljs-variable language_">self</span>.m - <span class="hljs-variable language_">self</span>.a[i] &gt;= k:<br>                <span class="hljs-variable language_">self</span>.a[i] += k<br>                k = <span class="hljs-number">0</span><br>            <span class="hljs-keyword">else</span>:<br>                k -= <span class="hljs-variable language_">self</span>.m - <span class="hljs-variable language_">self</span>.a[i]<br>                <span class="hljs-variable language_">self</span>.a[i] = <span class="hljs-variable language_">self</span>.m<br>            i += <span class="hljs-number">1</span><br>        <span class="hljs-keyword">return</span> <span class="hljs-literal">True</span><br></code></pre></td></tr></tbody></table></figure><p>线段树上二分代码：</p><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs python"><br></code></pre></td></tr></tbody></table></figure>]]></content>
     
     
     <categories>
@@ -1606,11 +1606,11 @@
   
   
   <entry>
-    <title>divide-and-conquer</title>
-    <link href="/Algorithm/divide-and-conquer/"/>
-    <url>/Algorithm/divide-and-conquer/</url>
+    <title>binary-search</title>
+    <link href="/Algorithm/binary-search/"/>
+    <url>/Algorithm/binary-search/</url>
     
-    <content type="html"><![CDATA[<h3 id="分治">分治</h3><p>将大问题转化为等价小问题进行求解。</p><h3 id="【分治】随机排列">【分治】随机排列</h3><p><a href="https://www.acwing.com/problem/content/5469/">https://www.acwing.com/problem/content/5469/</a></p><blockquote><p>题意：给定一个 n 个数的全排列序列，并将其进行一定的对换，问是对换了 3n 次还是 7n+1 次</p><p>思路：可以发现对于两种情况，就对应对换次数的奇偶性。当 n 为奇数：3n 为奇数，7n+1 为偶数；当 n 为偶数：3n 为偶数，7n+1 为奇数。故我们只需要判断序列的逆序数即可。为了求解逆序数，我们可以采用归并排序的 combine 过程进行统计即可</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">1000010</span>;<br><br><span class="hljs-type">int</span> n, a[N], t[N];<br><span class="hljs-type">int</span> cnt; <span class="hljs-comment">// 逆序数 </span><br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">MergeSort</span><span class="hljs-params">(<span class="hljs-type">int</span> l, <span class="hljs-type">int</span> r)</span> </span>{<br><span class="hljs-keyword">if</span> (l &gt;= r) <span class="hljs-keyword">return</span>;<br><br><span class="hljs-type">int</span> mid = (l + r) &gt;&gt; <span class="hljs-number">1</span>;<br><br><span class="hljs-built_in">MergeSort</span>(l, mid), <span class="hljs-built_in">MergeSort</span>(mid + <span class="hljs-number">1</span>, r);<br><br><span class="hljs-type">int</span> i = l, j = mid + <span class="hljs-number">1</span>, idx = <span class="hljs-number">0</span>;<br><br><span class="hljs-keyword">while</span> (i &lt;= mid &amp;&amp; j &lt;= r) {<br><span class="hljs-keyword">if</span> (a[i] &lt; a[j]) t[idx++] = a[i++];<br><span class="hljs-keyword">else</span> {<br>cnt += mid - i + <span class="hljs-number">1</span>;<br>t[idx++] = a[j++];<br>}<br>}<br><br><span class="hljs-keyword">while</span> (i &lt;= mid) t[idx++] = a[i++];<br><span class="hljs-keyword">while</span> (j &lt;= r) t[idx++] = a[j++];<br><br><span class="hljs-keyword">for</span> (i = l, idx = <span class="hljs-number">0</span>; i &lt;= r; i++, idx++) a[i] = t[idx];<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>cin &gt;&gt; n;<br><br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) cin &gt;&gt; a[i];<br><br><span class="hljs-built_in">MergeSort</span>(<span class="hljs-number">0</span>, n - <span class="hljs-number">1</span>);<br><br><span class="hljs-type">int</span> res;<br><br><span class="hljs-keyword">if</span> (n % <span class="hljs-number">2</span> == <span class="hljs-number">1</span>) {<br><span class="hljs-keyword">if</span> (cnt % <span class="hljs-number">2</span>) res = <span class="hljs-number">1</span>;<br><span class="hljs-keyword">else</span> res = <span class="hljs-number">2</span>;<br>} <span class="hljs-keyword">else</span> {<br><span class="hljs-keyword">if</span> (cnt % <span class="hljs-number">2</span>) res = <span class="hljs-number">2</span>;<br><span class="hljs-keyword">else</span> res = <span class="hljs-number">1</span>;<br>}<br><br>cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>; <br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br><span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//cin &gt;&gt; T;</span><br><span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br><span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure>]]></content>
+    <content type="html"><![CDATA[<h3 id="二分">二分</h3><p>二分本质上是一个线性的算法思维，只是比线性思维更进一步的是，二分思维需要提炼出题面中两个线性相关的变量，即单调变化的两个变量，从而采用二分加速检索。</p><h3 id="【二分答案】Building-an-Aquarium">【二分答案】Building an Aquarium</h3><p><a href="https://codeforces.com/contest/1873/problem/E">https://codeforces.com/contest/1873/problem/E</a></p><blockquote><p>题意：想象有一个二维平面，现在有一个数列，每一个数表示平面对应列的高度，现在要给这个平面在两边加上护栏，问护栏最高可以设置为多高，可以使得在完全填满的情况下，使用的水量不会超过给定的用水量。已知最大用水量为k</p><p>思路：对于一个护栏高度，水池高度低于护栏高度的地方都需要被水填满。为了便于分析，我们可以将水池高度进行排序。那么就会很显然的一个二分题目了，我们需要二分的就是护栏的高度（最小为1，最大需要考虑一下，就是只有一列的情况下，最大高度就是最高水池高度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>max</mi><mo>⁡</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>+</mo><mi>m</mi><mi>a</mi><mi>x</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\max(a_i)+max(k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">max</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">ma</span><span class="mord mathnormal">x</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span>），check的条件就是当前h的护栏高度时，消耗的水量与最大用水量之间的大小关系，如果超过了，那么高度就要下降，反之可以上升。由于是求最大高度，因此要使用的是求右边界的二分板子</p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-keyword">typedef</span> <span class="hljs-type">long</span> <span class="hljs-type">long</span> ll;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span></span><br><span class="hljs-function"></span>{<br>    <span class="hljs-type">int</span> n, w;<br>    cin &gt;&gt; n &gt;&gt; w;<br>    <span class="hljs-function">vector&lt;ll&gt; <span class="hljs-title">a</span><span class="hljs-params">(n)</span></span>;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i ++)<br>        cin &gt;&gt; a[i];<br>    <br>    <span class="hljs-built_in">sort</span>(a.<span class="hljs-built_in">begin</span>(), a.<span class="hljs-built_in">end</span>());<br>    <br>    ll l = <span class="hljs-number">0</span>, r = <span class="hljs-number">2e9</span> + <span class="hljs-number">1</span>;<br>    <span class="hljs-keyword">while</span> (l &lt; r)<br>    {<br>        ll h = (l + r + <span class="hljs-number">1</span>) &gt;&gt; <span class="hljs-number">1</span>;<br>        <br>        ll t = <span class="hljs-number">0</span>;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i ++)<br>            <span class="hljs-keyword">if</span> (a[i] &lt; h)<br>                t += h - a[i];<br>            <span class="hljs-keyword">else</span> <span class="hljs-keyword">break</span>;<br>        <br>        <span class="hljs-keyword">if</span> (t &lt;= w) l = h;<br>        <span class="hljs-keyword">else</span> r = h - <span class="hljs-number">1</span>;<br>    }<br>    cout &lt;&lt; r &lt;&lt; endl;<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span></span><br><span class="hljs-function"></span>{<br>    <span class="hljs-type">int</span> T; cin &gt;&gt; T;<br>    <span class="hljs-keyword">while</span> (T --) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【二分答案】分组">【二分答案】分组</h3><p><a href="https://www.lanqiao.cn/problems/5129/learning/">https://www.lanqiao.cn/problems/5129/learning/</a></p><blockquote><p>题意：给定一个序列，现在需要将这个数列分为k组，如何分组可以使得每一组的极差中，最大值最小</p><p>最开始想到的思路：很容易联想到的一种方法其实就是高中组合数学中学到的“隔板法”，现在有n个数，需要分成k组，则方案数就是在n-1个空档中插入k-1个隔板，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi>C</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">C_{n-1}^{k-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2139em;vertical-align:-0.3246em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8892em;"><span style="top:-2.4337em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.1031em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3246em;"><span></span></span></span></span></span></span></span></span></span> 种方案</p><p>时间复杂度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></p><p>优化思路：上述思路是正向思维，即对于构思分组情况计算极差。我们不妨逆向思维，即枚举极差的情况，判断此时的分组情况。如果对于当前的极差lim，我们显然可以分成n组，即有一个最大分组值；我们也可以求一个最小分组值cnt，即如果再少分一组那么此时的极差就会超过当前约束的极差值lim。因此对于当前约束的极差值lim，我们可以求一个最小分组值cnt</p><ul><li>如果当前的最小分组值cnt &gt; k，那么 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">[</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo separator="true">,</mo><mi>n</mi><mo fence="true">]</mo></mrow><annotation encoding="application/x-tex">\left [ cnt,n \right ]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose delimcenter" style="top:0em;">]</span></span></span></span></span> 就无法包含k，也就是说当前约束的极差lim不符合条件，lim偏小</li><li>如果当前的最小分组值cnt &lt;= k，那么 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo fence="true">[</mo><mi>c</mi><mi>n</mi><mi>t</mi><mo separator="true">,</mo><mi>n</mi><mo fence="true">]</mo></mrow><annotation encoding="application/x-tex">\left [ cnt,n \right ]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">[</span><span class="mord mathnormal">c</span><span class="mord mathnormal">n</span><span class="mord mathnormal">t</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose delimcenter" style="top:0em;">]</span></span></span></span></span> 就一定包含k，且当前分组的最小极差一定是 &lt;= 约束的极差值lim，lim偏大</li></ul><p>于是二分极差的思路就跃然纸上了。我们二分极差，然后根据可以分组的最小数量cnt判断二分的结果进行左右约束调整即可。</p><p>时间复杂度 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><br><span class="hljs-function"><span class="hljs-type">bool</span> <span class="hljs-title">check</span><span class="hljs-params">(<span class="hljs-type">int</span> lim, vector&lt;<span class="hljs-type">int</span>&gt;&amp; a, <span class="hljs-type">int</span> n, <span class="hljs-type">int</span> k)</span> </span>{<br>    <span class="hljs-type">int</span> cnt = <span class="hljs-number">1</span>; <span class="hljs-comment">// 当前可以分的最小组数</span><br>    <span class="hljs-type">int</span> pre = a[<span class="hljs-number">0</span>];<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        <span class="hljs-keyword">if</span> (a[i] - pre &gt; lim) {<br>            pre = a[i];<br>            cnt++;<br>        }<br>    }<br>    <span class="hljs-keyword">return</span> cnt &lt;= k;<br>}<br><br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-type">int</span> n, k;<br>    cin &gt;&gt; n &gt;&gt; k;<br><br>    <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">a</span><span class="hljs-params">(n)</span></span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        cin &gt;&gt; a[i];<br>    }<br><br>    <span class="hljs-built_in">sort</span>(a.<span class="hljs-built_in">begin</span>(), a.<span class="hljs-built_in">begin</span>() + n);<br><br>    <span class="hljs-type">int</span> l = <span class="hljs-number">0</span>, r = a[n - <span class="hljs-number">1</span>] - a[<span class="hljs-number">0</span>];<br>    <span class="hljs-keyword">while</span> (l &lt; r) {<br>        <span class="hljs-type">int</span> mid = (l + r) &gt;&gt; <span class="hljs-number">1</span>;<br>        <span class="hljs-keyword">if</span> (<span class="hljs-built_in">check</span>(mid, a, n, k)) {<br>            <span class="hljs-comment">// 分的最小组数 &lt;= k，则当前极差大了</span><br>            r = mid;<br>        } <span class="hljs-keyword">else</span> {<br>            <span class="hljs-comment">// 分的最小组数 &gt;  k，则当前极差小了</span><br>            l = mid + <span class="hljs-number">1</span>;<br>        }<br>    }<br><br>    cout &lt;&lt; r &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-number">0</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-number">0</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br>    <span class="hljs-comment">// cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) {<br>        <span class="hljs-built_in">solve</span>();<br>    }    <br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【二分答案】木材加工">【二分答案】木材加工</h3><p><a href="https://www.luogu.com.cn/problem/P2440">https://www.luogu.com.cn/problem/P2440</a></p><blockquote><p>题意：给定一个序列，现在需要将这个序列切分为等长的 k 段，长度必须为整数且尽可能的长，如果无法切分可以将多余的长度丢弃，问最长的长度是多少</p><p>思路：可以发现切分长度与切分的段数具有单调性，即切分的长度越长，切出来的段数就越少，可以进行二分。二分的思路就是直接二分答案，根据长度判断可切得的段数，最终套右边界的模板找到最大的长度即可。需要注意的是，对于无法切割的情况，就是需要切出的段数 k 超过了序列之和</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mrow><mo stretchy="false">(</mo><mn>1</mn><mi>e</mi><mn>8</mn><mo stretchy="false">)</mo></mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\log {(1e8)})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">1</span><span class="mord mathnormal">e</span><span class="mord">8</span><span class="mclose">)</span></span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long </span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">1e5</span> + <span class="hljs-number">10</span>;<br><br><span class="hljs-type">int</span> n, k;<br><span class="hljs-type">int</span> a[N];<br><br><span class="hljs-function"><span class="hljs-type">bool</span> <span class="hljs-title">chk</span><span class="hljs-params">(<span class="hljs-type">int</span> x)</span> </span>{<br>    <span class="hljs-type">int</span> sum = <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++)<br>        sum += a[i] / x;<br>    <span class="hljs-keyword">return</span> sum &gt;= k;<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; k;<br>    <br>    <span class="hljs-type">int</span> sum = <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        cin &gt;&gt; a[i];<br>        sum += a[i];<br>    }<br>    <br>    <span class="hljs-type">int</span> l = <span class="hljs-number">1</span>, r = <span class="hljs-number">1e8</span>;<br>    <span class="hljs-keyword">while</span> (l &lt; r) {<br>        <span class="hljs-type">int</span> mid = (l + r + <span class="hljs-number">1</span>) &gt;&gt; <span class="hljs-number">1</span>;<br>        <span class="hljs-keyword">if</span> (<span class="hljs-built_in">chk</span>(mid)) l = mid;<br>        <span class="hljs-keyword">else</span> r = mid - <span class="hljs-number">1</span>;<br>    }<br>    <br>    <span class="hljs-keyword">if</span> (k &gt; sum) cout &lt;&lt; <span class="hljs-string">"0\n"</span>;<br>    <span class="hljs-keyword">else</span> cout &lt;&lt; r &lt;&lt; <span class="hljs-string">"\n"</span>;    <br>} <br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();    <br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【二分答案】跳石头">【二分答案】跳石头</h3><p><a href="https://www.luogu.com.cn/problem/P2678">https://www.luogu.com.cn/problem/P2678</a></p><blockquote><p>题意：给定 n 个递增的不重复数，现在可以从其中拿掉 k 个数，使得相邻数字之间的最小差值最大，问最大的最小差值是多少</p><p>思路：<strong>二分答案。将答案看做 y 轴，拿掉的数的数量看做 x 轴，就是二分 y 值。<strong>可以发现拿的数字越多，最小差值就越大，具有单调性，可以二分。我们直接二分答案，即直接二分最小差值的具体数值，通过判断当前的最小差值至少需要拿掉多少个数才能满足，进行 check 操作。至于如何计算至少要拿掉的数字，我们采用</strong>右贪心</strong>准则，即检查当前点与上一个点之间的距离是否满足最小差值的要求，如果不满足就需要记数，为了便于后续的计算，直接将当前的下标用上一个点的下标覆盖掉即可</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long </span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">5e4</span> + <span class="hljs-number">10</span>;<br><br><span class="hljs-type">int</span> lim, n, k;<br><span class="hljs-type">int</span> a[N], b[N];<br><span class="hljs-type">bool</span> del[N];<br><br><span class="hljs-function"><span class="hljs-type">bool</span> <span class="hljs-title">ok</span><span class="hljs-params">(<span class="hljs-type">int</span> x)</span> </span>{<br>    <span class="hljs-type">int</span> cnt = <span class="hljs-number">0</span>;<br>    <span class="hljs-built_in">memset</span>(del, <span class="hljs-literal">false</span>, <span class="hljs-keyword">sizeof</span> del);<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        b[i] = a[i];<br>    }<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-keyword">if</span> (b[i] - b[i - <span class="hljs-number">1</span>] &lt; x) {<br>            del[i] = <span class="hljs-literal">true</span>;<br>            b[i] = b[i - <span class="hljs-number">1</span>];<br>            cnt++;<br>        }<br>    }<br>    <br>    <span class="hljs-keyword">if</span> (lim - b[n] &lt; x) {<br>        cnt++;<br>    }<br>    <br>    <span class="hljs-keyword">return</span> cnt &lt;= k;<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; lim &gt;&gt; n &gt;&gt; k;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++)<br>        cin &gt;&gt; a[i];<br>    <br>    <span class="hljs-type">int</span> l = <span class="hljs-number">1</span>, r = lim;<br>    <span class="hljs-keyword">while</span> (l &lt; r) {<br>        <span class="hljs-type">int</span> mid = (l + r + <span class="hljs-number">1</span>) &gt;&gt; <span class="hljs-number">1</span>;<br>        <span class="hljs-keyword">if</span> (<span class="hljs-built_in">ok</span>(mid)) l = mid;<br>        <span class="hljs-keyword">else</span> r = mid - <span class="hljs-number">1</span>;<br>    }<br>    <br>    cout &lt;&lt; r &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();    <br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【二分答案】路标设置">【二分答案】路标设置</h3><p><a href="https://www.luogu.com.cn/problem/P3853">https://www.luogu.com.cn/problem/P3853</a></p><blockquote><p>题意：与第四题题面几乎一致，只是现在不是从序列中拿走 k 数个，而是往序列中插入 k 个数（插入数字后要保证序列仍然没有重复数且递增），问在插入一定数量数字的情况下，最小的最大差值是多少</p><p>思路：**二分答案。将答案看做 y 轴，拿掉的数的数量看做 x 轴，就是二分 y 值。**同样可以发现，插入的数字越多，最大差值就越小，具有单调性，可以二分。我们依然直接二分答案，即直接二分最大差值的具体数值，通过判断当前的最大差值需要插入多少个数来检查当前状态是否合理。需要插入的数字的个数为：</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mo fence="true">⌈</mo><mfrac><mrow><mi>a</mi><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">]</mo><mo>−</mo><mi>a</mi><mo stretchy="false">[</mo><mi>i</mi><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><mrow><mi>d</mi><mi>i</mi><mi>s</mi><msub><mi>t</mi><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></mrow></mfrac><mo fence="true">⌉</mo></mrow><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\left \lceil \frac{a[i]-a[i-1]}{dist_{max}}\right \rceil - 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">⌈</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">i</span><span class="mord mathnormal">s</span><span class="mord"><span class="mord mathnormal">t</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">ma</span><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">a</span><span class="mopen">[</span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span><span class="mclose">]</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.836em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">⌉</span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long </span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">1e6</span>;<br><br><span class="hljs-type">int</span> lim, n, k;<br><span class="hljs-type">int</span> a[N];<br><br><span class="hljs-function"><span class="hljs-type">bool</span> <span class="hljs-title">chk</span><span class="hljs-params">(<span class="hljs-type">int</span> x)</span> </span>{<br>    <span class="hljs-type">int</span> cnt = <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt; n; i++) {<br>        <span class="hljs-type">int</span> gap = a[i] - a[i - <span class="hljs-number">1</span>];<br>        <span class="hljs-keyword">if</span> (gap &gt; x) {<br>            cnt += (gap + x - <span class="hljs-number">1</span>) / x - <span class="hljs-number">1</span>;<br>        }<br>    }<br>    <span class="hljs-keyword">return</span> cnt &gt; k;<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; lim &gt;&gt; n &gt;&gt; k;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        cin &gt;&gt; a[i];<br>    }<br>    <br>    <span class="hljs-type">int</span> l = <span class="hljs-number">1</span>, r = lim;<br>    <span class="hljs-keyword">while</span> (l &lt; r) {<br>        <span class="hljs-type">int</span> mid = (l + r) &gt;&gt; <span class="hljs-number">1</span>;<br>        <span class="hljs-keyword">if</span> (<span class="hljs-built_in">chk</span>(mid)) l = mid + <span class="hljs-number">1</span>;<br>        <span class="hljs-keyword">else</span> r = mid;<br>    }<br>    <br>    cout &lt;&lt; r &lt;&lt; <span class="hljs-string">"\n"</span>;<br>} <br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();    <br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【二分答案】数列分段-Section-II">【二分答案】数列分段 Section II</h3><p><a href="https://www.luogu.com.cn/problem/P1182">https://www.luogu.com.cn/problem/P1182</a></p><blockquote><p>题意：给定一个无序的序列，现在需要将这个序列进行分段（连续的），分成指定的段数。问应该如何分段可以使得所有段的分段和的最大值最小，输出这个最小的最大值</p><p>思路：**二分答案，将答案看做 y 轴，拿掉的数的数量看做 x 轴，就是二分 y 值。**可以发现，分的段数越多，所有分段和的最大值就越小，具有单调性，可以二分。我们直接二分答案，即直接二分分段最大值，通过判断当前最大值的约束条件下可以分的组数进行判断。至于如何计算当前最大值条件下可分得的组数，直接线性扫描进行局部求和即可</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long </span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">1e5</span> + <span class="hljs-number">10</span>;<br><br><span class="hljs-type">int</span> n, k;<br><span class="hljs-type">int</span> a[N];<br><br><span class="hljs-comment">// 当前分组时最大子段和为 x </span><br><span class="hljs-function"><span class="hljs-type">bool</span> <span class="hljs-title">chk</span><span class="hljs-params">(<span class="hljs-type">int</span> x)</span> </span>{<br>    <span class="hljs-type">int</span> cnt = <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>, s = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        <span class="hljs-keyword">if</span> (a[i] &gt; x) <span class="hljs-keyword">return</span> <span class="hljs-literal">true</span>;<br>        <br>        <span class="hljs-keyword">if</span> (s + a[i] &lt;= x) s += a[i];<br>        <span class="hljs-keyword">else</span> {<br>            cnt++;<br>            s = a[i];<br>        }<br>    }<br>    cnt += <span class="hljs-number">1</span>;<br>    <br>    <span class="hljs-keyword">return</span> cnt &gt; k;<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; k;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        cin &gt;&gt; a[i];<br>    }<br>    <br>    <span class="hljs-type">int</span> l = <span class="hljs-number">0</span>, r = <span class="hljs-number">1e9</span>;<br>    <span class="hljs-keyword">while</span> (l &lt; r) {<br>        <span class="hljs-type">int</span> mid = (l + r) &gt;&gt; <span class="hljs-number">1</span>;<br>        <span class="hljs-keyword">if</span> (<span class="hljs-built_in">chk</span>(mid)) l = mid + <span class="hljs-number">1</span>;<br>        <span class="hljs-keyword">else</span> r = mid;<br>    }<br>    <br>    cout &lt;&lt; r &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();    <br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【二分答案】kotori的设备">【二分答案】kotori的设备</h3><p><a href="https://www.luogu.com.cn/problem/P3743">https://www.luogu.com.cn/problem/P3743</a></p><blockquote><p>题意：现在有一批电子设备，每一个电子设备有一个电量消耗速度与当前剩余电量，现在有一个充电器有一个确定的充电速度。问这批设备最久可以运作多久？当可以无限运作时输出 -1</p><p>思路：可以发现，想要运行的越久，需要补充的电量就越多，具有单调性，可以直接二分答案。很显然我们可以根据期望运行的时间进行 check 操作，通过比对当前期望时间可以充的电量与需要补充的电量进行比对来修改边界值。<strong>需要注意的是</strong>边界的选择，最长可运行时间为 1e10，具体 <a href="https://www.luogu.com.cn/discuss/468555">推导</a> 待定</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">(</mo><mi>log</mi><mo>⁡</mo><mrow><mn>1</mn><msup><mn>0</mn><mn>10</mn></msup><mo stretchy="false">)</mo></mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n(\log{10^{10})})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">100010</span>;<br><br><span class="hljs-type">int</span> n;<br><span class="hljs-type">double</span> p, v[N], s[N];<br><br><span class="hljs-function"><span class="hljs-type">bool</span> <span class="hljs-title">chk</span><span class="hljs-params">(<span class="hljs-type">double</span> x)</span> </span>{<br>    <span class="hljs-comment">// x 为当前状态期望使用的时间 </span><br>    <span class="hljs-type">double</span> need = <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++)<br>        <span class="hljs-keyword">if</span> (v[i] * x &gt; s[i])<br>            need += v[i] * x - s[i];<br>    <span class="hljs-keyword">return</span> need &lt;= x * p;<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; p;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        cin &gt;&gt; v[i] &gt;&gt; s[i];<br>    }<br>    <br>    <span class="hljs-type">double</span> l = <span class="hljs-number">0</span>, r = <span class="hljs-number">1e10</span> + <span class="hljs-number">10</span>;<br>    <span class="hljs-keyword">while</span> (r - l &gt; <span class="hljs-number">1e-6</span>) {<br>        <span class="hljs-type">double</span> mid = (l + r) / <span class="hljs-number">2</span>; <br>        <span class="hljs-keyword">if</span> (<span class="hljs-built_in">chk</span>(mid)) l = mid;<br>        <span class="hljs-keyword">else</span> r = mid;<br>    }<br>    <br>    <span class="hljs-keyword">if</span> (r &gt; <span class="hljs-number">1e10</span>) cout &lt;&lt; <span class="hljs-number">-1</span> &lt;&lt; <span class="hljs-string">"\n"</span>;<br>    <span class="hljs-keyword">else</span> cout &lt;&lt; r &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【二分答案-并查集】关押罪犯-fire">【二分答案/并查集】关押罪犯 <span class="github-emoji"><span>🔥</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/1f525.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span></h3><p><a href="https://www.luogu.com.cn/problem/P1525">https://www.luogu.com.cn/problem/P1525</a></p><blockquote><p>题意：给定一个无向图，没有重边和自环，边权为正。现在需要将图中所有的顶点分为两部分，使得两部分中最大的边权尽可能小，问该最小边权是多少</p><p>思路一：二分答案。</p><ul><li>思路：本题一眼二分图问题，但是有些变化的是左右部并非散点图，其中是有连边的。如何求出最小边权呢？我们可以这么想，首先答案一定出自左右部的连边中（除非左右部全是散点，那答案就是 0），之所以可以采用二分图将点集分为两部分，是因为我们在某种规则下忽略了其中奇数环（二分图定理）的一些边。当规则定义为 <strong>忽略边权不超过 x</strong> 的边时，该图若可以二分，那么忽略边权比 x 值更大的边时，该图同样一定也可以二分，反之则不一定可以二分（特性图如下）。具备单调性，于是我们可以通过 <strong>二分阈值、检查图是否可二分</strong> 的方法来计算出答案</li></ul><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202402230117928.png" alt="image-20240223011704197"></p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Θ</mi><mo stretchy="false">(</mo><mi>log</mi><mo>⁡</mo><msub><mi>C</mi><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>×</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta(\log C_{max} \times (n+e))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Θ</span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">ma</span><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">e</span><span class="mclose">))</span></span></span></span></p><p>思路二：并查集。</p><ul><li><p>思路：</p></li><li><p>时间复杂度：</p></li></ul></blockquote><p>二分判定二分图代码：</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br></pre></td><td class="code"><pre><code class="hljs cpp">```<br><br>并查集代码：<br><br>```cpp<br></code></pre></td></tr></tbody></table></figure><h3 id="【二分答案】摆放棋子">【二分答案】摆放棋子</h3><p><a href="https://www.acwing.com/problem/content/5562/">https://www.acwing.com/problem/content/5562/</a></p><blockquote><p>题意：给定一个 01 序列表示一个 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">1\times n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 的棋子序列，其中 1 表示有棋子，0 表示没有棋子。现在给定 k 个棋子，如何放置可以使得棋盘上连续的棋子长度尽可能长？给出一个合法的最长的序列</p><p>思路：可以发现，想要序列尽可能的长，那么需要放置的棋子就要尽可能的多，具备单调性，可以二分。我们二分答案，即最长序列长度。对于已知的 k 个可放置棋子，我们需要找到最大的序列长度，于是我们套用<strong>寻找右边界</strong>模板。检查方法就是判断对于当前的长度，通过前缀和与滑动窗口的形式计算当前窗口内需要放置多少颗棋子才能连为一体：</p><ul><li>若需要的棋子数 &lt; k，说明可以继续增大长度</li><li>若需要的棋子数 &gt; k，说明当前长度无法满足，要缩小长度</li><li>若需要的棋子数 = k，归属在 &lt; k 的类比中即可。</li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;cstring&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;vector&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;queue&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;stack&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;algorithm&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;unordered_map&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;set&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">300010</span>;<br><br><span class="hljs-type">int</span> n, k;<br><span class="hljs-type">int</span> a[N], s[N];<br><br><span class="hljs-function"><span class="hljs-type">bool</span> <span class="hljs-title">chk</span><span class="hljs-params">(<span class="hljs-type">int</span> x)</span> </span>{<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = x; i &lt;= n; i++) {<br>        <span class="hljs-keyword">if</span> (x - (s[i] - s[i - x]) &lt;= k) {<br>            <span class="hljs-keyword">return</span> <span class="hljs-literal">true</span>;<br>        }<br>    }<br>    <span class="hljs-keyword">return</span> <span class="hljs-literal">false</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; k;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        cin &gt;&gt; a[i];<br>        s[i] = s[i - <span class="hljs-number">1</span>] + a[i];<br>    }<br><br>    <span class="hljs-type">int</span> l = <span class="hljs-number">0</span>, r = n;<br>    <span class="hljs-keyword">while</span> (l &lt; r) {<br>        <span class="hljs-type">int</span> mid = (l + r + <span class="hljs-number">1</span>) &gt;&gt; <span class="hljs-number">1</span>;<br>        <span class="hljs-keyword">if</span> (<span class="hljs-built_in">chk</span>(mid)) l = mid;<br>        <span class="hljs-keyword">else</span> r = mid - <span class="hljs-number">1</span>;<br>    }<br><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = r; i &lt;= n; i++) {<br>        <span class="hljs-keyword">if</span> (r - (s[i] - s[i - r]) &lt;= k) {<br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = i - r + <span class="hljs-number">1</span>; j &lt;= i; j++) {<br>                a[j] = <span class="hljs-number">1</span>;<br>            }<br>            <span class="hljs-keyword">break</span>;<br>        }<br>    }<br><br>    cout &lt;&lt; r &lt;&lt; <span class="hljs-string">"\n"</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) cout &lt;&lt; a[i] &lt;&lt; <span class="hljs-string">" "</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h3 id="【二分答案】盖楼">【二分答案】盖楼</h3><p><a href="https://www.acwing.com/problem/content/description/5569/">https://www.acwing.com/problem/content/description/5569/</a></p><blockquote><p>题意：给定 H 个正整数分别为 1 到 H。现在要将这 H 个正整数分给两个人，其中一个人希望获得可以不被 x 整除的数，另一个人希望可以获得不被 y 整除的数。其中 x 和 y 均为质数，问最小的 H 是多少？</p><p>思路：</p><ul><li><p>很显然的一个二分答案。因为 H 越大，越有可能满足这两个人的要求，具备单调性。</p></li><li><p>那么现在的问题就是检查函数的设计。对于当前的高度 h，即 h 个正整数，很显然可以划分出下面的集合关系</p>  <img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202404091829325.png" alt="集合关系" style="zoom:50%;"><p>其中 h-p-q+a 的部分两个人都可以获得。最策略是，p-a 与 q-a 的都给另外的人，如果不够，从 h-p-q+a 中拿，如果还不够那就说明当前 h 无法满足，需要增大 h，反之说明 h 可以满足，继续寻找答案的左边界。</p></li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Θ</mi><mo stretchy="false">(</mo><mi>log</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>N</mi><mo>+</mo><mi>M</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta(\log (N+M))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Θ</span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="mclose">))</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><span class="hljs-keyword">typedef</span> <span class="hljs-type">long</span> <span class="hljs-type">long</span> ll;<br><br>ll n, m, x, y;<br><br><span class="hljs-function"><span class="hljs-type">bool</span> <span class="hljs-title">chk</span><span class="hljs-params">(ll h)</span> </span>{<br>    ll p = h / x, q = h / y, a = h / (x * y);<br><br>    <span class="hljs-keyword">return</span> <span class="hljs-built_in">max</span>(<span class="hljs-number">0ll</span>, n - (q - a)) + <span class="hljs-built_in">max</span>(<span class="hljs-number">0ll</span>, m - (p - a)) &lt;= h - p - q + a;<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; m &gt;&gt; x &gt;&gt; y;<br><br>    ll l = <span class="hljs-number">1</span>, r = <span class="hljs-number">1e15</span>;<br>    <span class="hljs-keyword">while</span> (l &lt; r) {<br>        ll mid = (l + r) &gt;&gt; <span class="hljs-number">1</span>;<br>        <span class="hljs-keyword">if</span> (<span class="hljs-built_in">chk</span>(mid)) r = mid;<br>        <span class="hljs-keyword">else</span> l = mid + <span class="hljs-number">1</span>;<br>    }<br><br>    cout &lt;&lt; r &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><p>小插曲。一开始写的检查函数 code 始终只能通过 9/10，检查函数的代码如下：</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-function"><span class="hljs-type">bool</span> <span class="hljs-title">chk</span><span class="hljs-params">(ll h)</span> </span>{<br>    ll p = h / x, q = h / y, a = h / (x * y);<br><br>    <span class="hljs-keyword">if</span> (h - p &gt;= n &amp;&amp; h - q &gt;= m) {<br>        <span class="hljs-comment">// 如果满足第一人，检查第二人是否满足</span><br>        <span class="hljs-keyword">if</span> (q - a &gt;= n) <span class="hljs-keyword">return</span> <span class="hljs-literal">true</span>;<br>        <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (h - (n - (q - a)) - q &gt;= m) <span class="hljs-keyword">return</span> <span class="hljs-literal">true</span>;<br><br>        <span class="hljs-comment">// 如果满足第二人，检查第一人是否满足</span><br>        <span class="hljs-keyword">if</span> (p - a &gt;= m) <span class="hljs-keyword">return</span> <span class="hljs-literal">true</span>;<br>        <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (h - (n - (p - a)) - p &gt;= n) <span class="hljs-keyword">return</span> <span class="hljs-literal">true</span>;<br>    }<br>    <span class="hljs-keyword">return</span> <span class="hljs-literal">false</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><p>其实是因为有逻辑错误，即对于当前的 h，必须两个人都能满足才行。而上述代码可能会出现第一个人不满足，但是第二个人满足，却返回 true 的情况，而这是错误的。因此上述代码改成两个人同时满足就可以通过了，即如果有一个人不满足，则返回 false：</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-function"><span class="hljs-type">bool</span> <span class="hljs-title">chk</span><span class="hljs-params">(ll h)</span> </span>{<br>    ll p = h / x, q = h / y, a = h / (x * y);<br><br>    <span class="hljs-keyword">if</span> (h - p &gt;= n &amp;&amp; h - q &gt;= m) {<br>        <span class="hljs-comment">// 如果满足第一人，检查第二人是否满足</span><br>        <span class="hljs-keyword">if</span> (q - a &gt;= n) <span class="hljs-keyword">return</span> <span class="hljs-literal">true</span>;<br>        <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (h - (n - (q - a)) - q &gt;= m) <span class="hljs-keyword">return</span> <span class="hljs-literal">true</span>;<br><br>        <span class="hljs-keyword">return</span> <span class="hljs-literal">false</span>;<br>    }<br>    <span class="hljs-keyword">return</span> <span class="hljs-literal">false</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><p>小结：过多的分支语句不如一个 max 来的更加清晰，也可以避免一定的逻辑错误。</p><h3 id="【二分答案】找出叠涂元素">【二分答案】找出叠涂元素</h3><p><a href="https://leetcode.cn/problems/first-completely-painted-row-or-column/description/">https://leetcode.cn/problems/first-completely-painted-row-or-column/description/</a></p><blockquote><p>题意：给定一个 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">m\times n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 的矩阵，以及一个存储矩阵中所有元素值的数组，现在从左往右将数组中的元素在对应矩阵中涂色，问序列中最左边使得矩阵中某一行或列全部涂色的下标是什么</p><p>思路：显然，下标越往右越有可能出现矩阵中某一行或列全部涂色，具备单调性，可以二分答案。我们直接二分序列下标，对于 chk 函数我们直接 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>×</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\times m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span> 模拟即可</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>×</mo><mi>m</mi><mi>log</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>n</mi><mo>×</mo><mi>m</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\times m \log (n\times m))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">m</span><span class="mclose">))</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span> {<br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">firstCompleteIndex</span><span class="hljs-params">(vector&lt;<span class="hljs-type">int</span>&gt;&amp; arr, vector&lt;vector&lt;<span class="hljs-type">int</span>&gt;&gt;&amp; mat)</span> </span>{<br>        <span class="hljs-type">int</span> m = mat.<span class="hljs-built_in">size</span>(), n = mat[<span class="hljs-number">0</span>].<span class="hljs-built_in">size</span>();<br><br>        <span class="hljs-keyword">auto</span> chk = [&amp;](<span class="hljs-type">int</span> idx) {<br>            <span class="hljs-type">bool</span> vis[<span class="hljs-number">100010</span>] {};<br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt;= idx; i++) {<br>                vis[arr[i]] = <span class="hljs-literal">true</span>;<br>            }<br><br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; m; i++) {<br>                <span class="hljs-type">int</span> s = <span class="hljs-number">0</span>;<br>                <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">0</span>; j &lt; n; j++) {<br>                    s += vis[mat[i][j]];<br>                }<br>                <span class="hljs-keyword">if</span> (s == n) <span class="hljs-keyword">return</span> <span class="hljs-literal">true</span>;<br>            }<br><br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">0</span>; j &lt; n; j++) {<br>                <span class="hljs-type">int</span> s = <span class="hljs-number">0</span>;<br>                <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; m; i++) {<br>                    s += vis[mat[i][j]];<br>                }<br>                <span class="hljs-keyword">if</span> (s == m) <span class="hljs-keyword">return</span> <span class="hljs-literal">true</span>;<br>            }<br><br>            <span class="hljs-keyword">return</span> <span class="hljs-literal">false</span>;<br>        };<br><br>        <span class="hljs-type">int</span> l = <span class="hljs-number">0</span>, r = m * n - <span class="hljs-number">1</span>;<br>        <span class="hljs-keyword">while</span> (l &lt; r) {<br>            <span class="hljs-type">int</span> mid = (l + r) &gt;&gt; <span class="hljs-number">1</span>;<br>            <span class="hljs-keyword">if</span> (<span class="hljs-built_in">chk</span>(mid)) r = mid;<br>            <span class="hljs-keyword">else</span> l = mid + <span class="hljs-number">1</span>;<br>        }<br>        <span class="hljs-keyword">return</span> r;<br>    }<br>};<br></code></pre></td></tr></tbody></table></figure><h3 id="【二分答案-双指针】找出唯一性数组的中位数">【二分答案/双指针】找出唯一性数组的中位数</h3><p><a href="https://leetcode.cn/problems/find-the-median-of-the-uniqueness-array/">https://leetcode.cn/problems/find-the-median-of-the-uniqueness-array/</a></p><blockquote><p>题意：给定一个数组，返回其「唯一性数组」的中位数。如果唯一性数组有奇数个元素，则返回中间元素，如果有偶数个元素，则返回两个中间元素中较小的那个，即左边那个中间元素。唯一性数组定义为：原数组的所有非空子数组中不同元素的个数组成的序列。</p><p>思路：</p><ul><li>首先根据题意可以明确的计算出这个唯一性数组的元素个数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mi>o</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">tot</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span><span class="mord mathnormal">o</span><span class="mord mathnormal">t</span></span></span></span>。假设原数组有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个元素，则其唯一性数组一定有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mi>t</mi><mi>o</mi><mi>t</mi><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mi>n</mi><mo stretchy="false">)</mo><mo>×</mo><mi>n</mi></mrow><mn>2</mn></mfrac></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle tot=\frac{(1+n) \times n}{2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6151em;"></span><span class="mord mathnormal">t</span><span class="mord mathnormal">o</span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">n</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 个元素，且最小值一定是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，最大值 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≤</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">\le n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>。最暴力的做法就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span> 的维护出这个唯一性数组然后返回其中位数，但这对于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">10^5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span> 级别的数据显然是不合适的。观察其他性质。</li><li>不难发现，对于「不同元素的个数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>」而言，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 越大，满足条件的子数组数量就越多，反之 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> 越小，则满足条件的子数组数量就越少，具备单调性，考虑二分。</li><li>问题就是在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">[</mo><mn>1</mn><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[1,n]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">]</span></span></span></span> 内找到一个尽可能小的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mi>p</mi><mi>p</mi><mi>e</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">upper</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">u</span><span class="mord mathnormal">pp</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span></span></span></span> 使得满足「不同元素的个数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≤</mo><mi>u</mi><mi>p</mi><mi>p</mi><mi>e</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">\le upper</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">u</span><span class="mord mathnormal">pp</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span></span></span></span> 的子数组」的数量 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mo>≥</mo><mo stretchy="false">⌈</mo><mfrac><mrow><mi>t</mi><mi>o</mi><mi>t</mi></mrow><mn>2</mn></mfrac><mo stretchy="false">⌉</mo></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle \ge \lceil \frac{tot}{2} \rceil</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.9781em;vertical-align:-0.686em;"></span><span class="mopen">⌈</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2921em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">t</span><span class="mord mathnormal">o</span><span class="mord mathnormal">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌉</span></span></span></span>。</li><li>我们能否在指定不同元素个数上限为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi><mi>p</mi><mi>p</mi><mi>e</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">upper</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">u</span><span class="mord mathnormal">pp</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span></span></span></span> 的情况下，快速计算合法子数组的个数呢？答案是可以的，我们引入双指针算法进行 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi><mi>h</mi><mi>e</mi><mi>c</mi><mi>k</mi></mrow><annotation encoding="application/x-tex">check</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">c</span><span class="mord mathnormal">h</span><span class="mord mathnormal">ec</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span>。我们枚举子数组的右边界并利用哈希表维护子数组的左边界进行统计即可。</li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><figcaption><span>[]</span></figcaption><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span> {<br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">medianOfUniquenessArray</span><span class="hljs-params">(vector&lt;<span class="hljs-type">int</span>&gt;&amp; nums)</span> </span>{<br>        <span class="hljs-keyword">using</span> ll = <span class="hljs-type">long</span> <span class="hljs-type">long</span>;<br>        <span class="hljs-type">int</span> n = nums.<span class="hljs-built_in">size</span>();<br>        <br>        ll tot = <span class="hljs-number">1ll</span> * (<span class="hljs-number">1</span> + n) * n / <span class="hljs-number">2</span>;<br>        <span class="hljs-type">int</span> ma = *<span class="hljs-built_in">max_element</span>(nums.<span class="hljs-built_in">begin</span>(), nums.<span class="hljs-built_in">end</span>());<br>        <span class="hljs-type">int</span> f[ma + <span class="hljs-number">1</span>];<br>        <span class="hljs-built_in">memset</span>(f, <span class="hljs-number">0</span>, <span class="hljs-keyword">sizeof</span> f);<br><br>        <span class="hljs-comment">// 双指针 check</span><br>        <span class="hljs-keyword">auto</span> chk = [&amp;](<span class="hljs-type">int</span> upper) -&gt; <span class="hljs-type">bool</span> {<br>            ll cnt = <span class="hljs-number">0</span>;<br>            <span class="hljs-type">int</span> size = <span class="hljs-number">0</span>; <span class="hljs-comment">// 哈希表大小</span><br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> l = <span class="hljs-number">0</span>, r = <span class="hljs-number">0</span>; r &lt; n; r++) {<br>                size += f[nums[r]] == <span class="hljs-number">0</span>;<br>                f[nums[r]]++;<br>                <span class="hljs-keyword">while</span> (size &gt; upper) {<br>                    --f[nums[l]];<br>                    size -= f[nums[l]] == <span class="hljs-number">0</span>;<br>                    l++;<br>                }<br>                <span class="hljs-comment">// 此时以 nums[r] 结尾且 l 为左边界的所有子数组的不同元素的数量均 &lt;= upper</span><br>                cnt += r - l + <span class="hljs-number">1</span>;<br>            }<br>            <span class="hljs-built_in">memset</span>(f, <span class="hljs-number">0</span>, <span class="hljs-keyword">sizeof</span> f);<br>            <span class="hljs-keyword">return</span> cnt &gt;= (tot + <span class="hljs-number">1</span>) / <span class="hljs-number">2</span>; <span class="hljs-comment">// 上取整写法</span><br>        };<br><br>        <span class="hljs-type">int</span> l = <span class="hljs-number">1</span>, r = n;<br>        <span class="hljs-keyword">while</span> (l &lt; r) {<br>            <span class="hljs-type">int</span> mid = (l + r) &gt;&gt; <span class="hljs-number">1</span>;<br>            <span class="hljs-keyword">if</span> (<span class="hljs-built_in">chk</span>(mid)) r = mid;<br>            <span class="hljs-keyword">else</span> l = mid + <span class="hljs-number">1</span>;<br>        }<br>        <span class="hljs-keyword">return</span> r;<br>    }<br>};<br></code></pre></td></tr></tbody></table></figure><figure class="highlight plaintext"><figcaption><span>[]</span></figcaption><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br></pre></td><td class="code"><pre><code class="hljs python3">class Solution:<br>    def medianOfUniquenessArray(self, nums: List[int]) -&gt; int:<br>        n = len(nums)<br>        ma = max(nums)<br>        tot = (1 + n) * n &gt;&gt; 1<br><br>        def chk(upper: int) -&gt; int:<br>            f = [0] * (ma + 1)<br>            cnt, l, size = 0, 0, 0<br>            for r in range(n):<br>                size += f[nums[r]] == 0<br>                f[nums[r]] += 1<br>                while size &gt; upper:<br>                    f[nums[l]] -= 1<br>                    size -= f[nums[l]] == 0<br>                    l += 1<br>                cnt += r - l + 1<br>            return cnt &gt;= (tot + 1) &gt;&gt; 1<br><br>        l, r = 1, n<br>        while l &lt; r:<br>            mid = (l + r) &gt;&gt; 1<br>            if chk(mid): r = mid<br>            else: l = mid + 1<br>        return r<br></code></pre></td></tr></tbody></table></figure><h3 id="【二分查找】Bomb">【二分查找】Bomb</h3><p><a href="https://codeforces.com/contest/1996/problem/F">https://codeforces.com/contest/1996/problem/F</a></p><blockquote><p>题意：给定两个序列 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">b</span></span></span></span> 以及最大操作次数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span>，问在不超过最大操作次数的情况下，最多可以获得多少收益？收益的计算方法为：每次选择 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> 中一个数 <code>a[i]</code> 加入收益，并将该数减去 <code>b[i]</code> 直到为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span>。</p><p>思路：</p><ul><li>暴力抬手：最暴力的做法是我们每次操作时选择 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> 中的最大值进行计数并对其修改，但是显然 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(nk)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">nk</span><span class="mclose">)</span></span></span></span> 会超时，我们考虑加速这个过程。</li><li>加速进程：由于最终结果中每次操作都是最优的，因此操作次数越多获得的收益就越大，具备单调性。如何利用这个单调性呢？关键在于全体数据的操作次数，我们记作 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>k</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">k'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span>。假设我们对 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span> 中全体数据可取的范围设置一个下限 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">lower</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span></span></span></span>，那么显然的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">lower</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span></span></span></span> 越高，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>k</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">k'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> 越小，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">lower</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span></span></span></span> 越低，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>k</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">k'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> 越大。于是根据单调的传递性可知 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">lower</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span></span></span></span> 也和最终的收益具备单调性，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">lower</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span></span></span></span> 越高，收益越小，<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">lower</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span></span></span></span> 越低，收益越大。因此我们二分 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">lower</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span></span></span></span>，使得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>k</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">k'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> 左逼近 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span>。</li><li>细节处理：二分的边界是什么？这需要从计算最终结果的角度来思考。对于二分出的下界 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">lower</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span></span></span></span>，即 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span></span></span></span>，此时扫描一遍计算出来的操作次数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>k</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">k'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> 一定是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>k</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo>≤</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">k'\le k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8879em;vertical-align:-0.136em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span>。也就是说我们还需要操作 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo>−</mo><msup><mi>k</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">k-k'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> 次，显然我们可以将每个数操作后维护到最终的结果，然后执行上述的暴力思路，但是这样仍然会超时，因为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo>−</mo><msup><mi>k</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">k-k'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> 的计算结果最大是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span>，此时进行暴力仍然会达到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>。正难则反，我们不妨将上述二分出的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">lower</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span></span></span></span> 减一进行最终答案的计算，这样对应的操作次数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>k</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">k'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> 一定会超过 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span>，并且对于超过 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 的操作，累加到答案的数值一定都是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">lower-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，这一步也是本题最精妙的一步。从上述分析不难发现，二分的下界需要设定为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，因为后续累加答案时会对 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">lower</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span></span></span></span> 进行减一操作；二分的上界需要设定为严格大于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mn>9</mn></msup></mrow><annotation encoding="application/x-tex">10^9</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span></span></span></span> 的数，比如 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn><msup><mn>0</mn><mn>9</mn></msup><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">10^9+1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8974em;vertical-align:-0.0833em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">9</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，因为我们需要保证 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>o</mi><mi>w</mi><mi>e</mi><mi>r</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">lower-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7778em;vertical-align:-0.0833em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mord mathnormal" style="margin-right:0.02778em;">er</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 后对应的操作次数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>k</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup></mrow><annotation encoding="application/x-tex">k'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7519em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7519em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span> 严格大于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span>。</li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\log k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><br><span class="hljs-keyword">using</span> ll = <span class="hljs-type">long</span> <span class="hljs-type">long</span>;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-type">int</span> n, k;<br>    std::cin &gt;&gt; n &gt;&gt; k;<br>    <br>    <span class="hljs-function">std::vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">a</span><span class="hljs-params">(n)</span>, <span class="hljs-title">b</span><span class="hljs-params">(n)</span></span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) std::cin &gt;&gt; a[i];<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) std::cin &gt;&gt; b[i];<br>    <br>    <span class="hljs-keyword">auto</span> chk = [&amp;](<span class="hljs-type">int</span> x) {<br>        ll cnt = <span class="hljs-number">0</span>;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>            <span class="hljs-keyword">if</span> (a[i] &lt; x) <span class="hljs-keyword">continue</span>;<br>            cnt += (a[i] - x) / b[i] + <span class="hljs-number">1</span>;<br>        }<br>        <span class="hljs-keyword">return</span> cnt;<br>    };<br>    <br>    <span class="hljs-type">int</span> l = <span class="hljs-number">1</span>, r = <span class="hljs-number">1e9</span> + <span class="hljs-number">10</span>;<br>    <span class="hljs-keyword">while</span> (l &lt; r) {<br>        <span class="hljs-type">int</span> mid = (l + r) &gt;&gt; <span class="hljs-number">1</span>;<br>        <span class="hljs-keyword">if</span> (<span class="hljs-built_in">chk</span>(mid) &lt;= k) r = mid;<br>        <span class="hljs-keyword">else</span> l = mid + <span class="hljs-number">1</span>;<br>    }<br>    <br>    <span class="hljs-type">int</span> best = r - <span class="hljs-number">1</span>;<br>    ll res = <span class="hljs-number">0</span>, cnt = <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        <span class="hljs-keyword">if</span> (a[i] &lt; best) <span class="hljs-keyword">continue</span>;<br>        ll t = (a[i] - best) / b[i] + <span class="hljs-number">1</span>;<br>        cnt += t;<br>        res += t * a[i] - t * (t - <span class="hljs-number">1</span>) / <span class="hljs-number">2</span> * b[i];<br>    }<br>    res -= (cnt - k) * best;<br>    <br>    std::cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    std::ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    std::cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br>    std::cin &gt;&gt; T;<br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure>]]></content>
     
     
     <categories>
@@ -1644,11 +1644,11 @@
   
   
   <entry>
-    <title>data-structure</title>
-    <link href="/Algorithm/data-structure/"/>
-    <url>/Algorithm/data-structure/</url>
+    <title>divide-and-conquer</title>
+    <link href="/Algorithm/divide-and-conquer/"/>
+    <url>/Algorithm/divide-and-conquer/</url>
     
-    <content type="html"><![CDATA[<h2 id="数据结构">数据结构</h2><p>数据结构由 <strong>数据</strong> 和 <strong>结构</strong> 两部分组成。我们主要讨论的是后者，即结构部分。</p><p>按照 <strong>逻辑结构</strong> 可以将其区分为 <strong>线性结构</strong> 和 <strong>非线性结构</strong>。</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202408301527018.png" alt="线性数据结构 与 非线性数据结构"></p><p>按照 <strong>物理结构</strong> 可以将其区分为 <strong>连续结构</strong> 和 <strong>分散结构</strong>。</p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202408301528541.png" alt="连续空间存储 与 分散空间存储"></p><h2 id="【模板】双链表">【模板】双链表</h2><p><a href="https://www.acwing.com/problem/content/829/">https://www.acwing.com/problem/content/829/</a></p><blockquote><p>思路：用两个空结点作为起始状态的边界，避免所有边界讨论。</p><p>时间复杂度：插入、删除结点均为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span></span></span></span>，遍历为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br><span class="line">82</span><br><span class="line">83</span><br><span class="line">84</span><br><span class="line">85</span><br><span class="line">86</span><br><span class="line">87</span><br><span class="line">88</span><br><span class="line">89</span><br><span class="line">90</span><br><span class="line">91</span><br><span class="line">92</span><br><span class="line">93</span><br><span class="line">94</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><br><span class="hljs-keyword">using</span> ll = <span class="hljs-type">long</span> <span class="hljs-type">long</span>;<br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-keyword">template</span>&lt;<span class="hljs-keyword">class</span> <span class="hljs-title class_">T</span>&gt;<br><span class="hljs-keyword">class</span> <span class="hljs-title class_">myList</span> {<br><span class="hljs-keyword">private</span>:<br>    <span class="hljs-type">int</span> idx;<br>    std::vector&lt;T&gt; val;<br>    std::vector&lt;<span class="hljs-type">int</span>&gt; left, right;<br><br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-built_in">myList</span>(<span class="hljs-type">const</span> <span class="hljs-type">int</span> n) {<br>        idx = <span class="hljs-number">2</span>;<br>        val.<span class="hljs-built_in">resize</span>(n + <span class="hljs-number">10</span>);<br>        left.<span class="hljs-built_in">resize</span>(n + <span class="hljs-number">10</span>);<br>        right.<span class="hljs-built_in">resize</span>(n + <span class="hljs-number">10</span>);<br>        left[<span class="hljs-number">1</span>] = <span class="hljs-number">0</span>, right[<span class="hljs-number">0</span>] = <span class="hljs-number">1</span>;<br>    }<br><br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">push_back</span><span class="hljs-params">(T x)</span> </span>{<br>        <span class="hljs-built_in">insert_left</span>(<span class="hljs-number">1</span>, x);<br>    }<br><br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">push_front</span><span class="hljs-params">(T x)</span> </span>{<br>        <span class="hljs-built_in">insert_right</span>(<span class="hljs-number">0</span>, x);<br>    }<br><br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">insert_left</span><span class="hljs-params">(<span class="hljs-type">int</span> k, T x)</span> </span>{<br>        <span class="hljs-built_in">insert_right</span>(left[k], x);<br>    }<br><br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">insert_right</span><span class="hljs-params">(<span class="hljs-type">int</span> k, T x)</span> </span>{<br>        val[idx] = x;<br>        right[idx] = right[k];<br>        left[right[k]] = idx;<br>        left[idx] = k;<br>        right[k] = idx++;<br>    }<br><br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">erase</span><span class="hljs-params">(<span class="hljs-type">int</span> k)</span> </span>{<br>        right[left[k]] = right[k];<br>        left[right[k]] = left[k];<br>    }<br><br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">output</span><span class="hljs-params">()</span> </span>{<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = right[<span class="hljs-number">0</span>]; i != <span class="hljs-number">1</span>; i = right[i]) {<br>            cout &lt;&lt; val[i] &lt;&lt; <span class="hljs-string">" \n"</span>[i == <span class="hljs-number">1</span>];<br>        }<br>    }<br>};<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-type">int</span> n;<br>    cin &gt;&gt; n;<br><br>    <span class="hljs-function">myList&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">ls</span><span class="hljs-params">(n)</span></span>;<br><br>    <span class="hljs-keyword">while</span> (n--) {<br>        string op;<br>        cin &gt;&gt; op;<br><br>        <span class="hljs-type">int</span> k, x;<br><br>        <span class="hljs-keyword">if</span> (op == <span class="hljs-string">"L"</span>) {<br>            cin &gt;&gt; x;<br>            ls.<span class="hljs-built_in">push_front</span>(x);<br>        } <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (op == <span class="hljs-string">"R"</span>) {<br>            cin &gt;&gt; x;<br>            ls.<span class="hljs-built_in">push_back</span>(x);<br>        } <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (op == <span class="hljs-string">"D"</span>) {<br>            cin &gt;&gt; k;<br>            ls.<span class="hljs-built_in">erase</span>(k + <span class="hljs-number">1</span>);<br>        } <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (op == <span class="hljs-string">"IL"</span>) {<br>            cin &gt;&gt; k &gt;&gt; x;<br>            ls.<span class="hljs-built_in">insert_left</span>(k + <span class="hljs-number">1</span>, x);<br>        } <span class="hljs-keyword">else</span> {<br>            cin &gt;&gt; k &gt;&gt; x;<br>            ls.<span class="hljs-built_in">insert_right</span>(k + <span class="hljs-number">1</span>, x);<br>        }<br>    }<br><br>    ls.<span class="hljs-built_in">output</span>();<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    std::ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    std::cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    std::cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">import</span> heapq<br><span class="hljs-keyword">from</span> collections <span class="hljs-keyword">import</span> defaultdict<br><span class="hljs-keyword">from</span> typing <span class="hljs-keyword">import</span> <span class="hljs-type">List</span>, <span class="hljs-type">Tuple</span><br><span class="hljs-keyword">import</span> math<br><span class="hljs-keyword">from</span> itertools <span class="hljs-keyword">import</span> combinations<br><br>II = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">int</span>(<span class="hljs-built_in">input</span>())<br>FI = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">float</span>(<span class="hljs-built_in">input</span>())<br>MII = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">tuple</span>(<span class="hljs-built_in">map</span>(<span class="hljs-built_in">int</span>, <span class="hljs-built_in">input</span>().split()))<br>LII = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">list</span>(<span class="hljs-built_in">map</span>(<span class="hljs-built_in">int</span>, <span class="hljs-built_in">input</span>().split()))<br><br><br><span class="hljs-keyword">class</span> <span class="hljs-title class_">myList</span>:<br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">__init__</span>(<span class="hljs-params">self, n: <span class="hljs-built_in">int</span></span>) -&gt; <span class="hljs-literal">None</span>:<br>        <span class="hljs-variable language_">self</span>.val = [<span class="hljs-number">0</span>] * (n + <span class="hljs-number">10</span>)<br>        <span class="hljs-variable language_">self</span>.left = [<span class="hljs-number">0</span>] * (n + <span class="hljs-number">10</span>)<br>        <span class="hljs-variable language_">self</span>.right = [<span class="hljs-number">0</span>] * (n + <span class="hljs-number">10</span>)<br>        <span class="hljs-variable language_">self</span>.idx = <span class="hljs-number">2</span><br>        <span class="hljs-variable language_">self</span>.right[<span class="hljs-number">0</span>] = <span class="hljs-number">1</span><br>        <span class="hljs-variable language_">self</span>.left[<span class="hljs-number">1</span>] = <span class="hljs-number">0</span><br><br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">push_front</span>(<span class="hljs-params">self, x: <span class="hljs-built_in">int</span></span>):<br>        <span class="hljs-variable language_">self</span>.insert_right(<span class="hljs-number">0</span>, x)<br><br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">push_back</span>(<span class="hljs-params">self, x: <span class="hljs-built_in">int</span></span>):<br>        <span class="hljs-variable language_">self</span>.insert_left(<span class="hljs-number">1</span>, x)<br><br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">insert_left</span>(<span class="hljs-params">self, k: <span class="hljs-built_in">int</span>, x: <span class="hljs-built_in">int</span></span>):<br>        <span class="hljs-variable language_">self</span>.insert_right(<span class="hljs-variable language_">self</span>.left[k], x)<br><br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">insert_right</span>(<span class="hljs-params">self, k: <span class="hljs-built_in">int</span>, x: <span class="hljs-built_in">int</span></span>):<br>        <span class="hljs-variable language_">self</span>.val[<span class="hljs-variable language_">self</span>.idx] = x<br>        <span class="hljs-variable language_">self</span>.right[<span class="hljs-variable language_">self</span>.idx] = <span class="hljs-variable language_">self</span>.right[k]<br>        <span class="hljs-variable language_">self</span>.left[<span class="hljs-variable language_">self</span>.right[k]] = <span class="hljs-variable language_">self</span>.idx<br>        <span class="hljs-variable language_">self</span>.left[<span class="hljs-variable language_">self</span>.idx] = k<br>        <span class="hljs-variable language_">self</span>.right[k] = <span class="hljs-variable language_">self</span>.idx<br>        <span class="hljs-variable language_">self</span>.idx += <span class="hljs-number">1</span><br><br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">erase</span>(<span class="hljs-params">self, k: <span class="hljs-built_in">int</span></span>):<br>        <span class="hljs-variable language_">self</span>.left[<span class="hljs-variable language_">self</span>.right[k]] = <span class="hljs-variable language_">self</span>.left[k]<br>        <span class="hljs-variable language_">self</span>.right[<span class="hljs-variable language_">self</span>.left[k]] = <span class="hljs-variable language_">self</span>.right[k]<br><br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">output</span>(<span class="hljs-params">self</span>) -&gt; <span class="hljs-literal">None</span>:<br>        i = <span class="hljs-variable language_">self</span>.right[<span class="hljs-number">0</span>]<br>        <span class="hljs-keyword">while</span> i != <span class="hljs-number">1</span>:<br>            <span class="hljs-built_in">print</span>(<span class="hljs-variable language_">self</span>.val[i], end=<span class="hljs-string">' '</span>)<br>            i = <span class="hljs-variable language_">self</span>.right[i]<br><br><br><span class="hljs-keyword">def</span> <span class="hljs-title function_">solve</span>() -&gt; <span class="hljs-literal">None</span>:<br>    n = II()<br><br>    ls = myList(n)<br><br>    <span class="hljs-keyword">for</span> _ <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(n):<br>        op = <span class="hljs-built_in">input</span>().split()<br><br>        <span class="hljs-keyword">if</span> op[<span class="hljs-number">0</span>] == <span class="hljs-string">'L'</span>:<br>            ls.push_front(<span class="hljs-built_in">int</span>(op[-<span class="hljs-number">1</span>]))<br>        <span class="hljs-keyword">elif</span> op[<span class="hljs-number">0</span>] == <span class="hljs-string">'R'</span>:<br>            ls.push_back(<span class="hljs-built_in">int</span>(op[-<span class="hljs-number">1</span>]))<br>        <span class="hljs-keyword">elif</span> op[<span class="hljs-number">0</span>] == <span class="hljs-string">'D'</span>:<br>            ls.erase(<span class="hljs-built_in">int</span>(op[-<span class="hljs-number">1</span>]) + <span class="hljs-number">1</span>)<br>        <span class="hljs-keyword">elif</span> op[<span class="hljs-number">0</span>] == <span class="hljs-string">'IL'</span>:<br>            ls.insert_left(<span class="hljs-built_in">int</span>(op[<span class="hljs-number">1</span>]) + <span class="hljs-number">1</span>, <span class="hljs-built_in">int</span>(op[-<span class="hljs-number">1</span>]))<br>        <span class="hljs-keyword">else</span>:<br>            ls.insert_right(<span class="hljs-built_in">int</span>(op[<span class="hljs-number">1</span>]) + <span class="hljs-number">1</span>, <span class="hljs-built_in">int</span>(op[-<span class="hljs-number">1</span>]))<br><br>    ls.output()<br><br><br><span class="hljs-keyword">if</span> __name__ == <span class="hljs-string">'__main__'</span>:<br>    T = <span class="hljs-number">1</span><br>    <span class="hljs-comment"># T = II()</span><br>    <span class="hljs-keyword">while</span> T: solve(); T -= <span class="hljs-number">1</span><br></code></pre></td></tr></tbody></table></figure><h2 id="【模板】单调栈">【模板】单调栈</h2><p><a href="https://www.acwing.com/problem/content/832/">https://www.acwing.com/problem/content/832/</a></p><blockquote><p>题意：对于一个序列中的每一个元素，寻找每一个元素左侧最近的比其小的元素。</p><p>思路一：暴力枚举</p><ul><li>显然对于每一个元素 <code>nums[i]</code>，我们可以枚举倒序 <code>[0, i-1]</code> 直到找到第一个 <code>nums[j] &lt; nums[i]</code></li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></li></ul><p>思路二：单调栈</p><ul><li><p>可以发现时间开销主要在倒序枚举上，我们能少枚举一些元素吗？答案是可以的。我们定义「寻找 <code>[i-1, 0]</code> 中比当前元素小的第一个元素」的行为叫做「寻找合法对象」。显然我们在枚举每一个元素时都需要 <strong>查询</strong> 和 <strong>维护</strong> 这样的合法对象线性序列，可以理解为记忆化从而加速查询。那么如何高效查询和维护这样的线性序列呢？不妨考虑每一个合法对象对曾经的合法对象的影响：</p><ul><li>若当前元素 <code>nums[i]</code> 可以成为后续 <code>[i+1, n-1]</code> 元素的合法对象。则从 <code>i-1</code> 开始一直往左，只要比当前大的元素，都不可能成为 <code>[i+1, n-1]</code> 的合法对象，肯定都被 <code>nums[i]</code> “拦住了”。那么在「合法对象序列」中插入当前合法对象之前，需要不断尾弹出比当前大的元素</li><li>若当前元素 <code>nums[i]</code> 不能成为后续 <code>[i+1, n-1]</code> 元素的合法对象。表明当前元素过大，此时就不用将当前元素插入「合法对象序列」</li></ul></li><li><p>经过上述两个角度的讨论，很容易发现这样维护出来的的合法序列是严格单调递增的。于是，在查询操作时仅需要进行尾比较与尾弹出即可，在维护操作时，仅需要尾插入即可。满足这样的线性数据结构有很多，如栈、队列、数组、链表，我们就使用栈来演示，与标题遥相呼应</p></li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><br><span class="hljs-keyword">using</span> ll = <span class="hljs-type">long</span> <span class="hljs-type">long</span>;<br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-type">int</span> n;<br>    cin &gt;&gt; n;<br><br>    <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">a</span><span class="hljs-params">(n)</span></span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        cin &gt;&gt; a[i];<br>    }<br><br>    stack&lt;<span class="hljs-type">int</span>&gt; s;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        <span class="hljs-comment">// 查询</span><br>        <span class="hljs-keyword">while</span> (s.<span class="hljs-built_in">size</span>() &amp;&amp; s.<span class="hljs-built_in">top</span>() &gt;= a[i]) {<br>            s.<span class="hljs-built_in">pop</span>();<br>        }<br>        cout &lt;&lt; (s.<span class="hljs-built_in">size</span>() ? s.<span class="hljs-built_in">top</span>() : <span class="hljs-number">-1</span>) &lt;&lt; <span class="hljs-string">" "</span>;<br>        <br>        <span class="hljs-comment">// 维护</span><br>        s.<span class="hljs-built_in">push</span>(a[i]);<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    std::ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    std::cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    std::cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><p>相似题：</p><p><a href="https://leetcode.cn/problems/next-greater-element-ii/">下一个更大元素 II</a></p><h2 id="【模板】单调队列">【模板】单调队列</h2><blockquote><p>标签：双端队列、滑动窗口、单调队列</p><p>题意：给定一个含有 n 个元素的序列，求解其中每个长度为 k 的子数组中的最值。</p><p>思路：显然我们可以 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(nk)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">nk</span><span class="mclose">)</span></span></span></span> 暴力求解，有没有什么方法可以将「求解子数组中最值」的时间开销从 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span> 降低为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 呢？有的！我们重新定义一个队列就好了。为了做到线性时间复杂度的优化，我们对队列做以下自定义，以「求解子数组最小值」为例：</p><ol><li>插入元素到队尾：此时和单调栈的逻辑类似。如果当前元素可以作为当前子数组或后续子数组的最小值，则需要从当前队尾开始依次弹出比当前元素严格大的元素，最后再将当前元素入队。<strong>注意</strong>：当遇到和当前元素值相等的元素时不能出队，因为每一个元素都会经历入队和出队的操作，一旦此时出队了，后续进行出队判定时会提前弹出本不应该出队的与其等值的元素。</li><li>弹出队头元素：如果队头元素和子数组左端点 <code>nums[i-k]</code> 的元素值相等，则弹出。</li><li>获得队头元素：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 的获取队头元素，即队列中的最小值。</li></ol><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br><span class="line">82</span><br><span class="line">83</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><br><span class="hljs-keyword">using</span> ll = <span class="hljs-type">long</span> <span class="hljs-type">long</span>;<br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-keyword">template</span>&lt;<span class="hljs-keyword">class</span> <span class="hljs-title class_">T</span>&gt;<br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">minQueue</span> {<br>    std::deque&lt;T&gt; q;<br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">pushBack</span><span class="hljs-params">(T x)</span> </span>{<br>        <span class="hljs-keyword">while</span> (q.<span class="hljs-built_in">size</span>() &amp;&amp; x &lt; q.<span class="hljs-built_in">back</span>()) {<br>            q.<span class="hljs-built_in">pop_back</span>();<br>        }<br>        q.<span class="hljs-built_in">push_back</span>(x);<br>    }<br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">popFront</span><span class="hljs-params">(T x)</span> </span>{<br>        <span class="hljs-keyword">if</span> (q.<span class="hljs-built_in">size</span>() &amp;&amp; q.<span class="hljs-built_in">front</span>() == x) {<br>            q.<span class="hljs-built_in">pop_front</span>();<br>        }<br>    }<br>    <span class="hljs-function">T <span class="hljs-title">getMinValue</span><span class="hljs-params">()</span> </span>{<br>        <span class="hljs-keyword">return</span> q.<span class="hljs-built_in">front</span>();<br>    }<br>};<br><br><span class="hljs-keyword">template</span>&lt;<span class="hljs-keyword">class</span> <span class="hljs-title class_">T</span>&gt;<br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">maxQueue</span> {<br>    std::deque&lt;T&gt; q;<br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">pushBack</span><span class="hljs-params">(T x)</span> </span>{<br>        <span class="hljs-keyword">while</span> (q.<span class="hljs-built_in">size</span>() &amp;&amp; x &gt; q.<span class="hljs-built_in">back</span>()) {<br>            q.<span class="hljs-built_in">pop_back</span>();<br>        }<br>        q.<span class="hljs-built_in">push_back</span>(x);<br>    }<br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">popFront</span><span class="hljs-params">(T x)</span> </span>{<br>        <span class="hljs-keyword">if</span> (q.<span class="hljs-built_in">size</span>() &amp;&amp; q.<span class="hljs-built_in">front</span>() == x) {<br>            q.<span class="hljs-built_in">pop_front</span>();<br>        }<br>    }<br>    <span class="hljs-function">T <span class="hljs-title">getMaxValue</span><span class="hljs-params">()</span> </span>{<br>        <span class="hljs-keyword">return</span> q.<span class="hljs-built_in">front</span>();<br>    }<br>};<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-type">int</span> n, k;<br>    cin &gt;&gt; n &gt;&gt; k;<br>    <br>    <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">nums</span><span class="hljs-params">(n)</span></span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        cin &gt;&gt; nums[i];<br>    }<br>    <br>    minQueue&lt;<span class="hljs-type">int</span>&gt; minq;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        minq.<span class="hljs-built_in">pushBack</span>(nums[i]);<br>        <span class="hljs-keyword">if</span> (i &gt;= k) {<br>            minq.<span class="hljs-built_in">popFront</span>(nums[i - k]);<br>        }<br>        <span class="hljs-keyword">if</span> (i &gt;= k - <span class="hljs-number">1</span>) {<br>            cout &lt;&lt; minq.<span class="hljs-built_in">getMinValue</span>() &lt;&lt; <span class="hljs-string">" \n"</span>[i == n - <span class="hljs-number">1</span>];<br>        }<br>    }<br>    <br>    maxQueue&lt;<span class="hljs-type">int</span>&gt; maxq;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>        maxq.<span class="hljs-built_in">pushBack</span>(nums[i]);<br>        <span class="hljs-keyword">if</span> (i &gt;= k) {<br>            maxq.<span class="hljs-built_in">popFront</span>(nums[i - k]);<br>        }<br>        <span class="hljs-keyword">if</span> (i &gt;= k - <span class="hljs-number">1</span>) {<br>            cout &lt;&lt; maxq.<span class="hljs-built_in">getMaxValue</span>() &lt;&lt; <span class="hljs-string">" \n"</span>[i == n - <span class="hljs-number">1</span>];<br>        }<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    std::ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    std::cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    std::cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">from</span> collections <span class="hljs-keyword">import</span> defaultdict, deque<br><span class="hljs-keyword">from</span> typing <span class="hljs-keyword">import</span> <span class="hljs-type">List</span>, <span class="hljs-type">Tuple</span><br><span class="hljs-keyword">from</span> itertools <span class="hljs-keyword">import</span> combinations, permutations<br><span class="hljs-keyword">import</span> math, heapq, queue<br><br>II = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">int</span>(<span class="hljs-built_in">input</span>())<br>FI = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">float</span>(<span class="hljs-built_in">input</span>())<br>MII = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">tuple</span>(<span class="hljs-built_in">map</span>(<span class="hljs-built_in">int</span>, <span class="hljs-built_in">input</span>().split()))<br>LII = <span class="hljs-keyword">lambda</span>: <span class="hljs-built_in">list</span>(<span class="hljs-built_in">map</span>(<span class="hljs-built_in">int</span>, <span class="hljs-built_in">input</span>().split()))<br><br><br><span class="hljs-keyword">def</span> <span class="hljs-title function_">solve</span>() -&gt; <span class="hljs-literal">None</span>:<br>    n, k = MII()<br>    nums = LII()<br>    <br>    qa, qb = deque(), deque()<br>    ra, rb = [], []<br>    <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(n):<br>        <span class="hljs-comment"># push back</span><br>        <span class="hljs-keyword">while</span> <span class="hljs-built_in">len</span>(qa) <span class="hljs-keyword">and</span> nums[i] &lt; qa[-<span class="hljs-number">1</span>]:<br>            qa.pop()<br>        qa.append(nums[i])<br>        <span class="hljs-keyword">while</span> <span class="hljs-built_in">len</span>(qb) <span class="hljs-keyword">and</span> nums[i] &gt; qb[-<span class="hljs-number">1</span>]:<br>            qb.pop()<br>        qb.append(nums[i])<br>        <span class="hljs-keyword">if</span> i &gt;= k:<br>            <span class="hljs-comment"># pop front</span><br>            <span class="hljs-keyword">if</span> <span class="hljs-built_in">len</span>(qa) <span class="hljs-keyword">and</span> qa[<span class="hljs-number">0</span>] == nums[i - k]:<br>                qa.popleft()<br>            <span class="hljs-keyword">if</span> <span class="hljs-built_in">len</span>(qb) <span class="hljs-keyword">and</span> qb[<span class="hljs-number">0</span>] == nums[i - k]:<br>                qb.popleft()<br>        <span class="hljs-keyword">if</span> i &gt;= k - <span class="hljs-number">1</span>:<br>            <span class="hljs-comment"># get ans</span><br>            ra.append(qa[<span class="hljs-number">0</span>])<br>            rb.append(qb[<span class="hljs-number">0</span>])<br>    <br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">' '</span>.join(<span class="hljs-built_in">map</span>(<span class="hljs-built_in">str</span>, ra)))<br>    <span class="hljs-built_in">print</span>(<span class="hljs-string">' '</span>.join(<span class="hljs-built_in">map</span>(<span class="hljs-built_in">str</span>, rb)))<br><br><br><span class="hljs-keyword">if</span> __name__ == <span class="hljs-string">'__main__'</span>:<br>    T = <span class="hljs-number">1</span><br>    <span class="hljs-comment"># T = II()</span><br>    <span class="hljs-keyword">while</span> T: solve(); T -= <span class="hljs-number">1</span><br></code></pre></td></tr></tbody></table></figure><h2 id="【模板】最近公共祖先">【模板】最近公共祖先</h2><p><a href="https://www.luogu.com.cn/problem/P3379">https://www.luogu.com.cn/problem/P3379</a></p><blockquote><p>题意：寻找树中指定两个结点的最近公共祖先 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mtext>(Lowest&nbsp;Common&nbsp;Ancestor,&nbsp;简称&nbsp;LCA)</mtext></mrow><annotation encoding="application/x-tex">\text{(Lowest Common Ancestor, 简称 LCA)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">(Lowest&nbsp;Common&nbsp;Ancestor,&nbsp;</span><span class="mord cjk_fallback">简称</span><span class="mord">&nbsp;LCA)</span></span></span></span></span>。</p><p>思路：对于每次查询，我们可以从指定的两个结点开始往上跳，第一个公共结点就是目标的 LCA，每一次询问的时间复杂度均为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>，为了加速查询，我们可以采用倍增法，预处理出往上跳的结果，即 <code>fa[i][j]</code> 数组，表示 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 号点向上跳 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>j</mi></msup></mrow><annotation encoding="application/x-tex">2^j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8247em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span></span></span></span></span></span></span> 步后到达的结点。接下来在往上跳跃的过程中，利用二进制拼凑的思路，即可在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 的时间内查询到 LCA。</p><p>预处理：可以发现，对于 <code>fa[i][j]</code>，我们可以通过递推的方式获得，即 <code>fa[i][j] = fa[fa[i][j-1]][j-1]</code>，当前结点向上跳跃 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>j</mi></msup></mrow><annotation encoding="application/x-tex">2^j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8247em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span></span></span></span></span></span></span> 步可以拆分为先向上 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">2^{j-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8247em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span> 步, 在此基础之上再向上 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">2^{j-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8247em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8247em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span></span></span></span> 步.于是我们可以采用宽搜 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>o</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">or</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">or</span></span></span></span> 深搜的顺序维护 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mi>a</mi></mrow><annotation encoding="application/x-tex">fa</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mord mathnormal">a</span></span></span></span> 数组。</p><p>跳跃：我们首先需要将两个结点按照倍增的思路向上跳到同一个深度，接下来两个结点同时按照倍增的思路向上跳跃，为了确保求出最近的，我们需要确保在跳跃的步调一致的情况下，两者的祖先始终不相同，那么倍增结束后，两者的父结点就是最近公共祖先，即 <code>fa[x][k]</code> 或 <code>fa[y][k]</code></p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Θ</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>+</mo><mi>m</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta(n \log n + m \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Θ</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p><ul><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">n \log n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span></span></span></span> 为预处理每一个结点向上跳跃抵达的情况</li><li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mi>log</mi><mo>⁡</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">m \log n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span></span></span></span> 为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span> 次询问的情况</li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">5e5</span> + <span class="hljs-number">10</span>;<br><br><span class="hljs-type">int</span> n, Q, root;<br>vector&lt;<span class="hljs-type">int</span>&gt; G[N];<br><span class="hljs-type">int</span> fa[N][<span class="hljs-number">20</span>], dep[N];<br>queue&lt;<span class="hljs-type">int</span>&gt; q;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">init</span><span class="hljs-params">()</span> </span>{<br>    dep[root] = <span class="hljs-number">1</span>;<br>    q.<span class="hljs-built_in">push</span>(root);<br><br>    <span class="hljs-keyword">while</span> (q.<span class="hljs-built_in">size</span>()) {<br>        <span class="hljs-type">int</span> now = q.<span class="hljs-built_in">front</span>();<br>        q.<span class="hljs-built_in">pop</span>();<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> ch: G[now]) {<br>            <span class="hljs-keyword">if</span> (!dep[ch]) {<br>                dep[ch] = dep[now] + <span class="hljs-number">1</span>;<br>                fa[ch][<span class="hljs-number">0</span>] = now;<br>                <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> k = <span class="hljs-number">1</span>; k &lt;= <span class="hljs-number">19</span>; k++) {<br>                    fa[ch][k] = fa[ fa[ch][k<span class="hljs-number">-1</span>] ][k<span class="hljs-number">-1</span>];<br>                }<br>                q.<span class="hljs-built_in">push</span>(ch);<br>            }<br>        }<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">lca</span><span class="hljs-params">(<span class="hljs-type">int</span> a, <span class="hljs-type">int</span> b)</span> </span>{<br>    <span class="hljs-keyword">if</span> (dep[a] &lt; dep[b]) <span class="hljs-built_in">swap</span>(a, b);<br><br>    <span class="hljs-comment">// 二进制拼凑从而跳到一样高</span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> k = <span class="hljs-number">19</span>; k &gt;= <span class="hljs-number">0</span>; k--)<br>        <span class="hljs-keyword">if</span> (dep[fa[a][k]] &gt;= dep[b])<br>            a = fa[a][k];<br><br>    <span class="hljs-keyword">if</span> (a == b) <span class="hljs-keyword">return</span> a;<br><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> k = <span class="hljs-number">19</span>; k &gt;= <span class="hljs-number">0</span>; k--)<br>        <span class="hljs-keyword">if</span> (fa[a][k] != fa[b][k])<br>            a = fa[a][k], b = fa[b][k];<br><br>    <span class="hljs-keyword">return</span> fa[a][<span class="hljs-number">0</span>];<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; Q &gt;&gt; root;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n - <span class="hljs-number">1</span>; ++i) {<br>        <span class="hljs-type">int</span> a, b;<br>        cin &gt;&gt; a &gt;&gt; b;<br>        G[a].<span class="hljs-built_in">push_back</span>(b);<br>        G[b].<span class="hljs-built_in">push_back</span>(a);<br>    }<br><br>    <span class="hljs-built_in">init</span>();<br><br>    <span class="hljs-keyword">while</span> (Q--) {<br>        <span class="hljs-type">int</span> a, b;<br>        cin &gt;&gt; a &gt;&gt; b;<br>        cout &lt;&lt; <span class="hljs-built_in">lca</span>(a, b) &lt;&lt; <span class="hljs-string">"\n"</span>;<br>    }<br>}    <br></code></pre></td></tr></tbody></table></figure><h2 id="【栈】验证栈序列">【栈】验证栈序列</h2><p><a href="https://www.luogu.com.cn/problem/P4387">https://www.luogu.com.cn/problem/P4387</a></p><blockquote><ul><li><p>题意：给定入栈序列与出栈序列，问出栈序列是否合法</p></li><li><p>思路：思路很简单，就是对于当前出栈的数，和入栈序列中最后已出栈的数之间，如果还有数没有出，那么就是不合法的出栈序列，反之合法。这是从入栈的结果来看的，如果这么判断就需要扫描入栈序列 n 次，时间复杂度为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>。我们按照入栈的顺序来看，对于当前待入栈的数，若与出栈序列的队头不等，则成功入栈等待后续出栈；若与出栈序列相等，则匹配成功直接出栈无需入栈，同时对已入栈的数与出栈序列队头不断匹配直到不相等。最后判断待入栈的数与出栈序列是否全部匹配掉了，如果全部匹配掉了说明该出栈序列合法，反之不合法</p><p>抽象总结上述思路：为了判断出栈序列是否合法，我们不妨思考：对于每一个出栈的数，出栈的时机是什么？可以发现出栈的时机无非两种：</p><ul><li>一入栈就出栈（对应于枚举待入栈序列时发现待入栈的数与出栈序列队头相等）</li><li>紧跟着刚出栈的数继续出栈（对应于枚举待入栈序列时发现待入栈的数与出栈序列队头相等之后，继续判断出栈序列队头与已入栈的数是否相等，若相等则不断判断并出栈）</li></ul></li><li><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></li></ul></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-comment">// #include &lt;bits/stdc++.h&gt;</span><br><span class="hljs-comment">// #define int long long</span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;unordered_map&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;stack&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;queue&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-type">int</span> n;<br>    cin &gt;&gt; n;<br>    <br>    <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">a</span><span class="hljs-params">(n)</span>, <span class="hljs-title">b</span><span class="hljs-params">(n)</span></span>;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) cin &gt;&gt; a[i];<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) cin &gt;&gt; b[i];<br>    <br>    stack&lt;<span class="hljs-type">int</span>&gt; stk;<br>    <span class="hljs-type">int</span> i = <span class="hljs-number">0</span>, j = <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">while</span> (i &lt; n) {<br>        <span class="hljs-keyword">if</span> (a[i] != b[j]) stk.<span class="hljs-built_in">push</span>(a[i++]);<br>        <span class="hljs-keyword">else</span> {<br>            i++, j++;<br>            <span class="hljs-keyword">while</span> (!stk.<span class="hljs-built_in">empty</span>() &amp;&amp; b[j] == stk.<span class="hljs-built_in">top</span>()) {<br>                stk.<span class="hljs-built_in">pop</span>();<br>                j++;<br>            }<br>        }<br>    }<br>    <br>    cout &lt;&lt; (stk.<span class="hljs-built_in">empty</span>() ? <span class="hljs-string">"Yes"</span> : <span class="hljs-string">"No"</span>) &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br>    cin &gt;&gt; T;<br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h2 id="二叉搜索树-平衡树">二叉搜索树 &amp; 平衡树</h2><p>如果我们想要在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 时间复杂度内对数据进行增删查改的操作，就可以引入 <strong>二叉搜索树 (Binary Search Tree)</strong> 这一数据结构。然而，在某些极端的情况下，例如当插入的数据是单调不减或不增时，这棵树就会退化为一条链从而导致所有的增删查改操作退化到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>，这是我们不愿意看到的。因此我们引入 <strong>平衡二叉搜索树 (Balanced Binary Search Tree)</strong> 这一数据结构。</p><p>关于平衡二叉搜索树，有非常多的变种与实现，不同的应用场景会选择不同的平衡树类型。<strong>Treap</strong> 更灵活，通过随机化优先级实现预期的平衡，但在最坏情况下可能退化。<strong>AVL 树</strong> 严格保持平衡，保证了 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 的性能，但在频繁插入和删除的场景下可能有较大的旋转开销。<strong>红黑树</strong> 通过较宽松的平衡条件实现了较好的插入和删除性能，通常被广泛用于需要高效插入删除操作的系统（如 STL 中的 <code>map</code> 和 <code>set</code>）。一般来说，红黑树是一个较为通用的选择，而在需要严格平衡性时，AVL 树可能是更好的选择。</p><h3 id="Treap">Treap</h3><blockquote><p>Treap 是二叉搜索树和堆的结合体。它通过维护两种性质来保持平衡：</p><ul><li><strong>二叉搜索树性质</strong>：每个节点的左子树的所有节点值小于该节点的值，右子树的所有节点值大于该节点的值。</li><li><strong>堆性质</strong>：每个节点的优先级（通常随机生成）要大于或等于其子节点的优先级。</li></ul><p><strong>平衡机制</strong>：</p><ul><li>Treap 使用随机化优先级使得树的形状接近于理想的平衡树（期望树高为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>）。</li><li>通过旋转操作（左旋和右旋）在插入和删除时保持堆的性质。</li></ul><p><strong>优点</strong>：</p><ul><li>实现相对简单。</li><li>由于随机化的优先级，在期望情况下，树的高度是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>。</li><li>灵活性高，可以根据需要调整优先级函数。</li></ul><p><strong>缺点</strong>：</p><ul><li>最坏情况下，树的高度可能退化为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>（例如所有优先级相同或顺序生成的优先级），尽管发生概率很低。</li></ul></blockquote><h3 id="AVL-树">AVL 树</h3><blockquote><p>AVL 树是最早发明的自平衡二叉搜索树之一，1962 年由 Adelson-Velsky 和 Landis 发明。</p><ul><li><strong>平衡因子</strong>：每个节点的左右子树高度差不能超过 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，且需要记录每个节点的高度。</li></ul><p><strong>平衡机制</strong>：</p><ul><li>插入或删除节点后，如果某个节点的平衡因子不再为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span>、<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 或 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span>，就需要通过旋转（单旋转或双旋转）来恢复平衡。</li><li>旋转操作包括：左旋转、右旋转、左右双旋转和右左双旋转。</li></ul><p><strong>优点</strong>：</p><ul><li>严格的平衡条件保证了树的高度始终为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>，因此搜索、插入和删除操作的时间复杂度为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>。</li></ul><p><strong>缺点</strong>：</p><ul><li>由于平衡条件严格，每次插入和删除后可能需要较多的旋转操作，从而导致实现较复杂，插入和删除操作的常数时间开销较大。</li></ul></blockquote><h3 id="红黑树">红黑树</h3><blockquote><p>红黑树是一种较为宽松的自平衡二叉搜索树，由 Rudolf Bayer 于 1972 年发明。</p><ul><li><strong>颜色属性</strong>：每个节点都有红色或黑色两种颜色，通过这些颜色约束树的平衡性。</li></ul><p><strong>平衡机制</strong>：</p><ul><li>通过遵循红黑树的五个性质来保持平衡：<ol><li>每个节点要么是红色，要么是黑色。</li><li>根节点是黑色。</li><li>叶子节点（NIL 节点）是黑色。</li><li>如果一个节点是红色的，那么它的子节点必须是黑色（红节点不能连续出现）。</li><li>从任一节点到其每个叶子节点的所有路径都包含相同数量的黑色节点。</li></ol></li><li>插入和删除操作可能破坏红黑树的性质，需要通过重新着色和旋转来恢复平衡。</li></ul><p><strong>优点</strong>：</p><ul><li>红黑树的高度最多是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Θ</mi><mo stretchy="false">(</mo><mn>2</mn><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta (2  \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Θ</span><span class="mopen">(</span><span class="mord">2</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>，因此搜索、插入和删除操作的时间复杂度仍为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>。</li><li>由于平衡条件较为宽松，插入和删除操作需要的旋转操作通常比 AVL 树少，效率更高。</li></ul><p><strong>缺点</strong>：</p><ul><li>实现较复杂，特别是插入和删除的平衡修复过程。</li><li>虽然红黑树的搜索效率与 AVL 树相似，但由于平衡条件较宽松，实际应用中的树高度通常略高于 AVL 树，因此搜索操作的效率稍低。</li></ul></blockquote><h2 id="【二叉树】美国血统">【二叉树】美国血统</h2><p><a href="https://www.luogu.com.cn/problem/P1827">https://www.luogu.com.cn/problem/P1827</a></p><blockquote><p>题意：给定二叉树的中序和先序序列，输出后序序列</p><p>思路：经典二叉树的题目，主要用于巩固加强对于递归的理解。指针其实是没有必要的，为了得到后序序列，我们只需要有一个 dfs 序即可，为了得到 dfs 序，我们只需要根据给出的中序和前序序列即可得到 dfs 序</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><p>指针做法</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">30</span>;<br><br>string mid, pre;<br><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">Node</span> {<br>    <span class="hljs-type">char</span> data;<br>    Node* le, * ri;<br>    <span class="hljs-built_in">Node</span>(<span class="hljs-type">char</span> _data) : <span class="hljs-built_in">data</span>(_data), <span class="hljs-built_in">le</span>(<span class="hljs-literal">nullptr</span>), <span class="hljs-built_in">ri</span>(<span class="hljs-literal">nullptr</span>) {}<br>};<br><br><span class="hljs-function">Node* <span class="hljs-title">build</span><span class="hljs-params">(<span class="hljs-type">int</span> i, <span class="hljs-type">int</span> j, <span class="hljs-type">int</span> p, <span class="hljs-type">int</span> q)</span> </span>{<br>    <span class="hljs-keyword">if</span> (i &gt; j) <span class="hljs-keyword">return</span> <span class="hljs-literal">nullptr</span>;<br>    <br>    Node* root = <span class="hljs-keyword">new</span> <span class="hljs-built_in">Node</span>(pre[i]);<br>    <br>    <span class="hljs-type">int</span> k; <span class="hljs-comment">// 根结点在中序序列的下标 </span><br>    <span class="hljs-keyword">for</span> (k = p; k &lt;= q; k++)<br>        <span class="hljs-keyword">if</span> (mid[k] == root-&gt;data)<br>            <span class="hljs-keyword">break</span>;<br>    <br>    root-&gt;le = <span class="hljs-built_in">build</span>(i + <span class="hljs-number">1</span>, k - p + i, p, k - <span class="hljs-number">1</span>);<br>    root-&gt;ri = <span class="hljs-built_in">build</span>(k - p + i + <span class="hljs-number">1</span>, j, k + <span class="hljs-number">1</span>, q);<br>    <br>    cout &lt;&lt; root-&gt;data; <br>    <br>    <span class="hljs-keyword">return</span> root;<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; mid &gt;&gt; pre;<br>    <br>    <span class="hljs-type">int</span> i = <span class="hljs-number">0</span>, j = pre.<span class="hljs-built_in">size</span>() - <span class="hljs-number">1</span>;<br>    <span class="hljs-type">int</span> p = <span class="hljs-number">0</span>, q = mid.<span class="hljs-built_in">size</span>() - <span class="hljs-number">1</span>;<br>    <br>    <span class="hljs-built_in">build</span>(i, j, p, q);<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><p>构造出 dfs 序直接输出</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">30</span>;<br><br>string mid, pre;<br><br><span class="hljs-comment">// 前序起始 i，前序末尾 j，中序起始 p，中序末尾 q </span><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">build</span><span class="hljs-params">(<span class="hljs-type">int</span> i, <span class="hljs-type">int</span> j, <span class="hljs-type">int</span> p, <span class="hljs-type">int</span> q)</span> </span>{<br>    <span class="hljs-keyword">if</span> (i &gt; j) <span class="hljs-keyword">return</span>;<br>    <br>    <span class="hljs-type">char</span> root = pre[i];<br>    <br>    <span class="hljs-type">int</span> k;<br>    <span class="hljs-keyword">for</span> (k = p; k &lt;= q; k++)<br>        <span class="hljs-keyword">if</span> (mid[k] == root)<br>            <span class="hljs-keyword">break</span>;<br>            <br>    <span class="hljs-built_in">build</span>(i + <span class="hljs-number">1</span>, k - p + i, p, k - <span class="hljs-number">1</span>);<br>    <span class="hljs-built_in">build</span>(k - p + i + <span class="hljs-number">1</span>, j, k + <span class="hljs-number">1</span>, q);<br>    <br>    cout &lt;&lt; root;<br>} <br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; mid &gt;&gt; pre;<br>    <br>    <span class="hljs-type">int</span> i = <span class="hljs-number">0</span>, j = pre.<span class="hljs-built_in">size</span>() - <span class="hljs-number">1</span>;<br>    <span class="hljs-type">int</span> p = <span class="hljs-number">0</span>, q = mid.<span class="hljs-built_in">size</span>() - <span class="hljs-number">1</span>;<br>    <br>    <span class="hljs-built_in">build</span>(i, j, p, q);<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h2 id="【二叉树】新二叉树">【二叉树】新二叉树</h2><p><a href="https://www.luogu.com.cn/problem/P1305">https://www.luogu.com.cn/problem/P1305</a></p><blockquote><p>题意：给定一棵二叉树的 n 个结点信息，分别为当前结点的数据信息、左孩子结点信息和右结点信息，输出这棵二叉树的前序序列</p><p>思路：我们首先将这棵二叉树构建出来，接着遍历输出前序序列即可。关键在于如何构建二叉树？我们使用数组存储二叉树，对于每一个树上结点，我们将数组中元素的索引存储为树上结点信息，每一个结点再存储左孩子与右孩子的信息</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">int</span> n;<br>string s;<br><span class="hljs-type">char</span> root;<br><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">Node</span> {<br>    <span class="hljs-type">char</span> l, r;<br>} tree[<span class="hljs-number">200</span>];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">pre</span><span class="hljs-params">(<span class="hljs-type">char</span> now)</span> </span>{<br>    <span class="hljs-keyword">if</span> (now == <span class="hljs-string">'*'</span>) <span class="hljs-keyword">return</span>;<br>    cout &lt;&lt; now;<br>    <span class="hljs-built_in">pre</span>(tree[now].l);<br>    <span class="hljs-built_in">pre</span>(tree[now].r);<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        cin &gt;&gt; s;<br>        <span class="hljs-keyword">if</span> (i == <span class="hljs-number">1</span>) root = s[<span class="hljs-number">0</span>];<br>        tree[s[<span class="hljs-number">0</span>]].l = s[<span class="hljs-number">1</span>];<br>        tree[s[<span class="hljs-number">0</span>]].r = s[<span class="hljs-number">2</span>];<br>    }<br>    <br>    <span class="hljs-built_in">pre</span>(root);<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h2 id="【二叉树】遍历问题">【二叉树】遍历问题</h2><p><a href="https://www.luogu.com.cn/problem/P1229">https://www.luogu.com.cn/problem/P1229</a></p><blockquote><p>题意：给定一棵二叉树的前序序列与后序序列，问中序序列的可能情况有多少种</p><p>思路：我们采用从最小结构单元的思路进行考虑，即假设当前二叉树只有一个根结点与两个叶子结点，而非两棵子树。然后将题意进行等价变换，即问对于已经固定的前序和后序二叉树，该二叉树有多少种不同的形状？对于当前的最小结构二叉树，形状就是 <strong>左右根</strong> or <strong>根左右</strong>，现在的根可以直接确定，那么就只能从左右孩子进行变形，很显然只能进行左右交换的变形，但是问题是一旦左右变换，前序 or 后序都会变掉，说明这种左右孩子都存在的前后序固定的二叉树是唯一的，那么如何才是不唯一的呢？我们考虑减少孩子数量。假设没有孩子，那么很显然也只有一个形状，就是一个根结点，故排除。于是答案就呼之欲出了，就是当根结点只有一个孩子时，这个孩子无论是在左边还是右边，前后序都是相同的，但是中序序列就不同了，于是就产生了两种中序序列。于是最终的结论是：对于前后序确定的二叉树来说，中序序列的情况是就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mtext>单分支结点数</mtext></msup></mrow><annotation encoding="application/x-tex">2^{\text{单分支结点数}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8413em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">单分支结点数</span></span></span></span></span></span></span></span></span></span></span></span></span> 个。现在的问题就转变为了在给定前后序的二叉树中求解单分支结点个数的问题。</p><p>如何寻找单分支结点呢？根据下面的递归图可以发现，无论是左单分支还是右单分支，如果 pre 的连续两个结点与 post 的连续两个结点对称相同，那么就一定有一个单分支结点，故只需要寻找前后序序列中连续两个字符对称相同的情况数 cnt 即可。最终的答案数就是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mrow><mi>c</mi><mi>n</mi><mi>t</mi></mrow></msup></mrow><annotation encoding="application/x-tex">2^{cnt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7936em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7936em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">c</span><span class="mord mathnormal mtight">n</span><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span></span></span></span></p><p><img src="https://dwj-oss.oss-cn-nanjing.aliyuncs.com/images/202406061327025.png" alt="图例"></p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(nm)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">nm</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br>string pre, post;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; pre &gt;&gt; post;<br>    <br>    <span class="hljs-type">int</span> cnt = <span class="hljs-number">0</span>;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; pre.<span class="hljs-built_in">size</span>() - <span class="hljs-number">1</span>; i++)<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">0</span>; j &lt; post.<span class="hljs-built_in">size</span>(); j++)<br>            <span class="hljs-keyword">if</span> (pre[i] == post[j + <span class="hljs-number">1</span>] &amp;&amp; pre[i + <span class="hljs-number">1</span>] == post[j])<br>                cnt++;<br>    <br>    cout &lt;&lt; (<span class="hljs-number">1</span> &lt;&lt; cnt) &lt;&lt; <span class="hljs-string">"\n"</span>;<br>} <br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h2 id="【二叉树】医院设置-fire">【二叉树】医院设置 <span class="github-emoji"><span>🔥</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/1f525.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span></h2><p><a href="https://www.luogu.com.cn/problem/P1364">https://www.luogu.com.cn/problem/P1364</a></p><blockquote><p>题意：给定一棵二叉树，树中每一个结点存储了一个数值表示一个医院的人数，现在需要在所有的结点中将一个结点设置为医院使得其余结点中的所有人到达该医院走的路总和最小。路程为结点到医院的最短路，边权均为 1。给出最终的最短路径总和</p><p>思路一：暴力</p><ul><li><p>显然的对于已经设置好医院的局面，需要求解的路径总和就直接将树遍历一边即可。每一个结点都可以作为医院进行枚举，每次遍历是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 的</p></li><li><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></p></li></ul><p>思路二：带权树的重心</p><ul><li>TODO</li><li>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li></ul></blockquote><p>暴力代码</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">110</span>;<br><br><span class="hljs-type">int</span> n;<br><br>vector&lt;<span class="hljs-type">int</span>&gt; G[N];<br><span class="hljs-type">int</span> cnt[N];<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">bfs</span><span class="hljs-params">(<span class="hljs-type">int</span> v)</span> </span>{<br>    <span class="hljs-type">int</span> res = <span class="hljs-number">0</span>;<br>    <span class="hljs-function">vector&lt;<span class="hljs-type">bool</span>&gt; <span class="hljs-title">vis</span><span class="hljs-params">(n + <span class="hljs-number">1</span>, <span class="hljs-literal">false</span>)</span></span>;<br>    <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">d</span><span class="hljs-params">(n + <span class="hljs-number">1</span>, <span class="hljs-number">0</span>)</span></span>; <span class="hljs-comment">// d[i] 表示点 i 到点 v 的距离</span><br>        <br>    queue&lt;<span class="hljs-type">int</span>&gt; q;<br>    vis[v] = <span class="hljs-literal">true</span>;<br>    d[v] = <span class="hljs-number">0</span>;<br>    q.<span class="hljs-built_in">push</span>(v);<br>    <br>    <span class="hljs-keyword">while</span> (q.<span class="hljs-built_in">size</span>()) {<br>        <span class="hljs-type">int</span> now = q.<span class="hljs-built_in">front</span>();<br>        q.<span class="hljs-built_in">pop</span>();<br>        <br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; ch: G[now]) {<br>            <span class="hljs-keyword">if</span> (!vis[ch]) {<br>                vis[ch] = <span class="hljs-literal">true</span>;<br>                d[ch] = d[now] + <span class="hljs-number">1</span>;<br>                q.<span class="hljs-built_in">push</span>(ch);<br>                <br>                res += cnt[ch] * d[ch]; <br>            }<br>        }<br>    }<br>    <br>    <span class="hljs-keyword">return</span> res;<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-type">int</span> count, l, r;<br>        cin &gt;&gt; count &gt;&gt; l &gt;&gt; r;<br>        cnt[i] = count;<br>        <br>        <span class="hljs-keyword">if</span> (l) {<br>            G[i].<span class="hljs-built_in">push_back</span>(l);<br>            G[l].<span class="hljs-built_in">push_back</span>(i);<br>        }<br>        <br>        <span class="hljs-keyword">if</span> (r) {<br>            G[i].<span class="hljs-built_in">push_back</span>(r);<br>            G[r].<span class="hljs-built_in">push_back</span>(i);<br>        }<br>    }<br>    <br>    <span class="hljs-type">int</span> res = <span class="hljs-number">1e7</span> + <span class="hljs-number">10</span>;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        res = <span class="hljs-built_in">min</span>(res, <span class="hljs-built_in">bfs</span>(i));<br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>} <br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><p>优化代码</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><br></code></pre></td></tr></tbody></table></figure><h2 id="【二叉树】二叉树深度">【二叉树】二叉树深度</h2><p><a href="https://www.luogu.com.cn/problem/P4913">https://www.luogu.com.cn/problem/P4913</a></p><blockquote><p>题意：给定一棵二叉树，求解这棵二叉树的深度</p><p>思路：有两个考点，一个是如何根据给定的信息（从根结点开始依次给出已存在树上结点的左右孩子的编号）构建二叉树，一个是如何求解已经构建好的二叉树的深度。对于构建二叉树，我们沿用 T5 数组模拟构建的思路，直接定义结点类型即可；对于求解深度，很显然的一个递归求解，即左右子树深度值 +1 即可</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">1000010</span>;<br><br><span class="hljs-type">int</span> n;<br><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">Node</span> {<br>    <span class="hljs-type">int</span> l, r;<br>} t[N];<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">dep</span><span class="hljs-params">(<span class="hljs-type">int</span> now)</span> </span>{<br>    <span class="hljs-keyword">if</span> (!now) <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">return</span> <span class="hljs-built_in">max</span>(<span class="hljs-built_in">dep</span>(t[now].l), <span class="hljs-built_in">dep</span>(t[now].r)) + <span class="hljs-number">1</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n;<br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        <span class="hljs-type">int</span> x, y;<br>        cin &gt;&gt; x &gt;&gt; y;<br>        t[i].l = x, t[i].r = y;<br>    }<br>    <br>    cout &lt;&lt; <span class="hljs-built_in">dep</span>(<span class="hljs-number">1</span>);<br>} <br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h2 id="【完全二叉树】淘汰赛">【完全二叉树】淘汰赛</h2><p><a href="https://www.luogu.com.cn/problem/P1364">https://www.luogu.com.cn/problem/P1364</a></p><blockquote><p>题意：给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>2</mn><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">2^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6644em;"></span><span class="mord"><span class="mord">2</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span></span></span></span> 支球队的编号与能力值，进行淘汰赛，能力值者晋级下一轮直到赛出冠军。输出亚军编号</p><p>思路：很显然的一个完全二叉树的题目。我们都不需要进行递归操作，直接利用完全二叉树的下标性质利用数组模拟循环计算即可。给出的信息就是完全二叉树的全部叶子结点的信息，分别为球队编号 id 与球队能力值 val，我们从第 n-1 个结点开始循环枚举到第 1 个结点计算每一轮的胜者信息，最终输出最后一场的能力值较小者球队编号即可</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Θ</mi><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta(2n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Θ</span><span class="mopen">(</span><span class="mord">2</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">1</span> &lt;&lt; <span class="hljs-number">8</span>;<br><br><span class="hljs-keyword">struct</span> <span class="hljs-title class_">Node</span> {<br>    <span class="hljs-type">int</span> id, val;<br>} a[N];<br><br><span class="hljs-type">int</span> n;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n;<br>    <br>    n = <span class="hljs-number">1</span> &lt;&lt; n;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = n; i &lt;= <span class="hljs-number">2</span> * n - <span class="hljs-number">1</span>; i++) {<br>        a[i].id = i - n + <span class="hljs-number">1</span>;<br>        cin &gt;&gt; a[i].val;<br>    }<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = n - <span class="hljs-number">1</span>; i &gt;= <span class="hljs-number">1</span>; i--)<br>        <span class="hljs-keyword">if</span> (a[i * <span class="hljs-number">2</span>].val &gt; a[i * <span class="hljs-number">2</span> + <span class="hljs-number">1</span>].val) a[i] = a[i * <span class="hljs-number">2</span>];<br>        <span class="hljs-keyword">else</span> a[i] = a[i * <span class="hljs-number">2</span> + <span class="hljs-number">1</span>];<br>            <br>    <span class="hljs-keyword">if</span> (a[<span class="hljs-number">2</span>].val &gt; a[<span class="hljs-number">3</span>].val) cout &lt;&lt; a[<span class="hljs-number">3</span>].id;<br>    <span class="hljs-keyword">else</span> cout &lt;&lt; a[<span class="hljs-number">2</span>].id;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h2 id="【二叉树-LCA】二叉树问题">【二叉树/LCA】二叉树问题</h2><p><a href="https://www.luogu.com.cn/problem/P3884">https://www.luogu.com.cn/problem/P3884</a></p><blockquote><p>题意：给定一棵二叉树的结点关系信息，求出这棵二叉树的深度、宽度和两个指定结点之间的最短路径长度</p><p>思路：二叉树的构建直接采用有向图的构造方法。深度直接 dfs 即可，宽度直接在 dfs 遍历时哈希深度值即可。问题的关键在于如何求解两个给定结点之间的路径长度，很显然需要求解两个结点的 LCA，由于结点数 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≤</mo><mn>100</mn></mrow><annotation encoding="application/x-tex">\le 100</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">100</span></span></span></span> 故直接采用暴力的方法，可以重定义结点，增加父结点域。也可以通过比对根结点到两个指定结点的路径信息得到 LCA 即最后一个相同的结点编号（本题采用），通过在 dfs 遍历树时存储路径即可得到根结点到两个指定结点的路径信息。之后直接根据题中新定义的路径长度输出即可，即</p><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mtext>length</mtext><mo>=</mo><mn>2</mn><mo>×</mo><mo stretchy="false">(</mo><msub><mi>d</mi><mi>x</mi></msub><mo>−</mo><msub><mi>d</mi><mrow><mi>l</mi><mi>c</mi><mi>a</mi></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><msub><mi>d</mi><mi>y</mi></msub><mo>−</mo><msub><mi>d</mi><mrow><mi>l</mi><mi>c</mi><mi>a</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\text{length} = 2 \times (d_x - d_{lca}) + (d_y - d_{lca})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord text"><span class="mord">length</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.01968em;">l</span><span class="mord mathnormal mtight">c</span><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.01968em;">l</span><span class="mord mathnormal mtight">c</span><span class="mord mathnormal mtight">a</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p><p>其中 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>d</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">d_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 表示：根结点到第 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span> 号点之间的路径长度，在 dfs 时通过传递深度值维护得到</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">110</span>;<br><br><span class="hljs-type">int</span> n, x, y;<br>vector&lt;<span class="hljs-type">int</span>&gt; G[N];<br><span class="hljs-type">int</span> depth, width;<br>unordered_map&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt; ha; <span class="hljs-comment">// 将所有的深度值进行哈希</span><br><span class="hljs-type">int</span> d[N];                   <span class="hljs-comment">// d[i] 表示第 i 个点到根结点的边数</span><br>vector&lt;<span class="hljs-type">int</span>&gt; temp, rx, ry;   <span class="hljs-comment">// 根结点到 x 号点与 y 号点直接的路径结点编号</span><br><br><span class="hljs-comment">// 当前结点编号 now，当前深度 level</span><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">dfs</span><span class="hljs-params">(<span class="hljs-type">int</span> now, <span class="hljs-type">int</span> level)</span> </span>{<br>    depth = <span class="hljs-built_in">max</span>(depth, level);<br>    <br>    temp.<span class="hljs-built_in">push_back</span>(now);<br>    <span class="hljs-keyword">if</span> (now == x) rx = temp;<br>    <span class="hljs-keyword">if</span> (now == y) ry = temp;<br>    <br>    ha[level]++;<br>    d[now] = level - <span class="hljs-number">1</span>;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; ch: G[now]) {<br>        <span class="hljs-built_in">dfs</span>(ch, level + <span class="hljs-number">1</span>);<br>        temp.<span class="hljs-built_in">pop_back</span>();<br>    }<br>}<br><br><span class="hljs-comment">// 暴力 lca + 计算路径长度</span><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">len</span><span class="hljs-params">(<span class="hljs-type">int</span> x, <span class="hljs-type">int</span> y)</span> </span>{<br>    <span class="hljs-type">int</span> i = <span class="hljs-number">0</span>;<br>    <span class="hljs-keyword">while</span> (i &lt; rx.<span class="hljs-built_in">size</span>() &amp;&amp; i &lt; ry.<span class="hljs-built_in">size</span>() &amp;&amp; rx[i] == ry[i]) i++;<br>    <br>    <span class="hljs-type">int</span> lca = rx[--i];<br>    <br>    <span class="hljs-keyword">return</span> <span class="hljs-number">2</span> * (d[x] - d[lca]) + (d[y] - d[lca]);<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n;<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n - <span class="hljs-number">1</span>; i++) {<br>        <span class="hljs-type">int</span> a, b;<br>        cin &gt;&gt; a &gt;&gt; b;<br>        G[a].<span class="hljs-built_in">push_back</span>(b);<br>    }<br>    <br>    cin &gt;&gt; x &gt;&gt; y;<br>    <br>    <span class="hljs-comment">// 二叉树的深度 depth</span><br>    <span class="hljs-built_in">dfs</span>(<span class="hljs-number">1</span>, <span class="hljs-number">1</span>);<br>    cout &lt;&lt; depth &lt;&lt; <span class="hljs-string">"\n"</span>;<br>    <br>    <span class="hljs-comment">// 二叉树的宽度 width</span><br>    <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; item: ha) width = <span class="hljs-built_in">max</span>(width, item.second);<br>    cout &lt;&lt; width &lt;&lt; <span class="hljs-string">"\n"</span>;<br>    <br>    <span class="hljs-comment">// 两个结点之间的路径长度</span><br>    cout &lt;&lt; <span class="hljs-built_in">len</span>(x, y) &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h2 id="【set】营业额统计">【set】营业额统计</h2><p><a href="https://www.luogu.com.cn/problem/P2234">https://www.luogu.com.cn/problem/P2234</a></p><blockquote><p>题意：给定一个序列 a，需要计算 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mstyle scriptlevel="0" displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>2</mn><mo separator="true">,</mo><mn>1</mn><mo>≤</mo><mi>j</mi><mo>&lt;</mo><mi>i</mi></mrow><mi>n</mi></munderover><mi>min</mi><mo>⁡</mo><mrow><mi mathvariant="normal">∣</mi><msub><mi>a</mi><mi>i</mi></msub><mo>−</mo><msub><mi>a</mi><mi>j</mi></msub><mi mathvariant="normal">∣</mi></mrow></mstyle></mrow><annotation encoding="application/x-tex">a_1 + \displaystyle \sum_{i=2,1 \le j &lt;i}^{n} \min {|a_i - a_j|}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:3.0652em;vertical-align:-1.4138em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">2</span><span class="mpunct mtight">,</span><span class="mord mtight">1</span><span class="mrel mtight">≤</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">&lt;</span><span class="mord mathnormal mtight">i</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4138em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">min</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord">∣</span></span></span></span></span> ，即计算每一个数与序列中当前数之前的数的最小差值之和</p><p>思路：很显然的思路，对于每一个数，我们需要对之前的序列在短时间内找到一个数值最接近当前数的数。</p><ul><li>TLE：一开始的思路是每次对之前的序列进行排序，然后二分查找与当前值匹配的数，为了确保所有的情况都找到，就直接判断二分查到的数，查到的数之前的一个数，之后的一个数，但是时间复杂度极高（我居然没想到），是 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2 \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li><li>AC：后来看了题解才知道 <code>set</code> 的正确用法，就是一个 <strong>平衡树的 STL</strong>。我们对于之前的序列不断的插入平衡树中（默认升序排序），每次利用 <code>s.lower_bound(x)</code> 返回「集合 <code>s</code> 中第一个 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≥</mo></mrow><annotation encoding="application/x-tex">\ge</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≥</span></span></span></span> 当前数的迭代器」，然后进行判断即可。<code>lower_bound()</code> 的时间复杂度为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span> 。需要注意的是边界的判断，一开始的思路虽然会超时，但是二分后边界的判断很简单，使用 STL 后同样需要考虑边界的情况。分为三种（详情见代码）<ul><li>当前数比集合中所有的数都大，那么 <code>lower_bound</code> 就会返回 <code>s.end()</code> 答案就是当前数与集合中最后一个数的差值</li><li>当前数比集合中所有的数都小，那么 <code>lower_bound</code> 就会返回 <code>s.bigin()</code> 答案就是集合中第一个数与当前数的差值</li><li>当前数存在于集合中 or 集合中既有比当前数大的又有比当前数小的，那么就比较查到的数与查到的数前一个数和当前数的差值，取最小的差值即可</li></ul></li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><p>TLE 但逻辑清晰代码</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;algorithm&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">1</span> &lt;&lt; <span class="hljs-number">16</span>;<br><br><span class="hljs-type">int</span> n, a[N];<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n;<br>    <br>    <span class="hljs-type">int</span> res = <span class="hljs-number">0</span>;<br>    cin &gt;&gt; a[<span class="hljs-number">1</span>];<br>    res += a[<span class="hljs-number">1</span>];<br>    <br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">2</span>; i &lt;= n; i++) {<br>        <span class="hljs-comment">// 维护之前序列有序</span><br>        <span class="hljs-built_in">sort</span>(a + <span class="hljs-number">1</span>, a + i);<br>        cin &gt;&gt; a[i];<br><br>        <span class="hljs-comment">// 二分查找目标数</span><br>        <span class="hljs-type">int</span> l = <span class="hljs-number">1</span>, r = i - <span class="hljs-number">1</span>;<br>        <span class="hljs-keyword">while</span> (l &lt; r) {<br>            <span class="hljs-type">int</span> mid = (l + r) &gt;&gt; <span class="hljs-number">1</span>;<br>            <span class="hljs-keyword">if</span> (a[mid] &lt; a[i]) l = mid + <span class="hljs-number">1</span>;<br>            <span class="hljs-keyword">else</span> r = mid;<br>        }<br>        <br>        <span class="hljs-comment">// 边界判断</span><br>        <span class="hljs-type">int</span> ans = <span class="hljs-built_in">abs</span>(a[i] - a[r]);<br>        <span class="hljs-keyword">if</span> (r + <span class="hljs-number">1</span> &gt;= <span class="hljs-number">1</span> &amp;&amp; r + <span class="hljs-number">1</span> &lt;= i - <span class="hljs-number">1</span>) ans = <span class="hljs-built_in">min</span>(ans, <span class="hljs-built_in">abs</span>(a[i] - a[r + <span class="hljs-number">1</span>]));<br>        <span class="hljs-keyword">if</span> (r - <span class="hljs-number">1</span> &gt;= <span class="hljs-number">1</span> &amp;&amp; r - <span class="hljs-number">1</span> &lt;= i - <span class="hljs-number">1</span>) ans = <span class="hljs-built_in">min</span>(ans, <span class="hljs-built_in">abs</span>(a[i] - a[r - <span class="hljs-number">1</span>]));<br>        <br>        res += ans;<br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><p>AC 的 set 代码</p><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;algorithm&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;set&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">int</span> n, res;<br>set&lt;<span class="hljs-type">int</span>&gt; s;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n;<br>    <br>    <span class="hljs-type">int</span> x;<br>    cin &gt;&gt; x;<br>    res += x;<br>    s.<span class="hljs-built_in">insert</span>(x);<br>    <br>    <span class="hljs-keyword">while</span> (--n) {<br>        cin &gt;&gt; x;<br><br>        <span class="hljs-keyword">auto</span> it = s.<span class="hljs-built_in">lower_bound</span>(x);<br><br>        <span class="hljs-keyword">if</span> (it == s.<span class="hljs-built_in">end</span>()) {<br>            <span class="hljs-comment">// 没有比当前数大的</span><br>            res += x - *s.<span class="hljs-built_in">rbegin</span>();<br>        } <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (it == s.<span class="hljs-built_in">begin</span>()) {<br>            <span class="hljs-comment">// 没有比当前数小的</span><br>            res += *s.<span class="hljs-built_in">begin</span>() - x;<br>        } <span class="hljs-keyword">else</span> {<br>            <span class="hljs-comment">// 当前数已存在于集合中 or 既有比当前数大的也有比当前数小的</span><br>            <span class="hljs-keyword">auto</span> pre = it;<br>            pre--;<br>            res += <span class="hljs-built_in">min</span>(<span class="hljs-built_in">abs</span>(x - *it), <span class="hljs-built_in">abs</span>(x - *pre));<br>        }<br>        <br>        s.<span class="hljs-built_in">insert</span>(x);<br>    }<br>    <br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h2 id="【multiset】切蛋糕">【multiset】切蛋糕</h2><p><a href="https://www.acwing.com/problem/content/description/5581/">https://www.acwing.com/problem/content/description/5581/</a></p><blockquote><p>题意：给定一个矩形，由左下角和右上角的坐标确定。现在对这个矩形进行切割操作，要么竖着切，要么横着切。现在需要给出每次切割后最大子矩形的面积</p><p>思路：STL set。很容易想到，横纵坐标是相互独立的，最大子矩形面积一定产生于最大长度与最大宽度的乘积。因此我们只需要维护一个序列的最大值即可。对于二维可以直接当做一维来看，于是问题就变成了，需要在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>log</mi><mo>⁡</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">\log n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span></span></span></span> 的时间复杂度内，对一个序列完成下面三个操作：</p><ol><li>删除一个数</li><li>增加一个数（执行两次）</li><li>获取最大值</li></ol><p>如何实现呢？我们需要有序记录每一个子线段的长度，并且子线段的长度可能重复，因此我们用 <code>std::multiset</code> 来存储所有子线段的长度</p><ol><li>使用 <code>M.erase(M.find(value))</code> 实现：删除一个子线段长度值</li><li>使用 <code>M.insert(value)</code> 实现：增加子线段一个长度值</li><li>使用 <code>*M.rbegin()</code> 实现：获取当前所有子线段长度的最大值</li></ol><p>由于给的是切割的位置坐标 <code>x</code>，因此上述操作 1 不能直接实现，我们需要利用给定的切割坐标 <code>x</code> 计算出当前切割位置对应子线段的长度。如何实现呢？我们知道，对于当前切割的坐标 <code>x</code>，对应的子线段的长度取决于当前切割坐标左右两个切割的位置 <code>rp, lp</code>，因此我们只需要存储每一个切割的坐标即可。由于切割位置不会重复，并且需要在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>log</mi><mo>⁡</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">\log n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span></span></span></span> 的时间复杂度内查询到，因此我们还是可以使用 <code>std::set</code> 来存储切割位置</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;iostream&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;set&gt;</span></span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><span class="hljs-keyword">using</span> ll = <span class="hljs-type">long</span> <span class="hljs-type">long</span>;<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">work</span><span class="hljs-params">(<span class="hljs-type">int</span> x, set&lt;<span class="hljs-type">int</span>&gt;&amp; S, multiset&lt;<span class="hljs-type">int</span>&gt;&amp; M)</span> </span>{<br>    set&lt;<span class="hljs-type">int</span>&gt;::iterator rp = S.<span class="hljs-built_in">upper_bound</span>(x), lp = rp;<br>    lp--;<br>    S.<span class="hljs-built_in">insert</span>(x);<br><br>    M.<span class="hljs-built_in">erase</span>(M.<span class="hljs-built_in">find</span>(*rp - *lp));<br>    M.<span class="hljs-built_in">insert</span>(*rp - x);<br>    M.<span class="hljs-built_in">insert</span>(x - *lp);<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    <span class="hljs-type">int</span> w, h, n;<br>    cin &gt;&gt; w &gt;&gt; h &gt;&gt; n;<br><br>    set&lt;<span class="hljs-type">int</span>&gt; S1, S2;<br>    multiset&lt;<span class="hljs-type">int</span>&gt; M1, M2;<br>    S<span class="hljs-number">1.</span><span class="hljs-built_in">insert</span>(<span class="hljs-number">0</span>), S<span class="hljs-number">1.</span><span class="hljs-built_in">insert</span>(w), M<span class="hljs-number">1.</span><span class="hljs-built_in">insert</span>(w);<br>    S<span class="hljs-number">2.</span><span class="hljs-built_in">insert</span>(<span class="hljs-number">0</span>), S<span class="hljs-number">2.</span><span class="hljs-built_in">insert</span>(h), M<span class="hljs-number">2.</span><span class="hljs-built_in">insert</span>(h);<br><br>    <span class="hljs-keyword">while</span> (n--) {<br>        <span class="hljs-type">char</span> op;<br>        <span class="hljs-type">int</span> x;<br>        cin &gt;&gt; op &gt;&gt; x;<br>        <span class="hljs-keyword">if</span> (op == <span class="hljs-string">'X'</span>) <span class="hljs-built_in">work</span>(x, S1, M1);<br>        <span class="hljs-keyword">else</span> <span class="hljs-built_in">work</span>(x, S2, M2);<br><br>        cout &lt;&lt; (ll)*M<span class="hljs-number">1.</span><span class="hljs-built_in">rbegin</span>() * *M<span class="hljs-number">2.</span><span class="hljs-built_in">rbegin</span>() &lt;&lt; <span class="hljs-string">"\n"</span>;<br>    }<br>}<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>    ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>    cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br>    <span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//    cin &gt;&gt; T;</span><br>    <span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br>    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h2 id="【并查集】Milk-Visits-S">【并查集】Milk Visits S</h2><p><a href="https://www.luogu.com.cn/problem/P5836">https://www.luogu.com.cn/problem/P5836</a></p><blockquote><p>题意：给定一棵树，结点被标记成两种，一种是 H，一种是 G，在每一次查询中，需要知道指定的两个结点之间是否含有某一种标记</p><p>思路：对于树上标记，我们可以将相同颜色的分支连成一个连通块</p><ul><li>如果查询的两个结点在同一个连通块，则查询两个结点所在的颜色与所需的颜色是否匹配即可</li><li>如果查询的两个结点不在同一个连通块，两个结点之间的路径一定是覆盖了两种颜色的标记，则答案一定是 1</li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Θ</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta(n+m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Θ</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">m</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">100010</span>;<br><br><span class="hljs-type">int</span> n, m, p[N];<br><span class="hljs-type">char</span> col[N];<br><br><span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">find</span><span class="hljs-params">(<span class="hljs-type">int</span> x)</span> </span>{<br>    <span class="hljs-keyword">if</span> (p[x] != x) {<br>        p[x] = <span class="hljs-built_in">find</span>(p[x]);<br>    }<br>    <span class="hljs-keyword">return</span> p[x];<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>    cin &gt;&gt; n &gt;&gt; m;<br>    cin &gt;&gt; (col + <span class="hljs-number">1</span>);<br><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>        p[i] = i;<br>    }<br><br>    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n - <span class="hljs-number">1</span>; i++) {<br>        <span class="hljs-type">int</span> a, b;<br>        cin &gt;&gt; a &gt;&gt; b;<br>        <span class="hljs-keyword">if</span> (col[a] == col[b]) {<br>            p[<span class="hljs-built_in">find</span>(a)] = <span class="hljs-built_in">find</span>(b);<br>        }<br>    }<br><br>    string res;<br><br>    <span class="hljs-keyword">while</span> (m--) {<br>        <span class="hljs-type">int</span> u, v;<br>        cin &gt;&gt; u &gt;&gt; v;<br><br>        <span class="hljs-type">char</span> cow;<br>        cin &gt;&gt; cow;<br><br>        <span class="hljs-keyword">if</span> (<span class="hljs-built_in">find</span>(u) == <span class="hljs-built_in">find</span>(v)) {<br>            res += <span class="hljs-built_in">to_string</span>(col[u] == cow);<br>        } <span class="hljs-keyword">else</span> {<br>            res += <span class="hljs-string">'1'</span>;<br>        }<br>    }<br><br>    cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>;<br>}<br></code></pre></td></tr></tbody></table></figure><h2 id="【并查集】尽量减少恶意软件的传播">【并查集】尽量减少恶意软件的传播</h2><p><a href="https://leetcode.cn/problems/minimize-malware-spread/description/">https://leetcode.cn/problems/minimize-malware-spread/description/</a></p><blockquote><p>题意：给定一个由邻接矩阵存储的无向图，其中某些结点具备感染能力，可以感染相连的所有结点，问消除哪一个结点的感染能力可以使得最终不被感染的结点数量尽可能少，给出消除的有感染能力的最小结点编号</p><p>思路：很显然我们可以将当前无向图的多个连通分量，共三种感染情况：</p><ol><li><p>如果一个连通分量中含有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\ge 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span></span></span></span> 个感染结点，则当前连通分量一定会被全部感染；</p></li><li><p>如果一个连通块中含有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 个感染结点，则无需任何操作；</p></li><li><p>如果一个连通块中含有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 个感染结点，则最佳实践就是移除该连通块中唯一的那个感染结点。</p></li></ol><p>当然了，由于只能移走一个感染结点，我们需要从所有只含有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span> 个感染结点的连通块中移走连通块结点最多的那个感染结点。因此我们需要统计每一个连通分量中感染结点的数量以及总结点数，采用并查集进行统计。需要注意的是题目中的“索引最小”指的是结点编号最小而非结点在序列 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mi>n</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>a</mi><mi>l</mi></mrow><annotation encoding="application/x-tex">initial</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal">ini</span><span class="mord mathnormal">t</span><span class="mord mathnormal">ia</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span></span></span></span> 中的下标最小。算法流程见代码。</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>n</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0641em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span> {<br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-type">int</span> p[<span class="hljs-number">310</span>];<br><br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">Find</span><span class="hljs-params">(<span class="hljs-type">int</span> x)</span> </span>{<br>        <span class="hljs-keyword">if</span> (x != p[x]) p[x] = <span class="hljs-built_in">Find</span>(p[x]);<br>        <span class="hljs-keyword">return</span> p[x];<br>    }<br><br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">minMalwareSpread</span><span class="hljs-params">(vector&lt;vector&lt;<span class="hljs-type">int</span>&gt;&gt;&amp; graph, vector&lt;<span class="hljs-type">int</span>&gt;&amp; initial)</span> </span>{<br>        <span class="hljs-comment">// 1. 维护并查集数组：p[]</span><br>        <span class="hljs-type">int</span> n = graph.<span class="hljs-built_in">size</span>();<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) p[i] = i;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++)<br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">0</span>; j &lt; n; j++)<br>                <span class="hljs-keyword">if</span> (graph[i][j])<br>                    p[<span class="hljs-built_in">Find</span>(i)] = <span class="hljs-built_in">Find</span>(j);<br><br>        <span class="hljs-comment">// 2. 维护哈希表：每一个连通块中的感染结点数、总结点数</span><br>        unordered_map&lt;<span class="hljs-type">int</span>, pair&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt;&gt; ha;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; x: initial) ha[<span class="hljs-built_in">Find</span>(x)].first++;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) ha[<span class="hljs-built_in">Find</span>(i)].second++;<br><br>        <span class="hljs-comment">// 3. 排序：按照感染结点数升序，总结点数降序</span><br>        vector&lt;pair&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt;&gt; v;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; it: ha) v.<span class="hljs-built_in">push_back</span>(it.second);<br>        <span class="hljs-built_in">sort</span>(v.<span class="hljs-built_in">begin</span>(), v.<span class="hljs-built_in">end</span>(), [&amp;](pair&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt;&amp; x, pair&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt;&amp; y){<br>            <span class="hljs-keyword">if</span> (x.first == y.first) <span class="hljs-keyword">return</span> x.second &gt; y.second;<br>            <span class="hljs-keyword">return</span> x.first &lt; y.first;<br>        });<br><br>        <span class="hljs-comment">// 4. 寻找符合条件的连通块属性：找到序列中第一个含有 1 个感染结点的连通块祖宗结点编号 idx</span><br>        <span class="hljs-type">int</span> idx = <span class="hljs-number">-1</span>;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; v.<span class="hljs-built_in">size</span>(); i++) {<br>            <span class="hljs-keyword">if</span> (v[i].first == <span class="hljs-number">1</span>) {<br>                idx = i;<br>                <span class="hljs-keyword">break</span>;<br>            }<br>        }<br><br>        <span class="hljs-comment">// 5. 返回答案：比对感染结点所在的连通块属性与目标连通块属性</span><br>        <span class="hljs-keyword">if</span> (idx == <span class="hljs-number">-1</span>) {<br>            <span class="hljs-comment">// 特判没有连通块只含有 1 个感染结点的情况</span><br>            <span class="hljs-keyword">return</span> *<span class="hljs-built_in">min_element</span>(initial.<span class="hljs-built_in">begin</span>(), initial.<span class="hljs-built_in">end</span>());<br>        }<br>        <br>        <span class="hljs-type">int</span> res = n + <span class="hljs-number">10</span>;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; x: initial) {<br>            <span class="hljs-type">int</span> px = <span class="hljs-built_in">Find</span>(x);<br>            <span class="hljs-keyword">if</span> (ha[px].first == v[idx].first &amp;&amp; ha[px].second == v[idx].second) {<br>                res = <span class="hljs-built_in">min</span>(res, x);<br>            }<br>        }<br><br>        <span class="hljs-keyword">return</span> res;<br>    }<br>};<br></code></pre></td></tr></tbody></table></figure><h2 id="【并查集】账户合并">【并查集】账户合并</h2><p><a href="https://leetcode.cn/problems/accounts-merge/">https://leetcode.cn/problems/accounts-merge/</a></p><blockquote><p>题意：给定 n 个账户，每一个账户含有一个用户名和最多 m 个绑定的邮箱。由于一个用户可能注册多个账户，因此我们需要对所有的账户进行合并使得一个用户对应一个账户。合并的规则是将所有「含有相同邮箱的账户」视作同一个用户注册的账户。返回合并后的账户列表。</p><p>思路：这道题的需求很显然，我们需要合并含有相同邮箱的账户。显然有一个暴力的做法，我们直接枚举每一个账户中所有的邮箱，接着枚举剩余账户中的邮箱进行匹配，匹配上就进行合并，但这样做显然会造成大量的冗余匹配和冗余合并，我们不妨将这两个过程进行拆分。我们需要解决两个问题：</p><ul><li>哪些账户需要合并？很容易想到并查集这样的数据结构。我们使用哈希表存储每一个邮箱的账户编号，最后进行集合合并即可维护好每一个账号归属的集合编号。<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(nm)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">nm</span><span class="mclose">)</span></span></span></span></li><li>如何合并指定账户？对于上述维护好的集合编号，我们需要合并所有含有相同“祖先”的账户。排序去重或使用有序列表均可实现。<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><figcaption><span>[]</span></figcaption><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-keyword">struct</span> <span class="hljs-title class_">dsu</span> {<br>    <span class="hljs-type">int</span> n;<br>    std::vector&lt;<span class="hljs-type">int</span>&gt; p;<br>    <span class="hljs-built_in">dsu</span>(<span class="hljs-type">int</span> _n) { n = _n; p.<span class="hljs-built_in">resize</span>(n + <span class="hljs-number">1</span>); <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) p[i] = i; }<br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">find</span><span class="hljs-params">(<span class="hljs-type">int</span> x)</span> </span>{ <span class="hljs-keyword">return</span> (p[x] == x ? p[x] : p[x] = <span class="hljs-built_in">find</span>(p[x])); }<br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">merge</span><span class="hljs-params">(<span class="hljs-type">int</span> a, <span class="hljs-type">int</span> b)</span> </span>{ p[<span class="hljs-built_in">find</span>(a)] = <span class="hljs-built_in">find</span>(b); }<br>    <span class="hljs-function"><span class="hljs-type">bool</span> <span class="hljs-title">query</span><span class="hljs-params">(<span class="hljs-type">int</span> a, <span class="hljs-type">int</span> b)</span> </span>{ <span class="hljs-keyword">return</span> <span class="hljs-built_in">find</span>(a) == <span class="hljs-built_in">find</span>(b); }<br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">block</span><span class="hljs-params">()</span> </span>{ <span class="hljs-type">int</span> ret = <span class="hljs-number">0</span>; <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) ret += p[i] == i; <span class="hljs-keyword">return</span> ret; }<br>};<br><br><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span> {<br><span class="hljs-keyword">public</span>:<br>    vector&lt;vector&lt;string&gt;&gt; <span class="hljs-built_in">accountsMerge</span>(vector&lt;vector&lt;string&gt;&gt;&amp; accounts) {<br>        <span class="hljs-comment">// 维护每一个子账户归属的集合</span><br>        <span class="hljs-type">int</span> n = accounts.<span class="hljs-built_in">size</span>();<br>        unordered_map&lt;string, vector&lt;<span class="hljs-type">int</span>&gt;&gt; hash;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">1</span>; j &lt; accounts[i - <span class="hljs-number">1</span>].<span class="hljs-built_in">size</span>(); j++) {<br>                hash[accounts[i - <span class="hljs-number">1</span>][j]].<span class="hljs-built_in">push_back</span>(i);<br>            }<br>        }        <br>        <span class="hljs-function">dsu <span class="hljs-title">d</span><span class="hljs-params">(n)</span></span>;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span>&amp; it: hash) {<br>            vector&lt;<span class="hljs-type">int</span>&gt; v = it.second;<br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt; v.<span class="hljs-built_in">size</span>(); i++) {<br>                d.<span class="hljs-built_in">merge</span>(v[i - <span class="hljs-number">1</span>], v[i]);<br>            }<br>        }<br><br>        <span class="hljs-comment">// 按照子账户归属的集合合并出最终的账户</span><br>        unordered_set&lt;<span class="hljs-type">int</span>&gt; fa;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>            fa.<span class="hljs-built_in">insert</span>(d.<span class="hljs-built_in">find</span>(i));<br>        }<br>        vector&lt;vector&lt;string&gt;&gt; res;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span> p: fa) {<br>            set&lt;string&gt; se;<br>            vector&lt;string&gt; ans;<br>            <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">1</span>; i &lt;= n; i++) {<br>                <span class="hljs-keyword">if</span> (d.<span class="hljs-built_in">find</span>(i) == p) {<br>                    <span class="hljs-keyword">if</span> (ans.<span class="hljs-built_in">empty</span>()) {<br>                        ans.<span class="hljs-built_in">push_back</span>(accounts[i - <span class="hljs-number">1</span>][<span class="hljs-number">0</span>]);<br>                    }<br>                    <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> j = <span class="hljs-number">1</span>; j &lt; accounts[i - <span class="hljs-number">1</span>].<span class="hljs-built_in">size</span>(); j++) {<br>                        se.<span class="hljs-built_in">insert</span>(accounts[i - <span class="hljs-number">1</span>][j]);<br>                    }<br>                }<br>            }<br>            <span class="hljs-keyword">for</span> (<span class="hljs-keyword">auto</span> mail: se) {<br>                ans.<span class="hljs-built_in">push_back</span>(mail);<br>            }<br>            res.<span class="hljs-built_in">push_back</span>(ans);<br>        }<br><br>        <span class="hljs-keyword">return</span> res;<br>    }<br>};<br></code></pre></td></tr></tbody></table></figure><figure class="highlight python"><figcaption><span>[]</span></figcaption><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">class</span> <span class="hljs-title class_">dsu</span>:<br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">__init__</span>(<span class="hljs-params">self, n: <span class="hljs-built_in">int</span></span>) -&gt; <span class="hljs-literal">None</span>:<br>        <span class="hljs-variable language_">self</span>.n = n<br>        <span class="hljs-variable language_">self</span>.p = [i <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(n + <span class="hljs-number">1</span>)]<br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">find</span>(<span class="hljs-params">self, x: <span class="hljs-built_in">int</span></span>) -&gt; <span class="hljs-built_in">int</span>:<br>        <span class="hljs-keyword">if</span> <span class="hljs-variable language_">self</span>.p[x] != x: <span class="hljs-variable language_">self</span>.p[x] = <span class="hljs-variable language_">self</span>.find(<span class="hljs-variable language_">self</span>.p[x])<br>        <span class="hljs-keyword">return</span> <span class="hljs-variable language_">self</span>.p[x]<br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">merge</span>(<span class="hljs-params">self, a: <span class="hljs-built_in">int</span>, b: <span class="hljs-built_in">int</span></span>) -&gt; <span class="hljs-literal">None</span>:<br>        <span class="hljs-variable language_">self</span>.p[<span class="hljs-variable language_">self</span>.find(a)] = <span class="hljs-variable language_">self</span>.find(b)<br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">query</span>(<span class="hljs-params">self, a: <span class="hljs-built_in">int</span>, b: <span class="hljs-built_in">int</span></span>) -&gt; <span class="hljs-built_in">bool</span>:<br>        <span class="hljs-keyword">return</span> <span class="hljs-variable language_">self</span>.find(a) == <span class="hljs-variable language_">self</span>.find(b)<br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">block</span>(<span class="hljs-params">self</span>) -&gt; <span class="hljs-built_in">int</span>:<br>        <span class="hljs-keyword">return</span> <span class="hljs-built_in">sum</span>([<span class="hljs-number">1</span> <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">1</span>, <span class="hljs-variable language_">self</span>.n + <span class="hljs-number">1</span>) <span class="hljs-keyword">if</span> <span class="hljs-variable language_">self</span>.p[i] == i])<br><br><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span>:<br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">accountsMerge</span>(<span class="hljs-params">self, accounts: <span class="hljs-type">List</span>[<span class="hljs-type">List</span>[<span class="hljs-built_in">str</span>]]</span>) -&gt; <span class="hljs-type">List</span>[<span class="hljs-type">List</span>[<span class="hljs-built_in">str</span>]]:<br>        <span class="hljs-keyword">from</span> collections <span class="hljs-keyword">import</span> defaultdict<br>        <br>        n = <span class="hljs-built_in">len</span>(accounts)<br>        <span class="hljs-built_in">hash</span> = defaultdict(<span class="hljs-built_in">list</span>)<br>        <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">1</span>, n + <span class="hljs-number">1</span>):<br>            <span class="hljs-keyword">for</span> j <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">1</span>, <span class="hljs-built_in">len</span>(accounts[i - <span class="hljs-number">1</span>])):<br>                <span class="hljs-built_in">hash</span>[accounts[i - <span class="hljs-number">1</span>][j]].append(i)<br>        <br>        d = dsu(n)<br>        <span class="hljs-keyword">for</span> _, ids <span class="hljs-keyword">in</span> <span class="hljs-built_in">hash</span>.items():<br>            <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">1</span>, <span class="hljs-built_in">len</span>(ids)):<br>                d.merge(ids[i - <span class="hljs-number">1</span>], ids[i])<br>        <br>        fa = <span class="hljs-built_in">set</span>()<br>        <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">1</span>, n + <span class="hljs-number">1</span>):<br>            fa.add(d.find(i))<br>        <br>        res = []<br>        <span class="hljs-keyword">for</span> p <span class="hljs-keyword">in</span> fa:<br>            ans = []<br>            se = <span class="hljs-built_in">set</span>()<br>            <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">1</span>, n + <span class="hljs-number">1</span>):<br>                <span class="hljs-keyword">if</span> d.find(i) == p:<br>                    <span class="hljs-keyword">if</span> <span class="hljs-built_in">len</span>(ans) == <span class="hljs-number">0</span>:<br>                        ans.append(accounts[i - <span class="hljs-number">1</span>][<span class="hljs-number">0</span>])<br>                    <span class="hljs-keyword">for</span> j <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(<span class="hljs-number">1</span>, <span class="hljs-built_in">len</span>(accounts[i - <span class="hljs-number">1</span>])):<br>                        se.add(accounts[i - <span class="hljs-number">1</span>][j])<br>            ans += <span class="hljs-built_in">sorted</span>(se)<br>            res.append(ans)<br>        <br>        <span class="hljs-keyword">return</span> res<br></code></pre></td></tr></tbody></table></figure><h2 id="【树状数组】将元素分配到两个数组中-II">【树状数组】将元素分配到两个数组中 II</h2><p><a href="https://leetcode.cn/problems/distribute-elements-into-two-arrays-ii/description/">https://leetcode.cn/problems/distribute-elements-into-two-arrays-ii/description/</a></p><blockquote><p>题意：给定 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">n</span></span></span></span> 个数，现在需要将这些数按照某种规则分配到两个数组 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> 中。初始化分配 <code>nums[0]</code> 到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 中，<code>nums[1]</code> 到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> 中，接下来对于剩余的每个元素 <code>nums[i]</code>，分配取决于 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal">A</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> 中比当前元素 <code>nums[i]</code> 大的个数，最终返回两个分配好的数组</p><p>思路：首先每一个元素都需要进行枚举，那么本题需要考虑的就是如何在 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>log</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\log</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span></span></span></span> 时间复杂度内统计出两数组中比当前元素大的元素个数。针对 C++ 和 Python 分别讨论</p><ul><li>C++<ul><li>法一：<code>std::multiset&lt;int&gt;</code>。可惜不行，因为统计比当前元素大的个数时，<code>s.rbegin() - s.upper_bound(nums[i])</code> 是不合法的，因为 <code>std::multiset&lt;int&gt;</code> 的迭代器不是基于指针的，因此无法直接进行加减来计算地址差，遂作罢</li><li>法二：树状数组。很容易想到利用前缀和统计比当前数大的数字个数，但是由于此处需要对前缀和进行单点修改，因此时间复杂度肯定会寄。有什么数据结构支持「单点修改，区间更新」呢？我们引入树状数组。我们将数组元素哈希到 <code>[1, len(set(nums))]</code> 区间，定义哈希后的当前元素 <code>nums[i]</code> 为 <code>x</code>，对于当前哈希后的 <code>x</code> 而言想要知道两个数组中有多少数比当前数严格大，只需要计算前缀和数组 <code>arr</code> 中 <code>arr[n] - arr[x]</code> 的结果即可</li></ul></li><li>Python<ul><li><code>SortedList</code>。python 有一个 <code>sortedcontainers</code> 包其中有 <code>SortedList</code> 模块，可以实现 <code>std::multiset&lt;int&gt;</code> 所有 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>log</mi><mo>⁡</mo></mrow><annotation encoding="application/x-tex">\log</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span></span></span></span> 操作并且可以进行随机下标访问，于是就可以进行下标访问 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 计算比当前数大的元素个数</li><li>题外话。LeetCode 可以进行第三方库导入的操作，某些比赛不允许，需要手搓 <code>SortedList</code> 模块，<s>当然可以用树状数组 or 线段树解决</s>，板子链接：<a href="https://blog.dwj601.cn/Algorithm/a_template/#SortedList">https://blog.dwj601.cn/Algorithm/a_template/#SortedList</a></li></ul></li></ul><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><figcaption><span>[]</span></figcaption><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-keyword">template</span>&lt;<span class="hljs-keyword">class</span> <span class="hljs-title class_">T</span>&gt;<br><span class="hljs-keyword">class</span> <span class="hljs-title class_">BinaryIndexedTree</span> {<br><span class="hljs-keyword">private</span>:<br>    std::vector&lt;T&gt; _arr;<br>    <span class="hljs-type">int</span> _n;<br><br>    <span class="hljs-function"><span class="hljs-type">int</span> <span class="hljs-title">lowbit</span><span class="hljs-params">(<span class="hljs-type">int</span> x)</span> </span>{ <span class="hljs-keyword">return</span> x &amp; (-x); }<br><br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-built_in">BinaryIndexedTree</span>(<span class="hljs-type">int</span> n) :_n(n) {<br>        _arr.<span class="hljs-built_in">resize</span>(_n + <span class="hljs-number">1</span>, <span class="hljs-number">0</span>);<br>    }<br><br>    <span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">add</span><span class="hljs-params">(<span class="hljs-type">int</span> pos, T x)</span> </span>{<br>        <span class="hljs-keyword">while</span> (pos &lt;= _n) {<br>            _arr[pos] += x;<br>            pos += <span class="hljs-built_in">lowbit</span>(pos);<br>        }<br>    }<br><br>    <span class="hljs-function">T <span class="hljs-title">sum</span><span class="hljs-params">(<span class="hljs-type">int</span> pos)</span> </span>{<br>        T ret = <span class="hljs-number">0</span>;<br>        <span class="hljs-keyword">while</span> (pos) {<br>            ret += _arr[pos];<br>            pos -= <span class="hljs-built_in">lowbit</span>(pos);<br>        }<br>        <span class="hljs-keyword">return</span> ret;<br>    }<br>};<br><br><br><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span> {<br><span class="hljs-keyword">public</span>:<br>    <span class="hljs-function">vector&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">resultArray</span><span class="hljs-params">(vector&lt;<span class="hljs-type">int</span>&gt;&amp; nums)</span> </span>{<br>        vector&lt;<span class="hljs-type">int</span>&gt; copy = nums;<br>        <span class="hljs-built_in">sort</span>(copy.<span class="hljs-built_in">begin</span>(), copy.<span class="hljs-built_in">end</span>());<br>        copy.<span class="hljs-built_in">erase</span>(<span class="hljs-built_in">unique</span>(copy.<span class="hljs-built_in">begin</span>(), copy.<span class="hljs-built_in">end</span>()), copy.<span class="hljs-built_in">end</span>());<br><br>        <span class="hljs-type">int</span> n = copy.<span class="hljs-built_in">size</span>(), cnt = <span class="hljs-number">1</span>;<br>        unordered_map&lt;<span class="hljs-type">int</span>, <span class="hljs-type">int</span>&gt; a;<br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) {<br>            a[copy[i]] = cnt++;<br>        }<br><br>        vector&lt;<span class="hljs-type">int</span>&gt; v1, v2;<br>        v<span class="hljs-number">1.</span><span class="hljs-built_in">push_back</span>(nums[<span class="hljs-number">0</span>]);<br>        v<span class="hljs-number">2.</span><span class="hljs-built_in">push_back</span>(nums[<span class="hljs-number">1</span>]);<br><br>        <span class="hljs-function">BinaryIndexedTree&lt;<span class="hljs-type">int</span>&gt; <span class="hljs-title">t1</span><span class="hljs-params">(n)</span>, <span class="hljs-title">t2</span><span class="hljs-params">(n)</span></span>;<br>        t<span class="hljs-number">1.</span><span class="hljs-built_in">add</span>(a[nums[<span class="hljs-number">0</span>]], <span class="hljs-number">1</span>);<br>        t<span class="hljs-number">2.</span><span class="hljs-built_in">add</span>(a[nums[<span class="hljs-number">1</span>]], <span class="hljs-number">1</span>);<br><br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">2</span>; i &lt; nums.<span class="hljs-built_in">size</span>(); i++) {<br>            <span class="hljs-type">int</span> d1 = t<span class="hljs-number">1.</span><span class="hljs-built_in">sum</span>(n) - t<span class="hljs-number">1.</span><span class="hljs-built_in">sum</span>(a[nums[i]]);<br>            <span class="hljs-type">int</span> d2 = t<span class="hljs-number">2.</span><span class="hljs-built_in">sum</span>(n) - t<span class="hljs-number">2.</span><span class="hljs-built_in">sum</span>(a[nums[i]]);<br><br>            <span class="hljs-keyword">if</span> (d1 &gt; d2) {<br>                v<span class="hljs-number">1.</span><span class="hljs-built_in">push_back</span>(nums[i]);<br>                t<span class="hljs-number">1.</span><span class="hljs-built_in">add</span>(a[nums[i]], <span class="hljs-number">1</span>);                <br>            } <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (d1 &lt; d2) {<br>                v<span class="hljs-number">2.</span><span class="hljs-built_in">push_back</span>(nums[i]);<br>                t<span class="hljs-number">2.</span><span class="hljs-built_in">add</span>(a[nums[i]], <span class="hljs-number">1</span>);<br>            } <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (d1 == d2 &amp;&amp; v<span class="hljs-number">1.</span><span class="hljs-built_in">size</span>() &lt; v<span class="hljs-number">2.</span><span class="hljs-built_in">size</span>()) {<br>                v<span class="hljs-number">1.</span><span class="hljs-built_in">push_back</span>(nums[i]);<br>                t<span class="hljs-number">1.</span><span class="hljs-built_in">add</span>(a[nums[i]], <span class="hljs-number">1</span>);<br>            } <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span> (d1 == d2 &amp;&amp; v<span class="hljs-number">1.</span><span class="hljs-built_in">size</span>() &gt; v<span class="hljs-number">2.</span><span class="hljs-built_in">size</span>()) {<br>                v<span class="hljs-number">2.</span><span class="hljs-built_in">push_back</span>(nums[i]);<br>                t<span class="hljs-number">2.</span><span class="hljs-built_in">add</span>(a[nums[i]], <span class="hljs-number">1</span>);<br>            } <span class="hljs-keyword">else</span> {<br>                v<span class="hljs-number">1.</span><span class="hljs-built_in">push_back</span>(nums[i]);<br>                t<span class="hljs-number">1.</span><span class="hljs-built_in">add</span>(a[nums[i]], <span class="hljs-number">1</span>);<br>            }<br>        }<br><br>        <span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> x: v2) {<br>            v<span class="hljs-number">1.</span><span class="hljs-built_in">push_back</span>(x);<br>        }<br><br>        <span class="hljs-keyword">return</span> v1;<br>    }<br>};<br></code></pre></td></tr></tbody></table></figure><figure class="highlight python"><figcaption><span>[]</span></figcaption><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">class</span> <span class="hljs-title class_">BinaryIndexedTree</span>:<br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">__init__</span>(<span class="hljs-params">self, n: <span class="hljs-built_in">int</span></span>):<br>        <span class="hljs-variable language_">self</span>._n = n<br>        <span class="hljs-variable language_">self</span>._arr = [<span class="hljs-number">0</span>] * (n + <span class="hljs-number">1</span>)<br><br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">_lowbit</span>(<span class="hljs-params">self, x: <span class="hljs-built_in">int</span></span>) -&gt; <span class="hljs-built_in">int</span>:<br>        <span class="hljs-keyword">return</span> x &amp; (-x)<br><br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">add</span>(<span class="hljs-params">self, pos: <span class="hljs-built_in">int</span>, x: <span class="hljs-built_in">int</span></span>) -&gt; <span class="hljs-literal">None</span>:<br>        <span class="hljs-keyword">while</span> pos &lt;= <span class="hljs-variable language_">self</span>._n:<br>            <span class="hljs-variable language_">self</span>._arr[pos] += x<br>            pos += <span class="hljs-variable language_">self</span>._lowbit(pos)<br><br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">sum</span>(<span class="hljs-params">self, pos: <span class="hljs-built_in">int</span></span>) -&gt; <span class="hljs-built_in">int</span>:<br>        ret = <span class="hljs-number">0</span><br>        <span class="hljs-keyword">while</span> pos:<br>            ret += <span class="hljs-variable language_">self</span>._arr[pos]<br>            pos -= <span class="hljs-variable language_">self</span>._lowbit(pos)<br>        <span class="hljs-keyword">return</span> ret<br><br><br><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span>:<br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">resultArray</span>(<span class="hljs-params">self, nums: <span class="hljs-type">List</span>[<span class="hljs-built_in">int</span>]</span>) -&gt; <span class="hljs-type">List</span>[<span class="hljs-built_in">int</span>]:<br>        copy = <span class="hljs-built_in">sorted</span>(<span class="hljs-built_in">set</span>(nums))<br>        <br>        n, cnt, a = <span class="hljs-built_in">len</span>(copy), <span class="hljs-number">1</span>, {}<br>        <span class="hljs-keyword">for</span> x <span class="hljs-keyword">in</span> copy:<br>            a[x] = cnt<br>            cnt += <span class="hljs-number">1</span><br><br>        v1, v2 = [nums[<span class="hljs-number">0</span>]], [nums[<span class="hljs-number">1</span>]]<br>        t1, t2 = BinaryIndexedTree(n), BinaryIndexedTree(n)<br>        t1.add(a[nums[<span class="hljs-number">0</span>]], <span class="hljs-number">1</span>)<br>        t2.add(a[nums[<span class="hljs-number">1</span>]], <span class="hljs-number">1</span>)<br>        <br>        <span class="hljs-keyword">for</span> x <span class="hljs-keyword">in</span> nums[<span class="hljs-number">2</span>:]:<br>            d1, d2 = t1.<span class="hljs-built_in">sum</span>(n) - t1.<span class="hljs-built_in">sum</span>(a[x]), t2.<span class="hljs-built_in">sum</span>(n) - t2.<span class="hljs-built_in">sum</span>(a[x])<br>            <br>            <span class="hljs-keyword">if</span> d1 &gt; d2:<br>                v1.append(x)<br>                t1.add(a[x], <span class="hljs-number">1</span>)<br>            <span class="hljs-keyword">elif</span> d1 &lt; d2:<br>                v2.append(x)<br>                t2.add(a[x], <span class="hljs-number">1</span>)<br>            <span class="hljs-keyword">elif</span> d1 == d2 <span class="hljs-keyword">and</span> <span class="hljs-built_in">len</span>(v1) &lt; <span class="hljs-built_in">len</span>(v2):<br>                v1.append(x)<br>                t1.add(a[x], <span class="hljs-number">1</span>)<br>            <span class="hljs-keyword">elif</span> d1 == d2 <span class="hljs-keyword">and</span> <span class="hljs-built_in">len</span>(v1) &gt; <span class="hljs-built_in">len</span>(v2):<br>                v2.append(x)<br>                t2.add(a[x], <span class="hljs-number">1</span>)<br>            <span class="hljs-keyword">else</span>:<br>                v1.append(x)<br>                t1.add(a[x], <span class="hljs-number">1</span>)<br>        <br>        <span class="hljs-keyword">return</span> v1 + v2<br></code></pre></td></tr></tbody></table></figure><figure class="highlight python"><figcaption><span>[]</span></figcaption><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">class</span> <span class="hljs-title class_">Solution</span>:<br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">resultArray</span>(<span class="hljs-params">self, nums: <span class="hljs-type">List</span>[<span class="hljs-built_in">int</span>]</span>) -&gt; <span class="hljs-type">List</span>[<span class="hljs-built_in">int</span>]:<br>        <span class="hljs-keyword">from</span> sortedcontainers <span class="hljs-keyword">import</span> SortedList<br>        <br>        v1, v2 = copy.deepcopy(nums[:<span class="hljs-number">1</span>]), copy.deepcopy(nums[<span class="hljs-number">1</span>:<span class="hljs-number">2</span>])<br>        s1, s2 = SortedList(v1), SortedList(v2)<br><br>        <span class="hljs-keyword">for</span> x <span class="hljs-keyword">in</span> nums[<span class="hljs-number">2</span>:]:<br>            d1, d2 = <span class="hljs-built_in">len</span>(v1) - s1.bisect_right(x), <span class="hljs-built_in">len</span>(v2) - s2.bisect_right(x)<br>            <br>            <span class="hljs-keyword">if</span> d1 &gt; d2:<br>                v1.append(x)<br>                s1.add(x)<br>            <span class="hljs-keyword">elif</span> d1 &lt; d2:<br>                v2.append(x)<br>                s2.add(x)<br>            <span class="hljs-keyword">elif</span> d1 == d2 <span class="hljs-keyword">and</span> <span class="hljs-built_in">len</span>(v1) &lt; <span class="hljs-built_in">len</span>(v2):<br>                v1.append(x)<br>                s1.add(x)<br>            <span class="hljs-keyword">elif</span> d1 == d2 <span class="hljs-keyword">and</span> <span class="hljs-built_in">len</span>(v1) &gt; <span class="hljs-built_in">len</span>(v2):<br>                v2.append(x)<br>                s2.add(x)<br>            <span class="hljs-keyword">else</span>:<br>                v1.append(x)<br>                s1.add(x)<br>        <br>        <span class="hljs-keyword">return</span> v1 + v2<br></code></pre></td></tr></tbody></table></figure><h2 id="【线段树-二分】以组为单位订音乐会的门票-fire">【线段树/二分】以组为单位订音乐会的门票 <span class="github-emoji"><span>🔥</span><img src="https://github.githubassets.com/images/icons/emoji/unicode/1f525.png?v8" aria-hidden="true" onerror="this.parent.classList.add('github-emoji-fallback')"></span></h2><p><a href="https://leetcode.cn/problems/booking-concert-tickets-in-groups/">https://leetcode.cn/problems/booking-concert-tickets-in-groups/</a></p><blockquote><p>题意：给定一个长为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>n</mi><mo>≤</mo><mn>5</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mn>4</mn></msup></mrow><annotation encoding="application/x-tex">n\le 5 \times 10^4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719em;vertical-align:-0.136em;"></span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">5</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span> 且初始值均为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">0</span></span></span></span> 的数组 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">a</span></span></span></span>，数组中的每个元素最多增加到 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">m</span></span></span></span>。现在需要以这个数组为基础进行 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>q</mi><mo>≤</mo><mn>5</mn><mo>×</mo><mn>1</mn><msup><mn>0</mn><mn>4</mn></msup></mrow><annotation encoding="application/x-tex">q\le 5 \times 10^4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8304em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">5</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.8141em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span> 次询问，每次询问是以下两者之一：</p><ol><li>给定一个 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>i</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">lim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">im</span></span></span></span>，找到最小的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mi>l</mi><mi>i</mi><mi>m</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">i \in [0,lim]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6986em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">im</span><span class="mclose">]</span></span></span></span> 使得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>m</mi><mo>−</mo><msub><mi>a</mi><mi>i</mi></msub><mo>≥</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">m - a_i \ge k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.786em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span></li><li>给定一个 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> 和 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>i</mi><mi>m</mi></mrow><annotation encoding="application/x-tex">lim</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">im</span></span></span></span>，找到最小的 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>i</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo separator="true">,</mo><mi>l</mi><mi>i</mi><mi>m</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">i \in [0,lim]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6986em;vertical-align:-0.0391em;"></span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">im</span><span class="mclose">]</span></span></span></span> 使得 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle scriptlevel="0" displaystyle="true"><mi>m</mi><mo>×</mo><mo stretchy="false">(</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>0</mn></mrow><mi>i</mi></munderover><msub><mi>a</mi><mi>j</mi></msub><mo>≥</mo><mi>k</mi></mstyle></mrow><annotation encoding="application/x-tex">\displaystyle m\times (i+1) - \sum_{j=0}^i a_j \ge k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord mathnormal">m</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">i</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:3.2254em;vertical-align:-1.4138em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8117em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4138em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span></li></ol><p>思路一：<strong>暴力</strong>。对于询问 1，我们直接顺序遍历 a 数组直到找到第一个符合条件的即可；对于询问 2，同样直接顺序遍历 a 数组直到找到第一个符合条件的即可。时间复杂度为 <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>q</mi><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(qn)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span>。</p><p>思路二：<strong>线段树上二分</strong>。</p></blockquote><p>暴力代码：</p><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">class</span> <span class="hljs-title class_">BookMyShow</span>:<br><br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">__init__</span>(<span class="hljs-params">self, n: <span class="hljs-built_in">int</span>, m: <span class="hljs-built_in">int</span></span>):<br>        <span class="hljs-variable language_">self</span>.a = [<span class="hljs-number">0</span>] * n  <span class="hljs-comment"># a[i] 表示第 i 行已入座的人数</span><br>        <span class="hljs-variable language_">self</span>.n = n<br>        <span class="hljs-variable language_">self</span>.m = m<br><br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">gather</span>(<span class="hljs-params">self, k: <span class="hljs-built_in">int</span>, lim: <span class="hljs-built_in">int</span></span>) -&gt; <span class="hljs-type">List</span>[<span class="hljs-built_in">int</span>]:<br>        <span class="hljs-comment"># 在 [0, lim] 行中找到第一个可以容纳 k 人的行</span><br>        <span class="hljs-keyword">for</span> i <span class="hljs-keyword">in</span> <span class="hljs-built_in">range</span>(lim + <span class="hljs-number">1</span>):<br>            <span class="hljs-keyword">if</span> <span class="hljs-variable language_">self</span>.m - <span class="hljs-variable language_">self</span>.a[i] &gt;= k:<br>                l, r = i, <span class="hljs-variable language_">self</span>.a[i]<br>                <span class="hljs-variable language_">self</span>.a[i] += k<br>                <span class="hljs-keyword">return</span> [l, r]<br>        <span class="hljs-keyword">return</span> []<br><br>    <span class="hljs-keyword">def</span> <span class="hljs-title function_">scatter</span>(<span class="hljs-params">self, k: <span class="hljs-built_in">int</span>, lim: <span class="hljs-built_in">int</span></span>) -&gt; <span class="hljs-built_in">bool</span>:<br>        <span class="hljs-comment"># 在 [0, lim] 行中找到最小的 i 使得 [0, i] 行可以容纳 k 人</span><br>        <span class="hljs-keyword">if</span> <span class="hljs-variable language_">self</span>.m * (lim + <span class="hljs-number">1</span>) - <span class="hljs-built_in">sum</span>(<span class="hljs-variable language_">self</span>.a[:lim+<span class="hljs-number">1</span>]) &lt; k:<br>            <span class="hljs-keyword">return</span> <span class="hljs-literal">False</span><br><br>        i = <span class="hljs-number">0</span><br>        <span class="hljs-keyword">while</span> k &gt; <span class="hljs-number">0</span>:<br>            <span class="hljs-keyword">if</span> <span class="hljs-variable language_">self</span>.m - <span class="hljs-variable language_">self</span>.a[i] &gt;= k:<br>                <span class="hljs-variable language_">self</span>.a[i] += k<br>                k = <span class="hljs-number">0</span><br>            <span class="hljs-keyword">else</span>:<br>                k -= <span class="hljs-variable language_">self</span>.m - <span class="hljs-variable language_">self</span>.a[i]<br>                <span class="hljs-variable language_">self</span>.a[i] = <span class="hljs-variable language_">self</span>.m<br>            i += <span class="hljs-number">1</span><br>        <span class="hljs-keyword">return</span> <span class="hljs-literal">True</span><br></code></pre></td></tr></tbody></table></figure><p>线段树上二分代码：</p><figure class="highlight python"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><code class="hljs python"><br></code></pre></td></tr></tbody></table></figure>]]></content>
+    <content type="html"><![CDATA[<h3 id="分治">分治</h3><p>将大问题转化为等价小问题进行求解。</p><h3 id="【分治】随机排列">【分治】随机排列</h3><p><a href="https://www.acwing.com/problem/content/5469/">https://www.acwing.com/problem/content/5469/</a></p><blockquote><p>题意：给定一个 n 个数的全排列序列，并将其进行一定的对换，问是对换了 3n 次还是 7n+1 次</p><p>思路：可以发现对于两种情况，就对应对换次数的奇偶性。当 n 为奇数：3n 为奇数，7n+1 为偶数；当 n 为偶数：3n 为偶数，7n+1 为奇数。故我们只需要判断序列的逆序数即可。为了求解逆序数，我们可以采用归并排序的 combine 过程进行统计即可</p><p>时间复杂度：<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n \log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathnormal">n</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">n</span><span class="mclose">)</span></span></span></span></p></blockquote><figure class="highlight cpp"><table><tbody><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-keyword">include</span> <span class="hljs-string">&lt;bits/stdc++.h&gt;</span></span><br><span class="hljs-meta">#<span class="hljs-keyword">define</span> int long long</span><br><span class="hljs-keyword">using</span> <span class="hljs-keyword">namespace</span> std;<br><br><span class="hljs-type">const</span> <span class="hljs-type">int</span> N = <span class="hljs-number">1000010</span>;<br><br><span class="hljs-type">int</span> n, a[N], t[N];<br><span class="hljs-type">int</span> cnt; <span class="hljs-comment">// 逆序数 </span><br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">MergeSort</span><span class="hljs-params">(<span class="hljs-type">int</span> l, <span class="hljs-type">int</span> r)</span> </span>{<br><span class="hljs-keyword">if</span> (l &gt;= r) <span class="hljs-keyword">return</span>;<br><br><span class="hljs-type">int</span> mid = (l + r) &gt;&gt; <span class="hljs-number">1</span>;<br><br><span class="hljs-built_in">MergeSort</span>(l, mid), <span class="hljs-built_in">MergeSort</span>(mid + <span class="hljs-number">1</span>, r);<br><br><span class="hljs-type">int</span> i = l, j = mid + <span class="hljs-number">1</span>, idx = <span class="hljs-number">0</span>;<br><br><span class="hljs-keyword">while</span> (i &lt;= mid &amp;&amp; j &lt;= r) {<br><span class="hljs-keyword">if</span> (a[i] &lt; a[j]) t[idx++] = a[i++];<br><span class="hljs-keyword">else</span> {<br>cnt += mid - i + <span class="hljs-number">1</span>;<br>t[idx++] = a[j++];<br>}<br>}<br><br><span class="hljs-keyword">while</span> (i &lt;= mid) t[idx++] = a[i++];<br><span class="hljs-keyword">while</span> (j &lt;= r) t[idx++] = a[j++];<br><br><span class="hljs-keyword">for</span> (i = l, idx = <span class="hljs-number">0</span>; i &lt;= r; i++, idx++) a[i] = t[idx];<br>}<br><br><span class="hljs-function"><span class="hljs-type">void</span> <span class="hljs-title">solve</span><span class="hljs-params">()</span> </span>{<br>cin &gt;&gt; n;<br><br><span class="hljs-keyword">for</span> (<span class="hljs-type">int</span> i = <span class="hljs-number">0</span>; i &lt; n; i++) cin &gt;&gt; a[i];<br><br><span class="hljs-built_in">MergeSort</span>(<span class="hljs-number">0</span>, n - <span class="hljs-number">1</span>);<br><br><span class="hljs-type">int</span> res;<br><br><span class="hljs-keyword">if</span> (n % <span class="hljs-number">2</span> == <span class="hljs-number">1</span>) {<br><span class="hljs-keyword">if</span> (cnt % <span class="hljs-number">2</span>) res = <span class="hljs-number">1</span>;<br><span class="hljs-keyword">else</span> res = <span class="hljs-number">2</span>;<br>} <span class="hljs-keyword">else</span> {<br><span class="hljs-keyword">if</span> (cnt % <span class="hljs-number">2</span>) res = <span class="hljs-number">2</span>;<br><span class="hljs-keyword">else</span> res = <span class="hljs-number">1</span>;<br>}<br><br>cout &lt;&lt; res &lt;&lt; <span class="hljs-string">"\n"</span>; <br>}<br><br><span class="hljs-function"><span class="hljs-type">signed</span> <span class="hljs-title">main</span><span class="hljs-params">()</span> </span>{<br>ios::<span class="hljs-built_in">sync_with_stdio</span>(<span class="hljs-literal">false</span>);<br>cin.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>), cout.<span class="hljs-built_in">tie</span>(<span class="hljs-literal">nullptr</span>);<br><span class="hljs-type">int</span> T = <span class="hljs-number">1</span>;<br><span class="hljs-comment">//cin &gt;&gt; T;</span><br><span class="hljs-keyword">while</span> (T--) <span class="hljs-built_in">solve</span>();<br><span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;<br>}<br></code></pre></td></tr></tbody></table></figure>]]></content>
     
     
     <categories>
@@ -1685,33 +1685,33 @@
   
   
   <entry>
-    <title>关于作者</title>
+    <title>心语</title>
     <link href="/"/>
     <url>/</url>
     
-    <content type="html"><![CDATA[<h2 id="教育经历">教育经历</h2><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><menclose notation="left right"><mtable rowspacing="0.16em" columnalign="center center center center center" columnspacing="1em" rowlines="solid none"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext mathvariant="bold">层次</mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext mathvariant="bold">时间</mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext mathvariant="bold">学校</mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext mathvariant="bold">方向</mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext mathvariant="bold">地址</mtext></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>高中</mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>2019.09</mn><mo>−</mo><mn>2022.06</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>江苏省南菁高级中学</mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>物化生</mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>江苏省无锡市江阴市龙定路&nbsp;158&nbsp;号</mtext></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>本科</mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>2022.09</mn><mo>−</mo><mn>2026.06</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>南京师范大学</mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>人工智能</mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>江苏省南京市栖霞区文苑路&nbsp;1&nbsp;号</mtext></mstyle></mtd></mtr></mtable></menclose></mrow><annotation encoding="application/x-tex">\begin{array}{ccccc}\hline\textbf{层次} &amp; \textbf{时间} &amp; \textbf{学校} &amp; \textbf{方向} &amp; \textbf{地址} \\\hline\text{高中} &amp; 2019.09 - 2022.06 &amp; \text{江苏省南菁高级中学} &amp; \text{物化生} &amp; \text{江苏省无锡市江阴市龙定路 158 号} \\\text{本科} &amp; 2022.09 - 2026.06 &amp; \text{南京师范大学} &amp; \text{人工智能} &amp; \text{江苏省南京市栖霞区文苑路 1 号} \\\hline\end{array}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.64em;vertical-align:-1.55em;"></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.09em;"><span style="top:-4.05em;"><span class="pstrut" style="height:4.05em;"></span><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textbf cjk_fallback">层次</span></span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">高中</span></span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">本科</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textbf cjk_fallback">时间</span></span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2019.09</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2022.06</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2022.09</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2026.06</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textbf cjk_fallback">学校</span></span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">江苏省南菁高级中学</span></span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">南京师范大学</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textbf cjk_fallback">方向</span></span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">物化生</span></span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">人工智能</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textbf cjk_fallback">地址</span></span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">江苏省无锡市江阴市龙定路</span><span class="mord">&nbsp;158&nbsp;</span><span class="mord cjk_fallback">号</span></span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">江苏省南京市栖霞区文苑路</span><span class="mord">&nbsp;1&nbsp;</span><span class="mord cjk_fallback">号</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span style="top:-2.5em;"><span class="pstrut" style="height:4.05em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-4.9em;"><span class="pstrut" style="height:4.05em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-6.1em;"><span class="pstrut" style="height:4.05em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span></span></span></p><h2 id="网瘾青年">网瘾青年</h2><a href="https://codeforces.com/profile/dwj601">    <img src="https://img.shields.io/badge/codeforces-pupil_1378-008000?style=for-the-badge" alt="dwj601"></a><a href="https://leetcode.cn/u/dwj601/">    <img src="https://img.shields.io/badge/LeetCode-knight_1925-63bf88?style=for-the-badge" alt="dwj601"></a><h2 id="技术栈">技术栈</h2><p><img src="https://skillicons.dev/icons?i=flask,qt" alt="技术栈"></p><h2 id="编程语言">编程语言</h2><p><img src="https://skillicons.dev/icons?i=c,cpp,python,js,matlab,v" alt="编程语言"></p><h2 id="开发工具">开发工具</h2><p><img src="https://skillicons.dev/icons?i=md,git,clion,pycharm,webstorm,vscode" alt="开发工具"></p>]]></content>
+    <content type="html"><![CDATA[<div class="markdown-body"><div class="note note-light">            <p>每当独自听着歌，或者看到一些人、一些场景时，脑海中总会有小人在低语，有时倒也能让自己受到启发。但很多想法转瞬即逝，故借此处将一些内容及时记录下来，权当留念。</p>          </div></div>]]></content>
     
   </entry>
   
   
   
   <entry>
-    <title>心语</title>
+    <title></title>
     <link href="/"/>
     <url>/</url>
     
-    <content type="html"><![CDATA[<div class="markdown-body"><div class="note note-light">            <p>每当独自听着歌，或者看到一些人、一些场景时，脑海中总会有小人在低语，有时倒也能让自己受到启发。但很多想法转瞬即逝，故借此处将一些内容及时记录下来，权当留念。</p>          </div></div>]]></content>
+    <content type="html"><![CDATA[!(function () {var start = new Date("2024/02/18 00:00:00");function update() {var now = new Date();now.setTime(now.getTime() + 250);days = (now - start) / 1000 / 60 / 60 / 24;dnum = Math.floor(days);        hours = (now - start) / 1000 / 60 / 60 - 24 * dnum;hnum = Math.floor(hours);        if (String(hnum).length === 1) hnum = "0" + hnum;        minutes = (now - start) / 1000 / 60 - 24 * 60 * dnum - 60 * hnum;mnum = Math.floor(minutes);        if (String(mnum).length === 1) mnum = "0" + mnum;seconds = (now - start) / 1000 - 24 * 60 * 60 * dnum - 60 * 60 * hnum - 60 * mnum;snum = Math.round(seconds);if (String(snum).length === 1) snum = "0" + snum;document.getElementById("timeDate").innerHTML = "本站已安全运行&nbsp" + dnum + "&nbsp天";document.getElementById("times").innerHTML = hnum + "&nbsp小时&nbsp" + mnum + "&nbsp分&nbsp" + snum + "&nbsp秒";}update();setInterval(update, 1000);})();]]></content>
     
   </entry>
   
   
   
   <entry>
-    <title></title>
+    <title>关于作者</title>
     <link href="/"/>
     <url>/</url>
     
-    <content type="html"><![CDATA[!(function () {var start = new Date("2024/02/18 00:00:00");function update() {var now = new Date();now.setTime(now.getTime() + 250);days = (now - start) / 1000 / 60 / 60 / 24;dnum = Math.floor(days);        hours = (now - start) / 1000 / 60 / 60 - 24 * dnum;hnum = Math.floor(hours);        if (String(hnum).length === 1) hnum = "0" + hnum;        minutes = (now - start) / 1000 / 60 - 24 * 60 * dnum - 60 * hnum;mnum = Math.floor(minutes);        if (String(mnum).length === 1) mnum = "0" + mnum;seconds = (now - start) / 1000 - 24 * 60 * 60 * dnum - 60 * 60 * hnum - 60 * mnum;snum = Math.round(seconds);if (String(snum).length === 1) snum = "0" + snum;document.getElementById("timeDate").innerHTML = "本站已安全运行&nbsp" + dnum + "&nbsp天";document.getElementById("times").innerHTML = hnum + "&nbsp小时&nbsp" + mnum + "&nbsp分&nbsp" + snum + "&nbsp秒";}update();setInterval(update, 1000);})();]]></content>
+    <content type="html"><![CDATA[<h2 id="教育经历">教育经历</h2><p class="katex-block "><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><menclose notation="left right"><mtable rowspacing="0.16em" columnalign="center center center center center" columnspacing="1em" rowlines="solid none"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext mathvariant="bold">层次</mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext mathvariant="bold">时间</mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext mathvariant="bold">学校</mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext mathvariant="bold">方向</mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext mathvariant="bold">地址</mtext></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>高中</mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>2019.09</mn><mo>−</mo><mn>2022.06</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>江苏省南菁高级中学</mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>物化生</mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>江苏省无锡市江阴市龙定路&nbsp;158&nbsp;号</mtext></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>本科</mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>2022.09</mn><mo>−</mo><mn>2026.06</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>南京师范大学</mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>人工智能</mtext></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mtext>江苏省南京市栖霞区文苑路&nbsp;1&nbsp;号</mtext></mstyle></mtd></mtr></mtable></menclose></mrow><annotation encoding="application/x-tex">\begin{array}{ccccc}\hline\textbf{层次} &amp; \textbf{时间} &amp; \textbf{学校} &amp; \textbf{方向} &amp; \textbf{地址} \\\hline\text{高中} &amp; 2019.09 - 2022.06 &amp; \text{江苏省南菁高级中学} &amp; \text{物化生} &amp; \text{江苏省无锡市江阴市龙定路 158 号} \\\text{本科} &amp; 2022.09 - 2026.06 &amp; \text{南京师范大学} &amp; \text{人工智能} &amp; \text{江苏省南京市栖霞区文苑路 1 号} \\\hline\end{array}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.64em;vertical-align:-1.55em;"></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.09em;"><span style="top:-4.05em;"><span class="pstrut" style="height:4.05em;"></span><span class="mtable"><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textbf cjk_fallback">层次</span></span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">高中</span></span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">本科</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textbf cjk_fallback">时间</span></span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2019.09</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2022.06</span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2022.09</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">2026.06</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textbf cjk_fallback">学校</span></span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">江苏省南菁高级中学</span></span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">南京师范大学</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textbf cjk_fallback">方向</span></span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">物化生</span></span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">人工智能</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em;"><span style="top:-4.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord textbf cjk_fallback">地址</span></span></span></span><span style="top:-3.01em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">江苏省无锡市江阴市龙定路</span><span class="mord">&nbsp;158&nbsp;</span><span class="mord cjk_fallback">号</span></span></span></span><span style="top:-1.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord text"><span class="mord cjk_fallback">江苏省南京市栖霞区文苑路</span><span class="mord">&nbsp;1&nbsp;</span><span class="mord cjk_fallback">号</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span style="top:-2.5em;"><span class="pstrut" style="height:4.05em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-4.9em;"><span class="pstrut" style="height:4.05em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span><span style="top:-6.1em;"><span class="pstrut" style="height:4.05em;"></span><span class="hline" style="border-bottom-width:0.04em;"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55em;"><span></span></span></span></span></span></span></span></span></span></p><h2 id="网瘾青年">网瘾青年</h2><a href="https://codeforces.com/profile/dwj601">    <img src="https://img.shields.io/badge/codeforces-pupil_1378-008000?style=for-the-badge" alt="dwj601"></a><a href="https://leetcode.cn/u/dwj601/">    <img src="https://img.shields.io/badge/LeetCode-knight_1925-63bf88?style=for-the-badge" alt="dwj601"></a><h2 id="技术栈">技术栈</h2><img src="https://skillicons.dev/icons?i=flask,qt" alt="技术栈"><h2 id="编程语言">编程语言</h2><img src="https://skillicons.dev/icons?i=c,cpp,python,js,matlab,v" alt="编程语言"><h2 id="开发工具">开发工具</h2><img src="https://skillicons.dev/icons?i=md,git,clion,pycharm,webstorm,vscode" alt="开发工具">]]></content>
     
   </entry>
   
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@@ -318,15 +318,15 @@ <h2 class="index-header">
     <article class="col-12 col-md-12 mx-auto index-info">
       <h2 class="index-header">
         
-        <a href="/RoadMap/road-map-guide/" target="_blank">
-          road-map-guide
+        <a href="/Operation/operation-guide/" target="_blank">
+          operation-guide
         </a>
       </h2>
 
       
-      <a class="index-excerpt index-excerpt__noimg" href="/RoadMap/road-map-guide/" target="_blank">
+      <a class="index-excerpt index-excerpt__noimg" href="/Operation/operation-guide/" target="_blank">
         <div>
-          学习路线指南 参考：https://www.codefather.cn/学习路线/ 注：打星 (∗)(*)(∗) 内容为上述链接未涉及到的路线，侵权必删 计算机基础 计算机网络 参考：https://www.code-nav.cn/post/1640588119619551233 操作系统 参考：https://www.code-nav.cn/post/1640587909942099969 软件
+          运维指南 前言 本文旨在介绍运维的基本理念和路径规划。正在不断完善中。 云计算运维的相关知识包括但不限于：  Linux 基础：常用命令、文件及用户管理、文本处理、Vim 工具使用。 网络基础：网络基础知识、TCP/IP 协议、七层网络模型、Linux 网络管理与配置。 服务器运维：SSH 远程连接、文件上传下载、Nginx 和 MySQL 服务器搭建、LVS 负载均衡配置以及服务器优化经验。 自
         </div>
       </a>
 
@@ -350,7 +350,7 @@ <h2 class="index-header">
     
       <span class="category-chain">
         
-  <a href="/categories/RoadMap/" class="category-chain-item">RoadMap</a>
+  <a href="/categories/Operation/" class="category-chain-item">Operation</a>
   
   
 
@@ -372,15 +372,15 @@ <h2 class="index-header">
     <article class="col-12 col-md-12 mx-auto index-info">
       <h2 class="index-header">
         
-        <a href="/Operation/operation-guide/" target="_blank">
-          operation-guide
+        <a href="/RoadMap/road-map-guide/" target="_blank">
+          road-map-guide
         </a>
       </h2>
 
       
-      <a class="index-excerpt index-excerpt__noimg" href="/Operation/operation-guide/" target="_blank">
+      <a class="index-excerpt index-excerpt__noimg" href="/RoadMap/road-map-guide/" target="_blank">
         <div>
-          运维指南 前言 本文旨在介绍运维的基本理念和路径规划。正在不断完善中。 云计算运维的相关知识包括但不限于：  Linux 基础：常用命令、文件及用户管理、文本处理、Vim 工具使用。 网络基础：网络基础知识、TCP/IP 协议、七层网络模型、Linux 网络管理与配置。 服务器运维：SSH 远程连接、文件上传下载、Nginx 和 MySQL 服务器搭建、LVS 负载均衡配置以及服务器优化经验。 自
+          学习路线指南 参考：https://www.codefather.cn/学习路线/ 注：打星 (∗)(*)(∗) 内容为上述链接未涉及到的路线，侵权必删 计算机基础 计算机网络 参考：https://www.code-nav.cn/post/1640588119619551233 操作系统 参考：https://www.code-nav.cn/post/1640587909942099969 软件
         </div>
       </a>
 
@@ -404,7 +404,7 @@ <h2 class="index-header">
     
       <span class="category-chain">
         
-  <a href="/categories/Operation/" class="category-chain-item">Operation</a>
+  <a href="/categories/RoadMap/" class="category-chain-item">RoadMap</a>
   
   
 
diff --git a/page/16/index.html b/page/16/index.html
index 373b8c5d4..2a6fed9f1 100644
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@@ -311,15 +311,15 @@ <h2 class="index-header">
     <article class="col-12 col-md-12 mx-auto index-info">
       <h2 class="index-header">
         
-        <a href="/GPA/4th-term/PyAlgo/" target="_blank">
-          PyAlgo
+        <a href="/GPA/4th-term/OptMethod/" target="_blank">
+          OptMethod
         </a>
       </h2>
 
       
-      <a class="index-excerpt index-excerpt__noimg" href="/GPA/4th-term/PyAlgo/" target="_blank">
+      <a class="index-excerpt index-excerpt__noimg" href="/GPA/4th-term/OptMethod/" target="_blank">
         <div>
-          算法设计与分析 前言 学科地位：    主讲教师 学分配额 学科类别     段博佳 3 自发课    成绩组成：    平时（5次作业+签到） 期末（书本+讲义）     60% 40%    教材情况：    课程名称 选用教材 版次 作者 出版社 ISBN号     算法设计与分析 《算法设计与分析(python版)》 1 王秋芬 清华大学出版社 978-7-302-57072-1    一
+          最优化方法 前言 学科地位：    主讲教师 学分配额 学科类别     王启春 4 专业课    成绩组成：    平时（作业+考勤） 期中（大作业） 期末（闭卷）     20% 30% 50%    教材情况：    课程名称 选用教材 版次 作者 出版社 ISBN号     最优化算法 《最优化方法》 2 孙文瑜 高等教育出版社 978-7-04-029763-8    第一章 基本概念
         </div>
       </a>
 
diff --git a/page/17/index.html b/page/17/index.html
index 6c9d923c8..137377c55 100644
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@@ -250,15 +250,15 @@
     <article class="col-12 col-md-12 mx-auto index-info">
       <h2 class="index-header">
         
-        <a href="/GPA/4th-term/ProbAndStat/" target="_blank">
-          ProbAndStat
+        <a href="/GPA/4th-term/PyAlgo/" target="_blank">
+          PyAlgo
         </a>
       </h2>
 
       
-      <a class="index-excerpt index-excerpt__noimg" href="/GPA/4th-term/ProbAndStat/" target="_blank">
+      <a class="index-excerpt index-excerpt__noimg" href="/GPA/4th-term/PyAlgo/" target="_blank">
         <div>
-          概率论与数理统计 前言 学科情况：    主讲教师 学分配额 学科类别     周效尧 4 学科基础课    成绩组成：    平时 测验（×2） 期末     10% 30% 60%    教材情况：    课程名称 选用教材 版次 作者 出版社 ISBN号     概率论与数理统计Ⅰ 《概率论与数理统计》 第一版 刘国祥王晓谦     等主编 科学出版社 978-7-03-038317-4
+          算法设计与分析 前言 学科地位：    主讲教师 学分配额 学科类别     段博佳 3 自发课    成绩组成：    平时（5次作业+签到） 期末（书本+讲义）     60% 40%    教材情况：    课程名称 选用教材 版次 作者 出版社 ISBN号     算法设计与分析 《算法设计与分析(python版)》 1 王秋芬 清华大学出版社 978-7-302-57072-1    一
         </div>
       </a>
 
@@ -311,15 +311,15 @@ <h2 class="index-header">
     <article class="col-12 col-md-12 mx-auto index-info">
       <h2 class="index-header">
         
-        <a href="/GPA/4th-term/OptMethod/" target="_blank">
-          OptMethod
+        <a href="/GPA/4th-term/ProbAndStat/" target="_blank">
+          ProbAndStat
         </a>
       </h2>
 
       
-      <a class="index-excerpt index-excerpt__noimg" href="/GPA/4th-term/OptMethod/" target="_blank">
+      <a class="index-excerpt index-excerpt__noimg" href="/GPA/4th-term/ProbAndStat/" target="_blank">
         <div>
-          最优化方法 前言 学科地位：    主讲教师 学分配额 学科类别     王启春 4 专业课    成绩组成：    平时（作业+考勤） 期中（大作业） 期末（闭卷）     20% 30% 50%    教材情况：    课程名称 选用教材 版次 作者 出版社 ISBN号     最优化算法 《最优化方法》 2 孙文瑜 高等教育出版社 978-7-04-029763-8    第一章 基本概念
+          概率论与数理统计 前言 学科情况：    主讲教师 学分配额 学科类别     周效尧 4 学科基础课    成绩组成：    平时 测验（×2） 期末     10% 30% 60%    教材情况：    课程名称 选用教材 版次 作者 出版社 ISBN号     概率论与数理统计Ⅰ 《概率论与数理统计》 第一版 刘国祥王晓谦     等主编 科学出版社 978-7-03-038317-4
         </div>
       </a>
 
diff --git a/page/18/index.html b/page/18/index.html
index 921b03fca..cc807c895 100644
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+    var CONFIG = {"hostname":"blog.dwj601.cn","root":"/","version":"1.9.8","typing":{"enable":true,"typeSpeed":70,"cursorChar":"_","loop":false,"scope":[]},"anchorjs":{"enable":true,"element":"h1,h2,h3,h4,h5,h6","placement":"left","visible":"hover","icon":""},"progressbar":{"enable":true,"height_px":3,"color":"#29d","options":{"showSpinner":false,"trickleSpeed":100}},"code_language":{"enable":true,"default":"TEXT"},"copy_btn":true,"image_caption":{"enable":true},"image_zoom":{"enable":true,"img_url_replace":["",""]},"toc":{"enable":true,"placement":"left","headingSelector":"h1,h2,h3","collapseDepth":3},"lazyload":{"enable":true,"loading_img":"https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif","onlypost":false,"offset_factor":2},"web_analytics":{"enable":true,"follow_dnt":true,"baidu":"d08d23389220ad9b8f711cfb85757f4b","google":{"measurement_id":null},"tencent":{"sid":null,"cid":null},"leancloud":{"app_id":null,"app_key":null,"server_url":null,"path":"window.location.pathname","ignore_local":false},"umami":{"src":null,"website_id":null,"domains":null,"start_time":"2024-01-01T00:00:00.000Z","token":null,"api_server":null},"woyaola":null,"cnzz":null},"search_path":"/local-search.xml","include_content_in_search":true};
 
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@@ -311,15 +311,15 @@ <h2 class="index-header">
     <article class="col-12 col-md-12 mx-auto index-info">
       <h2 class="index-header">
         
-        <a href="/GPA/3rd-term/DataStructure/" target="_blank">
-          DataStructure
+        <a href="/GPA/3rd-term/DigitalLogicCircuit/" target="_blank">
+          DigitalLogicCircuit
         </a>
       </h2>
 
       
-      <a class="index-excerpt index-excerpt__noimg" href="/GPA/3rd-term/DataStructure/" target="_blank">
+      <a class="index-excerpt index-excerpt__noimg" href="/GPA/3rd-term/DigitalLogicCircuit/" target="_blank">
         <div>
-          score:\mathscr {score:}score:  平时 20%（出勤、作业、实验） 期中 20% 期末 60%   数据结构 完整实现代码：https://github.com/Explorer-Dong/DataStructure 一、绪论 1.1 数据分析+结构存储+算法计算 1.1.1 逻辑结构 对于当前的数据之间的关系进行分析，进而思考应该如何存储，有以下几种逻辑结构：集合、
+          联系方式： ✉ lijuncst@njnu.edu.cn ☎ 13770610040  数字逻辑电路 一、数字逻辑概论 1.1 数字信号与数字电路 1.1.1 数字技术的发展及其应用   电流控制器件：电子管、晶体管（二极管、三极管）、半导体集成电路   EDA(Electronic Design Automation)技术（硬件设计软件化）：设计：EWB or Verilog、仿真、下载、验证
         </div>
       </a>
 
@@ -372,15 +372,15 @@ <h2 class="index-header">
     <article class="col-12 col-md-12 mx-auto index-info">
       <h2 class="index-header">
         
-        <a href="/GPA/3rd-term/DataStructureClassDesign/" target="_blank">
-          DataStructureClassDesign
+        <a href="/GPA/3rd-term/DataStructure/" target="_blank">
+          DataStructure
         </a>
       </h2>
 
       
-      <a class="index-excerpt index-excerpt__noimg" href="/GPA/3rd-term/DataStructureClassDesign/" target="_blank">
+      <a class="index-excerpt index-excerpt__noimg" href="/GPA/3rd-term/DataStructure/" target="_blank">
         <div>
-          数据结构课程设计 效果演示    [!note] 以下为课设报告内容  一、必做题 题目：编程实现希尔、快速、堆排序、归并排序算法。要求随机产生 10000 个数据存入磁盘文件，然后读入数据文件，分别采用不同的排序方法进行排序，并将结果存入文件中。 1.1 数据结构设计与算法思想 本程序涉及到四种排序算法，其中：  希尔排序：通过倍增方法逐渐扩大选择插入数据量的规模，而达到减小数据比较的次数，时间
+          score:\mathscr {score:}score:  平时 20%（出勤、作业、实验） 期中 20% 期末 60%   数据结构 完整实现代码：https://github.com/Explorer-Dong/DataStructure 一、绪论 1.1 数据分析+结构存储+算法计算 1.1.1 逻辑结构 对于当前的数据之间的关系进行分析，进而思考应该如何存储，有以下几种逻辑结构：集合、
         </div>
       </a>
 
diff --git a/page/19/index.html b/page/19/index.html
index 7aab8b19e..f76a17e90 100644
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+    var CONFIG = {"hostname":"blog.dwj601.cn","root":"/","version":"1.9.8","typing":{"enable":true,"typeSpeed":70,"cursorChar":"_","loop":false,"scope":[]},"anchorjs":{"enable":true,"element":"h1,h2,h3,h4,h5,h6","placement":"left","visible":"hover","icon":""},"progressbar":{"enable":true,"height_px":3,"color":"#29d","options":{"showSpinner":false,"trickleSpeed":100}},"code_language":{"enable":true,"default":"TEXT"},"copy_btn":true,"image_caption":{"enable":true},"image_zoom":{"enable":true,"img_url_replace":["",""]},"toc":{"enable":true,"placement":"left","headingSelector":"h1,h2,h3","collapseDepth":3},"lazyload":{"enable":true,"loading_img":"https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif","onlypost":false,"offset_factor":2},"web_analytics":{"enable":true,"follow_dnt":true,"baidu":"d08d23389220ad9b8f711cfb85757f4b","google":{"measurement_id":null},"tencent":{"sid":null,"cid":null},"leancloud":{"app_id":null,"app_key":null,"server_url":null,"path":"window.location.pathname","ignore_local":false},"umami":{"src":null,"website_id":null,"domains":null,"start_time":"2024-01-01T00:00:00.000Z","token":null,"api_server":null},"woyaola":null,"cnzz":null},"search_path":"/local-search.xml","include_content_in_search":true};
 
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@@ -250,15 +250,15 @@
     <article class="col-12 col-md-12 mx-auto index-info">
       <h2 class="index-header">
         
-        <a href="/GPA/3rd-term/DigitalLogicCircuit/" target="_blank">
-          DigitalLogicCircuit
+        <a href="/GPA/3rd-term/DataStructureClassDesign/" target="_blank">
+          DataStructureClassDesign
         </a>
       </h2>
 
       
-      <a class="index-excerpt index-excerpt__noimg" href="/GPA/3rd-term/DigitalLogicCircuit/" target="_blank">
+      <a class="index-excerpt index-excerpt__noimg" href="/GPA/3rd-term/DataStructureClassDesign/" target="_blank">
         <div>
-          联系方式： ✉ lijuncst@njnu.edu.cn ☎ 13770610040  数字逻辑电路 一、数字逻辑概论 1.1 数字信号与数字电路 1.1.1 数字技术的发展及其应用   电流控制器件：电子管、晶体管（二极管、三极管）、半导体集成电路   EDA(Electronic Design Automation)技术（硬件设计软件化）：设计：EWB or Verilog、仿真、下载、验证
+          数据结构课程设计 效果演示    [!note] 以下为课设报告内容  一、必做题 题目：编程实现希尔、快速、堆排序、归并排序算法。要求随机产生 10000 个数据存入磁盘文件，然后读入数据文件，分别采用不同的排序方法进行排序，并将结果存入文件中。 1.1 数据结构设计与算法思想 本程序涉及到四种排序算法，其中：  希尔排序：通过倍增方法逐渐扩大选择插入数据量的规模，而达到减小数据比较的次数，时间
         </div>
       </a>
 
diff --git a/page/2/index.html b/page/2/index.html
index aee5a8383..1487ee484 100644
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@@ -70,7 +70,7 @@
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diff --git a/page/20/index.html b/page/20/index.html
index dce1b8f3d..25dc42ca1 100644
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+    var CONFIG = {"hostname":"blog.dwj601.cn","root":"/","version":"1.9.8","typing":{"enable":true,"typeSpeed":70,"cursorChar":"_","loop":false,"scope":[]},"anchorjs":{"enable":true,"element":"h1,h2,h3,h4,h5,h6","placement":"left","visible":"hover","icon":""},"progressbar":{"enable":true,"height_px":3,"color":"#29d","options":{"showSpinner":false,"trickleSpeed":100}},"code_language":{"enable":true,"default":"TEXT"},"copy_btn":true,"image_caption":{"enable":true},"image_zoom":{"enable":true,"img_url_replace":["",""]},"toc":{"enable":true,"placement":"left","headingSelector":"h1,h2,h3","collapseDepth":3},"lazyload":{"enable":true,"loading_img":"https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif","onlypost":false,"offset_factor":2},"web_analytics":{"enable":true,"follow_dnt":true,"baidu":"d08d23389220ad9b8f711cfb85757f4b","google":{"measurement_id":null},"tencent":{"sid":null,"cid":null},"leancloud":{"app_id":null,"app_key":null,"server_url":null,"path":"window.location.pathname","ignore_local":false},"umami":{"src":null,"website_id":null,"domains":null,"start_time":"2024-01-01T00:00:00.000Z","token":null,"api_server":null},"woyaola":null,"cnzz":null},"search_path":"/local-search.xml","include_content_in_search":true};
 
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@@ -311,15 +311,15 @@ <h2 class="index-header">
     <article class="col-12 col-md-12 mx-auto index-info">
       <h2 class="index-header">
         
-        <a href="/FrontEnd/Hexo/hexo-migrate/" target="_blank">
-          hexo-migrate
+        <a href="/FrontEnd/Hexo/hexo-learning-record/" target="_blank">
+          hexo-learning-record
         </a>
       </h2>
 
       
-      <a class="index-excerpt index-excerpt__noimg" href="/FrontEnd/Hexo/hexo-migrate/" target="_blank">
+      <a class="index-excerpt index-excerpt__noimg" href="/FrontEnd/Hexo/hexo-learning-record/" target="_blank">
         <div>
-          Hexo 迁移指南 前言 为了改变本地项目名称（强迫症），同时为了练习迁移编辑环境，特此记录。 一、克隆仓库 1git clone https://github.com/Explorer-Dong/explorer-dong.github.io.git HexoBlog 二、安装依赖 1npm install 三、编辑推送 123git add .git commit -m 'add: xxx'g
+          Hexo 学习记录 前言 操作系统：Windows 11 OS 建站第一枪，使用 Hexo 静态站点生成器搭建自己的博客网站。首先解释一下什么叫静态站点生成器，其实就是将自己写好的 Markdown   文件自动构建为 HTML 文件，然后进行浏览器渲染，而不需要自己编写前端样式文件。在开始介绍 Hexo 建站步骤之前，最好需要掌握一下技能（不掌握也没事，边建边学也行，我就是这样，不过可能会比较浪
         </div>
       </a>
 
@@ -372,15 +372,15 @@ <h2 class="index-header">
     <article class="col-12 col-md-12 mx-auto index-info">
       <h2 class="index-header">
         
-        <a href="/FrontEnd/Hexo/hexo-learning-record/" target="_blank">
-          hexo-learning-record
+        <a href="/FrontEnd/Hexo/hexo-migrate/" target="_blank">
+          hexo-migrate
         </a>
       </h2>
 
       
-      <a class="index-excerpt index-excerpt__noimg" href="/FrontEnd/Hexo/hexo-learning-record/" target="_blank">
+      <a class="index-excerpt index-excerpt__noimg" href="/FrontEnd/Hexo/hexo-migrate/" target="_blank">
         <div>
-          Hexo 学习记录 前言 操作系统：Windows 11 OS 建站第一枪，使用 Hexo 静态站点生成器搭建自己的博客网站。首先解释一下什么叫静态站点生成器，其实就是将自己写好的 Markdown   文件自动构建为 HTML 文件，然后进行浏览器渲染，而不需要自己编写前端样式文件。在开始介绍 Hexo 建站步骤之前，最好需要掌握一下技能（不掌握也没事，边建边学也行，我就是这样，不过可能会比较浪
+          Hexo 迁移指南 前言 为了改变本地项目名称（强迫症），同时为了练习迁移编辑环境，特此记录。 一、克隆仓库 1git clone https://github.com/Explorer-Dong/explorer-dong.github.io.git HexoBlog 二、安装依赖 1npm install 三、编辑推送 123git add .git commit -m 'add: xxx'g
         </div>
       </a>
 
diff --git a/page/21/index.html b/page/21/index.html
index 8b1217372..83e117e9a 100644
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@@ -70,7 +70,7 @@
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diff --git a/page/22/index.html b/page/22/index.html
index 76bdd27e1..845439dd9 100644
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@@ -70,7 +70,7 @@
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+    var CONFIG = {"hostname":"blog.dwj601.cn","root":"/","version":"1.9.8","typing":{"enable":true,"typeSpeed":70,"cursorChar":"_","loop":false,"scope":[]},"anchorjs":{"enable":true,"element":"h1,h2,h3,h4,h5,h6","placement":"left","visible":"hover","icon":""},"progressbar":{"enable":true,"height_px":3,"color":"#29d","options":{"showSpinner":false,"trickleSpeed":100}},"code_language":{"enable":true,"default":"TEXT"},"copy_btn":true,"image_caption":{"enable":true},"image_zoom":{"enable":true,"img_url_replace":["",""]},"toc":{"enable":true,"placement":"left","headingSelector":"h1,h2,h3","collapseDepth":3},"lazyload":{"enable":true,"loading_img":"https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif","onlypost":false,"offset_factor":2},"web_analytics":{"enable":true,"follow_dnt":true,"baidu":"d08d23389220ad9b8f711cfb85757f4b","google":{"measurement_id":null},"tencent":{"sid":null,"cid":null},"leancloud":{"app_id":null,"app_key":null,"server_url":null,"path":"window.location.pathname","ignore_local":false},"umami":{"src":null,"website_id":null,"domains":null,"start_time":"2024-01-01T00:00:00.000Z","token":null,"api_server":null},"woyaola":null,"cnzz":null},"search_path":"/local-search.xml","include_content_in_search":true};
 
     if (CONFIG.web_analytics.follow_dnt) {
       var dntVal = navigator.doNotTrack || window.doNotTrack || navigator.msDoNotTrack;
@@ -250,15 +250,15 @@
     <article class="col-12 col-md-12 mx-auto index-info">
       <h2 class="index-header">
         
-        <a href="/DevTools/DevCpp/devc-self-config/" target="_blank">
-          self-config
+        <a href="/DevTools/CLion/solve-clion-decoding-error/" target="_blank">
+          solve-clion-decoding-error
         </a>
       </h2>
 
       
-      <a class="index-excerpt index-excerpt__noimg" href="/DevTools/DevCpp/devc-self-config/" target="_blank">
+      <a class="index-excerpt index-excerpt__noimg" href="/DevTools/CLion/solve-clion-decoding-error/" target="_blank">
         <div>
-          DevCpp 偏好配置 前言 为了在比赛中不因为编辑器而拖后腿，写一个配置说明。编辑器为 Dev-C++ 5.11 一、编译标准 进入以下目录：  添加以下编译命令：  二、环境选项   三、编辑器选项 进入以下目录：   四、快捷键选项 进入以下目录：  设置注释配置：
+          CLion 解决中文输出乱码的问题 问题介绍 在 Clion 的默认设置下，输出中文会出现乱码，如下 1234567#include &lt;iostream&gt;using namespace std;int main() {    cout &lt;&lt; "你好" &lt;&lt; endl;    return 0;} 输出 123浣犲ソ	Process finished with e
         </div>
       </a>
 
@@ -287,7 +287,7 @@ <h2 class="index-header">
   
     <span>></span>
     
-  <a href="/categories/DevTools/DevCpp/" class="category-chain-item">DevCpp</a>
+  <a href="/categories/DevTools/CLion/" class="category-chain-item">CLion</a>
   
   
 
@@ -311,15 +311,15 @@ <h2 class="index-header">
     <article class="col-12 col-md-12 mx-auto index-info">
       <h2 class="index-header">
         
-        <a href="/DevTools/CLion/solve-clion-decoding-error/" target="_blank">
-          solve-clion-decoding-error
+        <a href="/DevTools/DevCpp/devc-self-config/" target="_blank">
+          self-config
         </a>
       </h2>
 
       
-      <a class="index-excerpt index-excerpt__noimg" href="/DevTools/CLion/solve-clion-decoding-error/" target="_blank">
+      <a class="index-excerpt index-excerpt__noimg" href="/DevTools/DevCpp/devc-self-config/" target="_blank">
         <div>
-          CLion 解决中文输出乱码的问题 问题介绍 在 Clion 的默认设置下，输出中文会出现乱码，如下 1234567#include &lt;iostream&gt;using namespace std;int main() {    cout &lt;&lt; "你好" &lt;&lt; endl;    return 0;} 输出 123浣犲ソ	Process finished with e
+          DevCpp 偏好配置 前言 为了在比赛中不因为编辑器而拖后腿，写一个配置说明。编辑器为 Dev-C++ 5.11 一、编译标准 进入以下目录：  添加以下编译命令：  二、环境选项   三、编辑器选项 进入以下目录：   四、快捷键选项 进入以下目录：  设置注释配置：
         </div>
       </a>
 
@@ -348,7 +348,7 @@ <h2 class="index-header">
   
     <span>></span>
     
-  <a href="/categories/DevTools/CLion/" class="category-chain-item">CLion</a>
+  <a href="/categories/DevTools/DevCpp/" class="category-chain-item">DevCpp</a>
   
   
 
diff --git a/page/23/index.html b/page/23/index.html
index 48abf9c61..35620bfe2 100644
--- a/page/23/index.html
+++ b/page/23/index.html
@@ -70,7 +70,7 @@
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     var Fluid = window.Fluid || {};
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+    var CONFIG = {"hostname":"blog.dwj601.cn","root":"/","version":"1.9.8","typing":{"enable":true,"typeSpeed":70,"cursorChar":"_","loop":false,"scope":[]},"anchorjs":{"enable":true,"element":"h1,h2,h3,h4,h5,h6","placement":"left","visible":"hover","icon":""},"progressbar":{"enable":true,"height_px":3,"color":"#29d","options":{"showSpinner":false,"trickleSpeed":100}},"code_language":{"enable":true,"default":"TEXT"},"copy_btn":true,"image_caption":{"enable":true},"image_zoom":{"enable":true,"img_url_replace":["",""]},"toc":{"enable":true,"placement":"left","headingSelector":"h1,h2,h3","collapseDepth":3},"lazyload":{"enable":true,"loading_img":"https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif","onlypost":false,"offset_factor":2},"web_analytics":{"enable":true,"follow_dnt":true,"baidu":"d08d23389220ad9b8f711cfb85757f4b","google":{"measurement_id":null},"tencent":{"sid":null,"cid":null},"leancloud":{"app_id":null,"app_key":null,"server_url":null,"path":"window.location.pathname","ignore_local":false},"umami":{"src":null,"website_id":null,"domains":null,"start_time":"2024-01-01T00:00:00.000Z","token":null,"api_server":null},"woyaola":null,"cnzz":null},"search_path":"/local-search.xml","include_content_in_search":true};
 
     if (CONFIG.web_analytics.follow_dnt) {
       var dntVal = navigator.doNotTrack || window.doNotTrack || navigator.msDoNotTrack;
@@ -311,15 +311,15 @@ <h2 class="index-header">
     <article class="col-12 col-md-12 mx-auto index-info">
       <h2 class="index-header">
         
-        <a href="/BackEnd/back-end-guide/" target="_blank">
-          back-end-guide
+        <a href="/Algorithm/z_logic/" target="_blank">
+          z_logic
         </a>
       </h2>
 
       
-      <a class="index-excerpt index-excerpt__noimg" href="/BackEnd/back-end-guide/" target="_blank">
+      <a class="index-excerpt index-excerpt__noimg" href="/Algorithm/z_logic/" target="_blank">
         <div>
-          后端开发指南 前言 本文旨在介绍后端开发的基本理念和路径规划。正在不断完善中。 对于后端开发来说，语言只是一个工具和基础。除了语言本身和对应的开发框架外，其他要学的技术在后端开发中往往是通用的，比如：数据库、缓存、消息队列、搜索引擎、Linux、分布式、高并发、设计模式、架构设计等等。 参考 Backend Beginner Roadmap Backend Roadmap
+          思维题 多角度切入。 【思维/哈希/排序】最长严格递增子序列 https://www.acwing.com/problem/content/5273/  题意：给定长度为 n 的序列，问将这个序列拼接 n 次后，最长严格递增子序列的长度为多少？ 思路：其实最终的思路很简单，将题目转化为在每个序列中选一个数，一共可以选出多少个不同的数。但是在产生这样的想法之前，先讲一下我的思考过程。我将序列脑补出一
         </div>
       </a>
 
@@ -343,7 +343,7 @@ <h2 class="index-header">
     
       <span class="category-chain">
         
-  <a href="/categories/BackEnd/" class="category-chain-item">BackEnd</a>
+  <a href="/categories/Algorithm/" class="category-chain-item">Algorithm</a>
   
   
 
@@ -365,15 +365,15 @@ <h2 class="index-header">
     <article class="col-12 col-md-12 mx-auto index-info">
       <h2 class="index-header">
         
-        <a href="/Algorithm/z_logic/" target="_blank">
-          z_logic
+        <a href="/BackEnd/back-end-guide/" target="_blank">
+          back-end-guide
         </a>
       </h2>
 
       
-      <a class="index-excerpt index-excerpt__noimg" href="/Algorithm/z_logic/" target="_blank">
+      <a class="index-excerpt index-excerpt__noimg" href="/BackEnd/back-end-guide/" target="_blank">
         <div>
-          思维题 多角度切入。 【思维/哈希/排序】最长严格递增子序列 https://www.acwing.com/problem/content/5273/  题意：给定长度为 n 的序列，问将这个序列拼接 n 次后，最长严格递增子序列的长度为多少？ 思路：其实最终的思路很简单，将题目转化为在每个序列中选一个数，一共可以选出多少个不同的数。但是在产生这样的想法之前，先讲一下我的思考过程。我将序列脑补出一
+          后端开发指南 前言 本文旨在介绍后端开发的基本理念和路径规划。正在不断完善中。 对于后端开发来说，语言只是一个工具和基础。除了语言本身和对应的开发框架外，其他要学的技术在后端开发中往往是通用的，比如：数据库、缓存、消息队列、搜索引擎、Linux、分布式、高并发、设计模式、架构设计等等。 参考 Backend Beginner Roadmap Backend Roadmap
         </div>
       </a>
 
@@ -397,7 +397,7 @@ <h2 class="index-header">
     
       <span class="category-chain">
         
-  <a href="/categories/Algorithm/" class="category-chain-item">Algorithm</a>
+  <a href="/categories/BackEnd/" class="category-chain-item">BackEnd</a>
   
   
 
diff --git a/page/24/index.html b/page/24/index.html
index feb579b45..fcfba7235 100644
--- a/page/24/index.html
+++ b/page/24/index.html
@@ -70,7 +70,7 @@
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     var Fluid = window.Fluid || {};
     Fluid.ctx = Object.assign({}, Fluid.ctx)
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+    var CONFIG = {"hostname":"blog.dwj601.cn","root":"/","version":"1.9.8","typing":{"enable":true,"typeSpeed":70,"cursorChar":"_","loop":false,"scope":[]},"anchorjs":{"enable":true,"element":"h1,h2,h3,h4,h5,h6","placement":"left","visible":"hover","icon":""},"progressbar":{"enable":true,"height_px":3,"color":"#29d","options":{"showSpinner":false,"trickleSpeed":100}},"code_language":{"enable":true,"default":"TEXT"},"copy_btn":true,"image_caption":{"enable":true},"image_zoom":{"enable":true,"img_url_replace":["",""]},"toc":{"enable":true,"placement":"left","headingSelector":"h1,h2,h3","collapseDepth":3},"lazyload":{"enable":true,"loading_img":"https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif","onlypost":false,"offset_factor":2},"web_analytics":{"enable":true,"follow_dnt":true,"baidu":"d08d23389220ad9b8f711cfb85757f4b","google":{"measurement_id":null},"tencent":{"sid":null,"cid":null},"leancloud":{"app_id":null,"app_key":null,"server_url":null,"path":"window.location.pathname","ignore_local":false},"umami":{"src":null,"website_id":null,"domains":null,"start_time":"2024-01-01T00:00:00.000Z","token":null,"api_server":null},"woyaola":null,"cnzz":null},"search_path":"/local-search.xml","include_content_in_search":true};
 
     if (CONFIG.web_analytics.follow_dnt) {
       var dntVal = navigator.doNotTrack || window.doNotTrack || navigator.msDoNotTrack;
@@ -250,15 +250,15 @@
     <article class="col-12 col-md-12 mx-auto index-info">
       <h2 class="index-header">
         
-        <a href="/Algorithm/geometry/" target="_blank">
-          geometry
+        <a href="/Algorithm/games/" target="_blank">
+          games
         </a>
       </h2>
 
       
-      <a class="index-excerpt index-excerpt__noimg" href="/Algorithm/geometry/" target="_blank">
+      <a class="index-excerpt index-excerpt__noimg" href="/Algorithm/games/" target="_blank">
         <div>
-          计算几何 【二维/数学】Minimum Manhattan Distance https://codeforces.com/gym/104639/problem/J  题意：给定两个圆的直径的两个点坐标，其中约束条件是两个圆一定是处在相离的两个角上。问如何在C2圆上或圆内找到一点p，使得点p到C1圆的所有点的曼哈顿距离的期望值最小 思路：   看似需要积分，其实我们可以发现，对于点p到C1中某个点
+          博弈论 思考如何必胜态和必败态是什么以及如何构造这样的局面。 【博弈/贪心/交互】Salyg1n and the MEX Game https://codeforces.com/contest/1867/problem/C  标签：博弈、贪心、交互 题面：对于给定n个数的数列，先手可以放入一个数列中不存在的数（0-1e9），后手可以从数列中拿掉一个数，但是这个数必须严格小于刚才先手放入的数。终止条
         </div>
       </a>
 
@@ -304,15 +304,15 @@ <h2 class="index-header">
     <article class="col-12 col-md-12 mx-auto index-info">
       <h2 class="index-header">
         
-        <a href="/Algorithm/hashing/" target="_blank">
-          hashing
+        <a href="/Algorithm/geometry/" target="_blank">
+          geometry
         </a>
       </h2>
 
       
-      <a class="index-excerpt index-excerpt__noimg" href="/Algorithm/hashing/" target="_blank">
+      <a class="index-excerpt index-excerpt__noimg" href="/Algorithm/geometry/" target="_blank">
         <div>
-          哈希 【哈希】分组 https://www.acwing.com/problem/content/5182/  存储不想同组和想同组的人员信息：存入数组，数据类型为一对字符串 存储所有的组队信息：存入哈希表，数据类型为“键:字符串”“值:一对字符串” 想要知道最终的分组情况，只需要查询数组中的队员情况与想同组 or 不想同组的成员名字是否一致即可 时间复杂度 O(n)O(n)O(n)，空间复杂度
+          计算几何 【二维/数学】Minimum Manhattan Distance https://codeforces.com/gym/104639/problem/J  题意：给定两个圆的直径的两个点坐标，其中约束条件是两个圆一定是处在相离的两个角上。问如何在C2圆上或圆内找到一点p，使得点p到C1圆的所有点的曼哈顿距离的期望值最小 思路：   看似需要积分，其实我们可以发现，对于点p到C1中某个点
         </div>
       </a>
 
@@ -358,15 +358,15 @@ <h2 class="index-header">
     <article class="col-12 col-md-12 mx-auto index-info">
       <h2 class="index-header">
         
-        <a href="/Algorithm/games/" target="_blank">
-          games
+        <a href="/Algorithm/graphs/" target="_blank">
+          graphs
         </a>
       </h2>
 
       
-      <a class="index-excerpt index-excerpt__noimg" href="/Algorithm/games/" target="_blank">
+      <a class="index-excerpt index-excerpt__noimg" href="/Algorithm/graphs/" target="_blank">
         <div>
-          博弈论 思考如何必胜态和必败态是什么以及如何构造这样的局面。 【博弈/贪心/交互】Salyg1n and the MEX Game https://codeforces.com/contest/1867/problem/C  标签：博弈、贪心、交互 题面：对于给定n个数的数列，先手可以放入一个数列中不存在的数（0-1e9），后手可以从数列中拿掉一个数，但是这个数必须严格小于刚才先手放入的数。终止条
+          图论 【拓扑】有向图的拓扑序列 https://www.acwing.com/problem/content/850/  题意：输出一个图的拓扑序，不存在则输出-1 思路：  首先我们要知道拓扑图的概念，感官上就是一张图可以从一个方向拓展到全图，用数学语言就是：若一个由图中所有点构成的序列 A 满足：对于图中的每条边 (x,y)，x 在 A 中都出现在 y 之前，则称 A 是该图的一个拓扑序列 接
         </div>
       </a>
 
diff --git a/page/25/index.html b/page/25/index.html
index 9e35616b9..b62608840 100644
--- a/page/25/index.html
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@@ -70,7 +70,7 @@
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@@ -250,15 +250,15 @@
     <article class="col-12 col-md-12 mx-auto index-info">
       <h2 class="index-header">
         
-        <a href="/Algorithm/graphs/" target="_blank">
-          graphs
+        <a href="/Algorithm/greedy/" target="_blank">
+          greedy
         </a>
       </h2>
 
       
-      <a class="index-excerpt index-excerpt__noimg" href="/Algorithm/graphs/" target="_blank">
+      <a class="index-excerpt index-excerpt__noimg" href="/Algorithm/greedy/" target="_blank">
         <div>
-          图论 【拓扑】有向图的拓扑序列 https://www.acwing.com/problem/content/850/  题意：输出一个图的拓扑序，不存在则输出-1 思路：  首先我们要知道拓扑图的概念，感官上就是一张图可以从一个方向拓展到全图，用数学语言就是：若一个由图中所有点构成的序列 A 满足：对于图中的每条边 (x,y)，x 在 A 中都出现在 y 之前，则称 A 是该图的一个拓扑序列 接
+          贪心 大胆猜测，小心求证（不会证也没事，做下一题吧）。证明方法总结了以下几种  反证法：假设取一种方案比贪心方案更好，得出相反的结论 边界法：从边界开始考虑，因为满足边界条件更加容易枚举，从而进行后续的贪心 直觉法：遵循社会法则（）  1. green_gold_dog, array and permutation https://codeforces.com/contest/1867/probl
         </div>
       </a>
 
@@ -304,15 +304,15 @@ <h2 class="index-header">
     <article class="col-12 col-md-12 mx-auto index-info">
       <h2 class="index-header">
         
-        <a href="/Algorithm/number-theory/" target="_blank">
-          number-theory
+        <a href="/Algorithm/hashing/" target="_blank">
+          hashing
         </a>
       </h2>
 
       
-      <a class="index-excerpt index-excerpt__noimg" href="/Algorithm/number-theory/" target="_blank">
+      <a class="index-excerpt index-excerpt__noimg" href="/Algorithm/hashing/" target="_blank">
         <div>
-          数论 整数问题。 【质数】Divide and Equalize  题意：给定 nnn 个数，问能否找到一个数 numnumnum，使得 numn=∏i=1nainum^n = \prod_{i=1}^{n}a_inumn=∏i=1n​ai​ 原始思路：起初我的思路是二分，我们需要寻找一个数使得n个数相乘为原数组所有元素之积，那么我们预计算出所有数之积，并且在数组最大值和最小值之间进行二分，每次二
+          哈希 【哈希】分组 https://www.acwing.com/problem/content/5182/  存储不想同组和想同组的人员信息：存入数组，数据类型为一对字符串 存储所有的组队信息：存入哈希表，数据类型为“键:字符串”“值:一对字符串” 想要知道最终的分组情况，只需要查询数组中的队员情况与想同组 or 不想同组的成员名字是否一致即可 时间复杂度 O(n)O(n)O(n)，空间复杂度
         </div>
       </a>
 
@@ -358,15 +358,15 @@ <h2 class="index-header">
     <article class="col-12 col-md-12 mx-auto index-info">
       <h2 class="index-header">
         
-        <a href="/Algorithm/greedy/" target="_blank">
-          greedy
+        <a href="/Algorithm/number-theory/" target="_blank">
+          number-theory
         </a>
       </h2>
 
       
-      <a class="index-excerpt index-excerpt__noimg" href="/Algorithm/greedy/" target="_blank">
+      <a class="index-excerpt index-excerpt__noimg" href="/Algorithm/number-theory/" target="_blank">
         <div>
-          贪心 大胆猜测，小心求证（不会证也没事，做下一题吧）。证明方法总结了以下几种  反证法：假设取一种方案比贪心方案更好，得出相反的结论 边界法：从边界开始考虑，因为满足边界条件更加容易枚举，从而进行后续的贪心 直觉法：遵循社会法则（）  1. green_gold_dog, array and permutation https://codeforces.com/contest/1867/probl
+          数论 整数问题。 【质数】Divide and Equalize  题意：给定 nnn 个数，问能否找到一个数 numnumnum，使得 numn=∏i=1nainum^n = \prod_{i=1}^{n}a_inumn=∏i=1n​ai​ 原始思路：起初我的思路是二分，我们需要寻找一个数使得n个数相乘为原数组所有元素之积，那么我们预计算出所有数之积，并且在数组最大值和最小值之间进行二分，每次二
         </div>
       </a>
 
diff --git a/page/26/index.html b/page/26/index.html
index 6b2616d85..2cfdaf5ec 100644
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+    var CONFIG = {"hostname":"blog.dwj601.cn","root":"/","version":"1.9.8","typing":{"enable":true,"typeSpeed":70,"cursorChar":"_","loop":false,"scope":[]},"anchorjs":{"enable":true,"element":"h1,h2,h3,h4,h5,h6","placement":"left","visible":"hover","icon":""},"progressbar":{"enable":true,"height_px":3,"color":"#29d","options":{"showSpinner":false,"trickleSpeed":100}},"code_language":{"enable":true,"default":"TEXT"},"copy_btn":true,"image_caption":{"enable":true},"image_zoom":{"enable":true,"img_url_replace":["",""]},"toc":{"enable":true,"placement":"left","headingSelector":"h1,h2,h3","collapseDepth":3},"lazyload":{"enable":true,"loading_img":"https://dwj-oss.oss-cn-nanjing.aliyuncs.com/web-imgs/img-static/loading.gif","onlypost":false,"offset_factor":2},"web_analytics":{"enable":true,"follow_dnt":true,"baidu":"d08d23389220ad9b8f711cfb85757f4b","google":{"measurement_id":null},"tencent":{"sid":null,"cid":null},"leancloud":{"app_id":null,"app_key":null,"server_url":null,"path":"window.location.pathname","ignore_local":false},"umami":{"src":null,"website_id":null,"domains":null,"start_time":"2024-01-01T00:00:00.000Z","token":null,"api_server":null},"woyaola":null,"cnzz":null},"search_path":"/local-search.xml","include_content_in_search":true};
 
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@@ -358,15 +358,15 @@ <h2 class="index-header">
     <article class="col-12 col-md-12 mx-auto index-info">
       <h2 class="index-header">
         
-        <a href="/Algorithm/binary-search/" target="_blank">
-          binary-search
+        <a href="/Algorithm/data-structure/" target="_blank">
+          data-structure
         </a>
       </h2>
 
       
-      <a class="index-excerpt index-excerpt__noimg" href="/Algorithm/binary-search/" target="_blank">
+      <a class="index-excerpt index-excerpt__noimg" href="/Algorithm/data-structure/" target="_blank">
         <div>
-          二分 二分本质上是一个线性的算法思维，只是比线性思维更进一步的是，二分思维需要提炼出题面中两个线性相关的变量，即单调变化的两个变量，从而采用二分加速检索。 【二分答案】Building an Aquarium https://codeforces.com/contest/1873/problem/E  题意：想象有一个二维平面，现在有一个数列，每一个数表示平面对应列的高度，现在要给这个平面在两边加
+          数据结构 数据结构由 数据 和 结构 两部分组成。我们主要讨论的是后者，即结构部分。 按照 逻辑结构 可以将其区分为 线性结构 和 非线性结构。  按照 物理结构 可以将其区分为 连续结构 和 分散结构。  【模板】双链表 https://www.acwing.com/problem/content/829/  思路：用两个空结点作为起始状态的边界，避免所有边界讨论。 时间复杂度：插入、删除结点均
         </div>
       </a>
 
diff --git a/page/27/index.html b/page/27/index.html
index 0a6fa79f4..cc7681ccb 100644
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@@ -250,15 +250,15 @@
     <article class="col-12 col-md-12 mx-auto index-info">
       <h2 class="index-header">
         
-        <a href="/Algorithm/divide-and-conquer/" target="_blank">
-          divide-and-conquer
+        <a href="/Algorithm/binary-search/" target="_blank">
+          binary-search
         </a>
       </h2>
 
       
-      <a class="index-excerpt index-excerpt__noimg" href="/Algorithm/divide-and-conquer/" target="_blank">
+      <a class="index-excerpt index-excerpt__noimg" href="/Algorithm/binary-search/" target="_blank">
         <div>
-          分治 将大问题转化为等价小问题进行求解。 【分治】随机排列 https://www.acwing.com/problem/content/5469/  题意：给定一个 n 个数的全排列序列，并将其进行一定的对换，问是对换了 3n 次还是 7n+1 次 思路：可以发现对于两种情况，就对应对换次数的奇偶性。当 n 为奇数：3n 为奇数，7n+1 为偶数；当 n 为偶数：3n 为偶数，7n+1 为奇数。
+          二分 二分本质上是一个线性的算法思维，只是比线性思维更进一步的是，二分思维需要提炼出题面中两个线性相关的变量，即单调变化的两个变量，从而采用二分加速检索。 【二分答案】Building an Aquarium https://codeforces.com/contest/1873/problem/E  题意：想象有一个二维平面，现在有一个数列，每一个数表示平面对应列的高度，现在要给这个平面在两边加
         </div>
       </a>
 
@@ -358,15 +358,15 @@ <h2 class="index-header">
     <article class="col-12 col-md-12 mx-auto index-info">
       <h2 class="index-header">
         
-        <a href="/Algorithm/data-structure/" target="_blank">
-          data-structure
+        <a href="/Algorithm/divide-and-conquer/" target="_blank">
+          divide-and-conquer
         </a>
       </h2>
 
       
-      <a class="index-excerpt index-excerpt__noimg" href="/Algorithm/data-structure/" target="_blank">
+      <a class="index-excerpt index-excerpt__noimg" href="/Algorithm/divide-and-conquer/" target="_blank">
         <div>
-          数据结构 数据结构由 数据 和 结构 两部分组成。我们主要讨论的是后者，即结构部分。 按照 逻辑结构 可以将其区分为 线性结构 和 非线性结构。  按照 物理结构 可以将其区分为 连续结构 和 分散结构。  【模板】双链表 https://www.acwing.com/problem/content/829/  思路：用两个空结点作为起始状态的边界，避免所有边界讨论。 时间复杂度：插入、删除结点均
+          分治 将大问题转化为等价小问题进行求解。 【分治】随机排列 https://www.acwing.com/problem/content/5469/  题意：给定一个 n 个数的全排列序列，并将其进行一定的对换，问是对换了 3n 次还是 7n+1 次 思路：可以发现对于两种情况，就对应对换次数的奇偶性。当 n 为奇数：3n 为奇数，7n+1 为偶数；当 n 为偶数：3n 为偶数，7n+1 为奇数。
         </div>
       </a>
 
diff --git a/page/28/index.html b/page/28/index.html
index b3287b4e7..a463e6fd6 100644
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@@ -258,7 +258,7 @@ <h2 class="index-header">
       
       <a class="index-excerpt index-excerpt__noimg" href="/GPA/5th-term/OperatingSystem/" target="_blank">
         <div>
-          操作系统 学科地位：    主讲教师 学分配额 学科类别     段博佳 4+1 专业课    成绩组成：    理论课  实验课      平时 10% 小组作业 100%   期中 30%     期末 60%      教材情况：    课程名称 选用教材 版次 作者 出版社 ISBN 号     操作系统原理 《操作系统教程》 6 骆斌 葛季栋 费翔林 高等教育出版社 978-7-04-0
+          操作系统 前言 学科地位：    主讲教师 学分配额 学科类别     段博佳 4+1 专业课    成绩组成：    理论课  实验课      平时 10% 小组作业 100%   期中 30%     期末 60%      教材情况：    课程名称 选用教材 版次 作者 出版社 ISBN 号     操作系统原理 《操作系统教程》 6 骆斌 葛季栋 费翔林 高等教育出版社 978-7-0
         </div>
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diff --git a/tags/index.html b/tags/index.html
index cb36097f0..b77545612 100644
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