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<?xml version="1.0" encoding="utf-8"?>
<feed xmlns="http://www.w3.org/2005/Atom">
<title>西工院数学第一</title>
<link href="http://sgnxotsmicf.com/atom.xml" rel="self"/>
<link href="http://sgnxotsmicf.com/"/>
<updated>2022-03-17T15:21:51.448Z</updated>
<id>http://sgnxotsmicf.com/</id>
<author>
<name>sgnxOtsmiCf</name>
</author>
<generator uri="https://hexo.io/">Hexo</generator>
<entry>
<title>数学笔记-必记</title>
<link href="http://sgnxotsmicf.com/2022/03/16/%E6%95%B0%E5%AD%A6%E7%AC%94%E8%AE%B0-%E5%BF%85%E8%AE%B0/"/>
<id>http://sgnxotsmicf.com/2022/03/16/%E6%95%B0%E5%AD%A6%E7%AC%94%E8%AE%B0-%E5%BF%85%E8%AE%B0/</id>
<published>2022-03-16T12:12:38.000Z</published>
<updated>2022-03-17T15:21:51.448Z</updated>
<content type="html"><![CDATA[<link rel="stylesheet" class="aplayer-secondary-style-marker" href="\assets\css\APlayer.min.css"><script src="\assets\js\APlayer.min.js" class="aplayer-secondary-script-marker"></script><script class="meting-secondary-script-marker" src="\assets\js\Meting.min.js"></script><h4 id="三月"><a href="#三月" class="headerlink" title="三月"></a>三月</h4><blockquote><p>3月16</p></blockquote><ol><li>去掉、增加或改变级数的有限个项并不改变级数的敛散性</li><li>在收敛级数的项中任意加括号，既不改变级数的敛散性，也不改变它的和</li></ol><blockquote><p>3月17</p></blockquote><ol><li><p>重点理解正项级数收敛的充要条件，及其衍生内容</p></li><li><p>注意：比式判别法和根式判别法的极限形式，对其所依赖的比值或常量为1时，无法适用，要具体境况具体判断</p><p>如调和级数和p级数，对应的极限形式值均为1，但调和级数明显发散，而p级数明显是收敛的</p></li><li><p>比式判别法的极限形式可应用上下限来判断，而根式判别法的极限形式只能应用上限来判断</p><p>但同时，均无法对1进行处理</p><p>但是，二者的正常形式在其值为1时，均发散、</p></li><li><p>凡能由比式判别法鉴别收敛性的级数，它也能由根式判别法来判断，而且可以说，根式判别法较之比式判别法更有效</p><p>比如某些级数用比式判别法会得到不一样的极限，显然发散，无从解决，但用根式判别法来考查其敛散性，往往时行之有效的</p></li><li><img src="/2022/03/16/%E6%95%B0%E5%AD%A6%E7%AC%94%E8%AE%B0-%E5%BF%85%E8%AE%B0/R5V1I781L95$@OM%60J70W_L.jpg" class="" title="img"></li></ol><p>很不错的阶乘题，其实很简单，但要掌握方法</p>]]></content>
<summary type="html"><link rel="stylesheet" class="aplayer-secondary-style-marker" href="\assets\css\APlayer.min.css"><script src="\assets\js\APlayer.min.js" cla</summary>
<category term="数学笔记" scheme="http://sgnxotsmicf.com/tags/%E6%95%B0%E5%AD%A6%E7%AC%94%E8%AE%B0/"/>
</entry>
<entry>
<title>需要证明的一些东西-加深记忆</title>
<link href="http://sgnxotsmicf.com/2022/03/15/%E9%9C%80%E8%A6%81%E8%AF%81%E6%98%8E%E7%9A%84%E4%B8%80%E4%BA%9B%E4%B8%9C%E8%A5%BF-%E5%8A%A0%E6%B7%B1%E8%AE%B0%E5%BF%86/"/>
<id>http://sgnxotsmicf.com/2022/03/15/%E9%9C%80%E8%A6%81%E8%AF%81%E6%98%8E%E7%9A%84%E4%B8%80%E4%BA%9B%E4%B8%9C%E8%A5%BF-%E5%8A%A0%E6%B7%B1%E8%AE%B0%E5%BF%86/</id>
<published>2022-03-15T13:51:37.000Z</published>
<updated>2022-03-15T13:54:39.557Z</updated>
<content type="html"><![CDATA[<link rel="stylesheet" class="aplayer-secondary-style-marker" href="\assets\css\APlayer.min.css"><script src="\assets\js\APlayer.min.js" class="aplayer-secondary-script-marker"></script><script class="meting-secondary-script-marker" src="\assets\js\Meting.min.js"></script><h4 id="三月"><a href="#三月" class="headerlink" title="三月"></a>三月</h4><blockquote><p>3月15</p></blockquote><ol><li>证明常数项级数收敛的必要条件是其通项极限为0</li></ol>]]></content>
<summary type="html"><link rel="stylesheet" class="aplayer-secondary-style-marker" href="\assets\css\APlayer.min.css"><script src="\assets\js\APlayer.min.js" cla</summary>
<category term="数学证明" scheme="http://sgnxotsmicf.com/categories/%E6%95%B0%E5%AD%A6%E8%AF%81%E6%98%8E/"/>
</entry>
<entry>
<title>英语单词笔记-必记</title>
<link href="http://sgnxotsmicf.com/2022/03/15/%E8%8B%B1%E8%AF%AD%E5%8D%95%E8%AF%8D%E7%AC%94%E8%AE%B0-%E5%BF%85%E8%AE%B0/"/>
<id>http://sgnxotsmicf.com/2022/03/15/%E8%8B%B1%E8%AF%AD%E5%8D%95%E8%AF%8D%E7%AC%94%E8%AE%B0-%E5%BF%85%E8%AE%B0/</id>
<published>2022-03-15T12:06:06.000Z</published>
<updated>2022-03-17T12:33:31.498Z</updated>
<content type="html"><![CDATA[<link rel="stylesheet" class="aplayer-secondary-style-marker" href="\assets\css\APlayer.min.css"><script src="\assets\js\APlayer.min.js" class="aplayer-secondary-script-marker"></script><script class="meting-secondary-script-marker" src="\assets\js\Meting.min.js"></script><h4 id="三月份英语单词笔记-必记"><a href="#三月份英语单词笔记-必记" class="headerlink" title="三月份英语单词笔记-必记"></a>三月份英语单词笔记-必记</h4><blockquote><p>3月15</p></blockquote><ol><li>march 行军,进行……</li><li>March 三月</li><li>neat 整洁的,干净的,优美的,精致的</li><li>queer 奇怪的,古怪的</li><li>terminal 电脑终端</li><li>thigh 大腿 股</li></ol><blockquote><p>3月16</p></blockquote><ol><li>haven 天堂,天空,上帝</li><li>match 比赛、竞赛；匹配,相配</li><li>May 五月</li><li>may 可能,也许</li><li>optical 眼的,视力的,光学的</li><li>civilian 平民,百姓</li></ol><blockquote><p>3月17</p></blockquote><ol><li>smash 打破,打碎,摧毁</li><li>beard 胡须</li><li>reap 收割,收获,获得,得到</li></ol>]]></content>
<summary type="html"><link rel="stylesheet" class="aplayer-secondary-style-marker" href="\assets\css\APlayer.min.css"><script src="\assets\js\APlayer.min.js" cla</summary>
<category term="英语单词" scheme="http://sgnxotsmicf.com/tags/%E8%8B%B1%E8%AF%AD%E5%8D%95%E8%AF%8D/"/>
</entry>
<entry>
<title>不等式总结</title>
<link href="http://sgnxotsmicf.com/2022/01/13/%E4%B8%8D%E7%AD%89%E5%BC%8F%E6%80%BB%E7%BB%93/"/>
<id>http://sgnxotsmicf.com/2022/01/13/%E4%B8%8D%E7%AD%89%E5%BC%8F%E6%80%BB%E7%BB%93/</id>
<published>2022-01-13T12:06:13.000Z</published>
<updated>2022-01-13T12:16:22.302Z</updated>
<content type="html"><![CDATA[<link rel="stylesheet" class="aplayer-secondary-style-marker" href="\assets\css\APlayer.min.css"><script src="\assets\js\APlayer.min.js" class="aplayer-secondary-script-marker"></script><script class="meting-secondary-script-marker" src="\assets\js\Meting.min.js"></script><span id="more"></span><img src="/2022/01/13/%E4%B8%8D%E7%AD%89%E5%BC%8F%E6%80%BB%E7%BB%93/image-20220113201242132.png" class="" title="image-20220113201242132"><p>…</p><p>基本不等式：</p>]]></content>
<summary type="html"><link rel="stylesheet" class="aplayer-secondary-style-marker" href="\assets\css\APlayer.min.css"><script src="\assets\js\APlayer.min.js" cla</summary>
<category term="公式" scheme="http://sgnxotsmicf.com/categories/%E5%85%AC%E5%BC%8F/"/>
<category term="不等式" scheme="http://sgnxotsmicf.com/tags/%E4%B8%8D%E7%AD%89%E5%BC%8F/"/>
</entry>
<entry>
<title>隐函数可微性定理</title>
<link href="http://sgnxotsmicf.com/2022/01/13/%E9%9A%90%E5%87%BD%E6%95%B0%E5%8F%AF%E5%BE%AE%E6%80%A7%E5%AE%9A%E7%90%86/"/>
<id>http://sgnxotsmicf.com/2022/01/13/%E9%9A%90%E5%87%BD%E6%95%B0%E5%8F%AF%E5%BE%AE%E6%80%A7%E5%AE%9A%E7%90%86/</id>
<published>2022-01-13T11:25:02.000Z</published>
<updated>2022-01-13T11:59:02.962Z</updated>
<content type="html"><![CDATA[<link rel="stylesheet" class="aplayer-secondary-style-marker" href="\assets\css\APlayer.min.css"><script src="\assets\js\APlayer.min.js" class="aplayer-secondary-script-marker"></script><script class="meting-secondary-script-marker" src="\assets\js\Meting.min.js"></script><p> 公式：${\Large{\dot f(x) = - { \frac {F_x(x,y)}{F_y(x,y)}}}}$</p><span id="more"></span><h3 id="隐函数可微性定理"><a href="#隐函数可微性定理" class="headerlink" title="隐函数可微性定理"></a>隐函数可微性定理</h3><p>设$F(x,y)$满足隐函数存在唯一性定理(待定)中的条件 i—>iv，又设在D上还存在连续性的偏导数$F_x(x,y)$,则由方程$F(x,y)=0$所确定的隐函数$y=f(x)$在其定义域上$(x_0-{\alpha},x_0+{\alpha})$上有连续导函数，且：</p><p>${\Large{\dot f(x) = - { \frac {F_x(x,y)}{F_y(x,y)}}}}$</p><h3 id="证明"><a href="#证明" class="headerlink" title="证明"></a>证明</h3><p>待定…….</p><h3 id="应用"><a href="#应用" class="headerlink" title="应用"></a>应用</h3><p>一般用来做应用求极值，如面积最值！</p>]]></content>
<summary type="html"><p> 公式：${\Large{\dot f(x) = - { \frac {F_x(x,y)}{F_y(x,y)}}}}$</p></summary>
<category term="函数" scheme="http://sgnxotsmicf.com/categories/%E5%87%BD%E6%95%B0/"/>
<category term="隐函数-01" scheme="http://sgnxotsmicf.com/tags/%E9%9A%90%E5%87%BD%E6%95%B0-01/"/>
</entry>
<entry>
<title>递归思想</title>
<link href="http://sgnxotsmicf.com/2022/01/11/%E9%80%92%E5%BD%92%E6%80%9D%E6%83%B3/"/>
<id>http://sgnxotsmicf.com/2022/01/11/%E9%80%92%E5%BD%92%E6%80%9D%E6%83%B3/</id>
<published>2022-01-11T09:01:42.000Z</published>
<updated>2022-01-13T12:00:42.654Z</updated>
<content type="html"><![CDATA[<link rel="stylesheet" class="aplayer-secondary-style-marker" href="\assets\css\APlayer.min.css"><script src="\assets\js\APlayer.min.js" class="aplayer-secondary-script-marker"></script><script class="meting-secondary-script-marker" src="\assets\js\Meting.min.js"></script><p> 递归已经不陌生了，最重要的是如何求解递归问题？</p><span id="more"></span><blockquote><p>递归公式</p></blockquote><p>首先要有递归公式，通过总结规律来找出n与n+1的转换公式（递归公式），只有拿到公式才能正确快速的找到递归方法，不然就是空中楼阁，对着一个递归问题一个劲的死想，一个劲的调试代码…..然而并没有什么暖用，就算解决出来了也已经晚了，时间就是效率！</p><blockquote><p>递归终点与递归方向</p></blockquote><p>递归终点显然是必须要有的，没有递归终点，不是成死循环了吗？同时递归终点的设立也为递归指明了方向，然最后一步递下去得终点是它就行了，然后通过递归终点一步步把数据归回来！</p><blockquote><p>一道例题</p></blockquote><img src="/2022/01/11/%E9%80%92%E5%BD%92%E6%80%9D%E6%83%B3/image-20220111170951290.png" class="" title="image-20220111170951290"><blockquote><p>最后，这只是有规律的递归，对于没有规律的递归，还应该具体问题具体分析，但不管是哪一种，递归终点与递归方向都是必不可少的！！</p></blockquote>]]></content>
<summary type="html"><p> 递归已经不陌生了，最重要的是如何求解递归问题？</p></summary>
<category term="数学思想" scheme="http://sgnxotsmicf.com/categories/%E6%95%B0%E5%AD%A6%E6%80%9D%E6%83%B3/"/>
<category term="递归" scheme="http://sgnxotsmicf.com/tags/%E9%80%92%E5%BD%92/"/>
</entry>
<entry>
<title>Java学习日记</title>
<link href="http://sgnxotsmicf.com/2022/01/02/Java%E5%AD%A6%E4%B9%A0%E6%97%A5%E8%AE%B0/"/>
<id>http://sgnxotsmicf.com/2022/01/02/Java%E5%AD%A6%E4%B9%A0%E6%97%A5%E8%AE%B0/</id>
<published>2022-01-02T12:17:14.000Z</published>
<updated>2022-01-03T13:03:06.538Z</updated>
<content type="html"><![CDATA[<link rel="stylesheet" class="aplayer-secondary-style-marker" href="\assets\css\APlayer.min.css"><script src="\assets\js\APlayer.min.js" class="aplayer-secondary-script-marker"></script><script class="meting-secondary-script-marker" src="\assets\js\Meting.min.js"></script><p> 用于记录Java日常学习，便于复习</p><span id="more"></span><blockquote><ol><li><p>Stream流：<a href="https://www.bilibili.com/video/BV1Cv411372m?p=141&spm_id_from=pageDriver">这里</a></p></li><li><p>继承和实现可以强转且编译时不报错，但运行时可能会报错！</p></li><li><p>如何自己处理异常：<a href="https://www.bilibili.com/video/BV1Cv411372m?p=143&spm_id_from=pageDriver">编译/运行时异常</a> </p></li><li><p>常用API :Object Object <a href="https://www.bilibili.com/video/BV1Cv411372m?p=117">这里</a></p></li><li><p>常用API :StringBuilder <a href="https://www.bilibili.com/video/BV1Cv411372m?p=118">这里</a></p></li><li><p>常用API :Math System BigDecimal <a href="https://www.bilibili.com/video/BV1Cv411372m?p=119">这里</a></p></li><li><p>常用快捷键：<a href="https://www.cnblogs.com/zhangpengshou/p/5366413.html">这里</a></p></li></ol></blockquote>]]></content>
<summary type="html"><p> 用于记录Java日常学习，便于复习</p></summary>
<category term="日记" scheme="http://sgnxotsmicf.com/categories/%E6%97%A5%E8%AE%B0/"/>
<category term="Java" scheme="http://sgnxotsmicf.com/tags/Java/"/>
</entry>
<entry>
<title>二元函数求导</title>
<link href="http://sgnxotsmicf.com/2021/12/30/%E4%BA%8C%E5%85%83%E5%87%BD%E6%95%B0%E6%B1%82%E5%AF%BC/"/>
<id>http://sgnxotsmicf.com/2021/12/30/%E4%BA%8C%E5%85%83%E5%87%BD%E6%95%B0%E6%B1%82%E5%AF%BC/</id>
<published>2021-12-30T05:26:55.000Z</published>
<updated>2021-12-30T05:32:03.436Z</updated>
<content type="html"><![CDATA[<link rel="stylesheet" class="aplayer-secondary-style-marker" href="\assets\css\APlayer.min.css"><script src="\assets\js\APlayer.min.js" class="aplayer-secondary-script-marker"></script><script class="meting-secondary-script-marker" src="\assets\js\Meting.min.js"></script><p> 单纯的无聊做的，实际一丁点用都没有！</p><span id="more"></span><figure class="highlight java"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br><span class="line">82</span><br><span class="line">83</span><br><span class="line">84</span><br><span class="line">85</span><br><span class="line">86</span><br><span class="line">87</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">package</span> cn.sgnxotsmicf.Demo;</span><br><span class="line"></span><br><span class="line"><span class="comment">//二元多项式求导</span></span><br><span class="line"><span class="keyword">import</span> java.util.ArrayList;</span><br><span class="line"><span class="keyword">import</span> java.util.Scanner;</span><br><span class="line"><span class="keyword">public</span> <span class="class"><span class="keyword">class</span> <span class="title">test0005</span> </span>{</span><br><span class="line"> <span class="function"><span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">main</span><span class="params">(String[] args)</span> </span>{</span><br><span class="line"> ArrayList<Double> <span class="keyword">var</span> = <span class="keyword">new</span> ArrayList<>();<span class="comment">//变量系数</span></span><br><span class="line"> ArrayList<Double> exp = <span class="keyword">new</span> ArrayList<>();<span class="comment">//幂次幂次</span></span><br><span class="line"> ArrayList<Double> varX1 = <span class="keyword">new</span> ArrayList<>();<span class="comment">//求x1时x2幂次</span></span><br><span class="line"> ArrayList<Double> varX2 = <span class="keyword">new</span> ArrayList<>();<span class="comment">//求x2时x1幂次</span></span><br><span class="line"> Scanner sc = <span class="keyword">new</span> Scanner(System.in);</span><br><span class="line"> x1_x2Judge(sc,<span class="keyword">var</span>,exp,varX1,varX2);</span><br><span class="line"> input(<span class="keyword">var</span>,exp,varX1,varX2);</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> <span class="function"><span class="keyword">private</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">x1_x2Judge</span><span class="params">(Scanner sc,ArrayList<Double> <span class="keyword">var</span>,ArrayList<Double> exp,</span></span></span><br><span class="line"><span class="params"><span class="function"> ArrayList<Double> varX1,ArrayList<Double> varX2)</span> </span>{</span><br><span class="line"> <span class="keyword">int</span> i =<span class="number">0</span>;</span><br><span class="line"> System.out.println(<span class="string">"输入1对x1求导:"</span>);</span><br><span class="line"> <span class="keyword">int</span> instruct1 = sc.nextInt();</span><br><span class="line"> System.out.println(<span class="string">"请输入关于x"</span>+instruct1+<span class="string">"的项数:"</span>);</span><br><span class="line"> <span class="keyword">int</span> number1 = sc.nextInt();</span><br><span class="line"> <span class="keyword">int</span> tempt =operation(sc,<span class="keyword">var</span>,exp,i,number1,instruct1,varX1,varX2);</span><br><span class="line"> System.out.println(<span class="string">"输入2对x2求导:"</span>);</span><br><span class="line"> <span class="keyword">int</span> instruct2= sc.nextInt();</span><br><span class="line"> System.out.println(<span class="string">"请输入关于x"</span>+instruct2+<span class="string">"的项数:"</span>);</span><br><span class="line"> <span class="keyword">int</span> number2 = sc.nextInt();</span><br><span class="line"> operation(sc,<span class="keyword">var</span>,exp,tempt,number2,instruct2,varX1,varX2);</span><br><span class="line"> }</span><br><span class="line"> <span class="function"><span class="keyword">private</span> <span class="keyword">static</span> <span class="keyword">int</span> <span class="title">operation</span><span class="params">(Scanner sc,ArrayList<Double> <span class="keyword">var</span>,ArrayList<Double> exp,<span class="keyword">int</span> i,<span class="keyword">int</span> number,<span class="keyword">int</span> instruct,</span></span></span><br><span class="line"><span class="params"><span class="function"> ArrayList<Double> varX1,ArrayList<Double> varX2)</span></span>{</span><br><span class="line"> <span class="keyword">int</span> tempt = <span class="number">0</span>;</span><br><span class="line"> <span class="keyword">do</span> {</span><br><span class="line"> System.out.println(<span class="string">"请输入第"</span> + (tempt + <span class="number">1</span>) + <span class="string">"个变量系数:"</span>);</span><br><span class="line"> <span class="keyword">var</span>.add(sc.nextDouble());</span><br><span class="line"> System.out.println(<span class="string">"请输入第"</span> + (tempt + <span class="number">1</span>) + <span class="string">"个变量幂次:"</span>);</span><br><span class="line"> exp.add(sc.nextDouble());</span><br><span class="line"> <span class="keyword">if</span> (instruct == <span class="number">1</span>) {</span><br><span class="line"> derivation_Operation(<span class="keyword">var</span>, exp, i);</span><br><span class="line"> }<span class="keyword">else</span> <span class="keyword">if</span> (instruct == <span class="number">2</span>){</span><br><span class="line"> derivation_Operation(<span class="keyword">var</span>,exp,i);</span><br><span class="line"> }</span><br><span class="line"> Judge(instruct,sc,varX1,varX2);</span><br><span class="line"> i++;</span><br><span class="line"> tempt++;</span><br><span class="line"> } <span class="keyword">while</span> (tempt < number);</span><br><span class="line"> <span class="keyword">return</span> i;</span><br><span class="line"> }</span><br><span class="line"> <span class="function"><span class="keyword">private</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">Judge</span><span class="params">(<span class="keyword">int</span> instruct,Scanner sc,ArrayList<Double> varX1,ArrayList<Double> varX2)</span> </span>{</span><br><span class="line"> <span class="keyword">if</span> (instruct==<span class="number">1</span>){</span><br><span class="line"> System.out.println(<span class="string">"该x"</span>+instruct+<span class="string">"项(有/无)x2项相乘？"</span>);</span><br><span class="line"> varX1_X2_Operation(sc,varX1);</span><br><span class="line"> }<span class="keyword">else</span> <span class="keyword">if</span> (instruct==<span class="number">2</span>){</span><br><span class="line"> System.out.println(<span class="string">"该x"</span>+instruct+<span class="string">"项(有/无)x1项相乘？"</span>);</span><br><span class="line"> varX1_X2_Operation(sc,varX2);</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> <span class="function"><span class="keyword">private</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">varX1_X2_Operation</span><span class="params">(Scanner sc,ArrayList<Double> varX)</span> </span>{</span><br><span class="line"> String answer = sc.next();</span><br><span class="line"> <span class="keyword">if</span> (answer.equals(<span class="string">"有"</span>)){</span><br><span class="line"> System.out.println(<span class="string">"幂次为多少？"</span>);</span><br><span class="line"> <span class="keyword">double</span> x = sc.nextInt();</span><br><span class="line"> varX.add(x);</span><br><span class="line"> }<span class="keyword">else</span> <span class="keyword">if</span> (answer.equals(<span class="string">"无"</span>)){</span><br><span class="line"> varX.add(<span class="number">0.0</span>);</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> <span class="function"><span class="keyword">private</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">input</span><span class="params">(ArrayList<Double> <span class="keyword">var</span>, ArrayList<Double> exp, ArrayList<Double> varX1,ArrayList<Double> varX2)</span> </span>{</span><br><span class="line"> System.out.println(<span class="string">"变量系数:"</span>+<span class="keyword">var</span>.toString());</span><br><span class="line"> System.out.println(<span class="string">"变量幂次:"</span>+exp.toString());</span><br><span class="line"> System.out.println(<span class="string">"求x1时x2幂次:"</span>+varX1.toString());</span><br><span class="line"> System.out.println(<span class="string">"求x2时x1幂次:"</span>+varX2.toString());</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> <span class="function"><span class="keyword">private</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">derivation_Operation</span><span class="params">(ArrayList<Double> <span class="keyword">var</span>,ArrayList<Double> exp,<span class="keyword">int</span> i)</span> </span>{</span><br><span class="line"> <span class="keyword">if</span> (<span class="keyword">var</span>.get(i)==<span class="number">0</span>||exp.get(i)==<span class="number">0</span>){</span><br><span class="line"> <span class="keyword">var</span>.set(i,<span class="number">0.0</span>);</span><br><span class="line"> exp.set(i,<span class="number">0.0</span>);</span><br><span class="line"> }<span class="keyword">else</span> {</span><br><span class="line"> <span class="keyword">var</span>.set(i,<span class="keyword">var</span>.get(i)*exp.get(i));</span><br><span class="line"> exp.set(i,exp.get(i)-<span class="number">1</span>);</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line">}</span><br></pre></td></tr></table></figure>]]></content>
<summary type="html"><p> 单纯的无聊做的，实际一丁点用都没有！</p></summary>
<category term="兴趣" scheme="http://sgnxotsmicf.com/categories/%E5%85%B4%E8%B6%A3/"/>
<category term="java" scheme="http://sgnxotsmicf.com/tags/java/"/>
<category term="求导" scheme="http://sgnxotsmicf.com/tags/%E6%B1%82%E5%AF%BC/"/>
</entry>
<entry>
<title>二次型与二次曲面</title>
<link href="http://sgnxotsmicf.com/2021/12/30/%E4%BA%8C%E6%AC%A1%E5%9E%8B%E4%B8%8E%E4%BA%8C%E6%AC%A1%E6%9B%B2%E9%9D%A2/"/>
<id>http://sgnxotsmicf.com/2021/12/30/%E4%BA%8C%E6%AC%A1%E5%9E%8B%E4%B8%8E%E4%BA%8C%E6%AC%A1%E6%9B%B2%E9%9D%A2/</id>
<published>2021-12-30T05:26:52.000Z</published>
<updated>2021-12-30T05:58:56.116Z</updated>
<content type="html"><![CDATA[<link rel="stylesheet" class="aplayer-secondary-style-marker" href="\assets\css\APlayer.min.css"><script src="\assets\js\APlayer.min.js" class="aplayer-secondary-script-marker"></script><script class="meting-secondary-script-marker" src="\assets\js\Meting.min.js"></script><p> 来源于网络！！！用来总结复习！！！</p><span id="more"></span> <div class="pdf-container" data-target="./1.pdf" data-height="550px"></div> <div class="pdf-container" data-target="./2.pdf" data-height="550px"></div> <div class="pdf-container" data-target="./3.pdf" data-height="550px"></div> <div class="pdf-container" data-target="./4.pdf" data-height="550px"></div> <div class="pdf-container" data-target="./5.pdf" data-height="550px"></div> ]]></content>
<summary type="html"><p> 来源于网络！！！用来总结复习！！！</p></summary>
<category term="高等代数" scheme="http://sgnxotsmicf.com/categories/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/"/>
<category term="二次型" scheme="http://sgnxotsmicf.com/tags/%E4%BA%8C%E6%AC%A1%E5%9E%8B/"/>
<category term="二次曲面" scheme="http://sgnxotsmicf.com/tags/%E4%BA%8C%E6%AC%A1%E6%9B%B2%E9%9D%A2/"/>
</entry>
<entry>
<title>可逆矩阵的一些性质(一)</title>
<link href="http://sgnxotsmicf.com/2021/12/30/%E5%8F%AF%E9%80%86%E7%9F%A9%E9%98%B5%E7%9A%84%E4%B8%80%E4%BA%9B%E6%80%A7%E8%B4%A8(%E4%B8%80)/"/>
<id>http://sgnxotsmicf.com/2021/12/30/%E5%8F%AF%E9%80%86%E7%9F%A9%E9%98%B5%E7%9A%84%E4%B8%80%E4%BA%9B%E6%80%A7%E8%B4%A8(%E4%B8%80)/</id>
<published>2021-12-30T05:17:44.000Z</published>
<updated>2021-12-30T05:27:17.238Z</updated>
<content type="html"><![CDATA[<link rel="stylesheet" class="aplayer-secondary-style-marker" href="\assets\css\APlayer.min.css"><script src="\assets\js\APlayer.min.js" class="aplayer-secondary-script-marker"></script><script class="meting-secondary-script-marker" src="\assets\js\Meting.min.js"></script><ol><li><p>可逆矩阵，说明该方阵各向量线性无关，因为如果各向量线性相关，就不可能是可逆矩阵。</p><span id="more"></span></li><li><p>如果一个方阵乘以非零向量，结果是0向量，那么说明以该非零向量各元素为系数，和该方阵各行向量相乘后相加，能得到0向量。<br>而非零向量的元素不能全部为0<br>所以就说明存在一组不全为0的系数，使得系数和行向量相乘后相加，结果为0向量。<br>这就说明行向量线性相关（线性相关的定义）<br>所以就不是可逆矩阵。<br><em>因此可逆矩阵乘以一个非零向量的结果不可能是0向量</em></p></li></ol>]]></content>
<summary type="html"><ol>
<li><p>可逆矩阵，说明该方阵各向量线性无关，因为如果各向量线性相关，就不可能是可逆矩阵。</p></summary>
<category term="高等代数" scheme="http://sgnxotsmicf.com/categories/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/"/>
<category term="矩阵" scheme="http://sgnxotsmicf.com/tags/%E7%9F%A9%E9%98%B5/"/>
</entry>
<entry>
<title>降阶公式与矩阵秩的性质</title>
<link href="http://sgnxotsmicf.com/2021/12/26/%E9%99%8D%E9%98%B6%E5%85%AC%E5%BC%8F%E4%B8%8E%E7%9F%A9%E9%98%B5%E7%A7%A9%E7%9A%84%E5%85%AC%E5%BC%8F/"/>
<id>http://sgnxotsmicf.com/2021/12/26/%E9%99%8D%E9%98%B6%E5%85%AC%E5%BC%8F%E4%B8%8E%E7%9F%A9%E9%98%B5%E7%A7%A9%E7%9A%84%E5%85%AC%E5%BC%8F/</id>
<published>2021-12-26T08:26:44.000Z</published>
<updated>2021-12-26T15:52:56.639Z</updated>
<content type="html"><![CDATA[<link rel="stylesheet" class="aplayer-secondary-style-marker" href="\assets\css\APlayer.min.css"><script src="\assets\js\APlayer.min.js" class="aplayer-secondary-script-marker"></script><script class="meting-secondary-script-marker" src="\assets\js\Meting.min.js"></script><h3 id="降阶公式"><a href="#降阶公式" class="headerlink" title="降阶公式"></a>降阶公式</h3><script type="math/tex; mode=display">\begin{vmatrix}{\lambda}{E_{m}-AB}\end{vmatrix}={\lambda}^{m-n}\begin{vmatrix}{\lambda}{E_{n}}{-BA}\end{vmatrix}</script><p>其中$E_{m}{}$ 、 $E_{n}$分别为$m阶、n阶$单位阵</p><span id="more"></span><h3 id="矩阵秩的性质"><a href="#矩阵秩的性质" class="headerlink" title="矩阵秩的性质"></a>矩阵秩的性质</h3><ol><li><p>$0{\le}{R(A)}{\le}{min}${$m,n$}</p></li><li><p>$\begin{cases}0,k=0 \\ {R(A)},k{\not=0}<br> \end{cases}$</p></li><li><p>$R(A_1){\le}{R(A)}$ 其中，$A_1是A的子矩阵$</p></li><li><p>$R(A^T)=R(A)$ 其中，$A^T是A的转置$</p></li><li><p>$A{\rightarrow}^{初等变换}{\rightarrow}B{\Rightarrow}{R(A)=R(B)}$</p></li><li><p>$R\begin{pmatrix}A&0 \\ 0&B\end{pmatrix}$记$R(A)=r_1{\quad} R(B)=r_2$</p><p>= $r_1+r_2$</p></li><li><p>$R(A,B)\le{R(A)+R(B)}$</p></li><li><p>$R(A+B)\le{R(A)+R(B)}$</p></li><li><p>$R(AB)\le{min\{ R\big(A),R\big(B)\}}$</p></li><li><p>$\cal<strong>重点</strong>:$</p><p>设$A\in{F^{m\times{n}}},B\in{F^{m\times{n}}}$</p><p>$R(A)+R(B)-n\le{R\big(AB)}\le{min\{R(A),R(B)\}}$</p><p>特别地，若$AB=0$,则</p><p>$R(A)+R(B)\le{n}$</p></li></ol>]]></content>
<summary type="html"><h3 id="降阶公式"><a href="#降阶公式" class="headerlink" title="降阶公式"></a>降阶公式</h3><script type="math/tex; mode=display">
\begin{vmatrix}{\lambda}{E_{m}-AB}\end{vmatrix}={\lambda}^{m-n}\begin{vmatrix}{\lambda}{E_{n}}{-BA}\end{vmatrix}</script><p>其中$E_{m}{}$ 、 $E_{n}$分别为$m阶、n阶$单位阵</p></summary>
<category term="高等代数" scheme="http://sgnxotsmicf.com/categories/%E9%AB%98%E7%AD%89%E4%BB%A3%E6%95%B0/"/>
<category term="矩阵" scheme="http://sgnxotsmicf.com/tags/%E7%9F%A9%E9%98%B5/"/>
<category term="公式" scheme="http://sgnxotsmicf.com/tags/%E5%85%AC%E5%BC%8F/"/>
<category term="秩" scheme="http://sgnxotsmicf.com/tags/%E7%A7%A9/"/>
</entry>
<entry>
<title>blog搭建日记</title>
<link href="http://sgnxotsmicf.com/2021/12/26/blog%E6%90%AD%E5%BB%BA%E6%97%A5%E8%AE%B0/"/>
<id>http://sgnxotsmicf.com/2021/12/26/blog%E6%90%AD%E5%BB%BA%E6%97%A5%E8%AE%B0/</id>
<published>2021-12-26T05:12:55.962Z</published>
<updated>2021-12-31T12:08:24.485Z</updated>
<content type="html"><![CDATA[<link rel="stylesheet" class="aplayer-secondary-style-marker" href="\assets\css\APlayer.min.css"><script src="\assets\js\APlayer.min.js" class="aplayer-secondary-script-marker"></script><script class="meting-secondary-script-marker" src="\assets\js\Meting.min.js"></script><p> 用来记录我对hexo+next的更改记录，方便以后查询……</p><span id="more"></span><ol><li>点击特效(爱心)</li></ol><p> 来源于： <a href="https://www.jianshu.com/p/924b3eea4fa6">这里</a></p><ol><li><p>看板娘</p><p>来源于：<a href="https://afuya.blog.csdn.net/article/details/107341035">这里</a></p></li><li><p>pdf</p><p>来源于：<a href="https://ugirc.blog.csdn.net/article/details/104574959">这里</a></p></li></ol>]]></content>
<summary type="html"><p> 用来记录我对hexo+next的更改记录，方便以后查询……</p></summary>
<category term="日记" scheme="http://sgnxotsmicf.com/categories/%E6%97%A5%E8%AE%B0/"/>
</entry>
<entry>
<title>0.618法</title>
<link href="http://sgnxotsmicf.com/2021/12/26/0.618%E6%B3%95/"/>
<id>http://sgnxotsmicf.com/2021/12/26/0.618%E6%B3%95/</id>
<published>2021-12-26T04:26:55.000Z</published>
<updated>2021-12-26T05:47:19.176Z</updated>
<content type="html"><![CDATA[<link rel="stylesheet" class="aplayer-secondary-style-marker" href="\assets\css\APlayer.min.css"><script src="\assets\js\APlayer.min.js" class="aplayer-secondary-script-marker"></script><script class="meting-secondary-script-marker" src="\assets\js\Meting.min.js"></script><p> 一种经典的一维搜索方法</p><span id="more"></span><p>用的集合装，实际应该更简单！</p><figure class="highlight java"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">package</span> cn.sgnxotsmicf.Demo;</span><br><span class="line"><span class="keyword">import</span> java.text.DecimalFormat;</span><br><span class="line"><span class="keyword">import</span> java.time.LocalDateTime;</span><br><span class="line"><span class="keyword">import</span> java.time.temporal.ChronoUnit;</span><br><span class="line"><span class="keyword">import</span> java.util.ArrayList;</span><br><span class="line"><span class="keyword">import</span> java.util.Scanner;</span><br><span class="line"><span class="keyword">public</span> <span class="class"><span class="keyword">class</span> <span class="title">Demo_Number</span> </span>{</span><br><span class="line"> <span class="keyword">static</span><span class="keyword">double</span> T = <span class="number">0.618</span>;</span><br><span class="line"> <span class="keyword">static</span> {</span><br><span class="line"> System.out.println(<span class="string">"======0.618法求解最优化问题======"</span>);</span><br><span class="line"> }</span><br><span class="line"> <span class="function"><span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">main</span><span class="params">(String[] args)</span> </span>{</span><br><span class="line"> Scanner sc = <span class="keyword">new</span> Scanner(System.in);</span><br><span class="line"> ArrayList<Double> Data = <span class="keyword">new</span> ArrayList<>();</span><br><span class="line"> inPut(sc,Data);</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> <span class="function"><span class="keyword">private</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">inPut</span><span class="params">(Scanner sc,ArrayList<Double> Data)</span> </span>{</span><br><span class="line"> System.out.println(<span class="string">"请输入允许误差ε:"</span>);</span><br><span class="line"> <span class="keyword">double</span> c = sc.nextDouble();</span><br><span class="line"> System.out.println(<span class="string">"给定搜索左区间a:"</span>);</span><br><span class="line"> <span class="keyword">double</span> a = sc.nextInt();</span><br><span class="line"> System.out.println(<span class="string">"给定搜索右区间b:"</span>);</span><br><span class="line"> <span class="keyword">double</span> b = sc.nextInt();</span><br><span class="line"> LocalDateTime startTime = LocalDateTime.now();</span><br><span class="line"> Init(Data,a,b);</span><br><span class="line"> Judge(Data,c);</span><br><span class="line"> LocalDateTime endTime = LocalDateTime.now();</span><br><span class="line"> System.out.println(<span class="string">"迭代时间大约为:"</span>+ChronoUnit.MICROS.between(startTime, endTime)+<span class="string">"微秒"</span>);</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> <span class="comment">/**</span></span><br><span class="line"><span class="comment"> ！！！注意：函数在此方法下修改</span></span><br><span class="line"><span class="comment"> */</span></span><br><span class="line"> <span class="function"><span class="keyword">private</span> <span class="keyword">static</span> <span class="keyword">double</span> <span class="title">function</span><span class="params">(<span class="keyword">double</span> tempt)</span></span>{</span><br><span class="line"> <span class="keyword">return</span> tempt*tempt-tempt+<span class="number">2</span>;<span class="comment">//0.554</span></span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> <span class="function"><span class="keyword">private</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">Init</span><span class="params">(ArrayList<Double> Data,<span class="keyword">double</span> a, <span class="keyword">double</span> b)</span> </span>{</span><br><span class="line"> <span class="keyword">double</span> L = a+(<span class="number">1</span>-T)*(b-a);</span><br><span class="line"> <span class="keyword">double</span> U = a+T*(b-a);</span><br><span class="line"> <span class="keyword">double</span> L1 = function(L);</span><br><span class="line"> <span class="keyword">double</span> U1 = function(U);</span><br><span class="line"> Data.add(L);<span class="comment">//0</span></span><br><span class="line"> Data.add(U);<span class="comment">//1</span></span><br><span class="line"> Data.add(L1);<span class="comment">//2</span></span><br><span class="line"> Data.add(U1);<span class="comment">//3</span></span><br><span class="line"> Data.add(a);<span class="comment">//4</span></span><br><span class="line"> Data.add(b);<span class="comment">//5</span></span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> <span class="function"><span class="keyword">private</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">Judge</span><span class="params">(ArrayList<Double> Data,<span class="keyword">double</span> c)</span> </span>{</span><br><span class="line"> <span class="keyword">while</span>(<span class="keyword">true</span>) {</span><br><span class="line"> <span class="keyword">if</span>(Math.abs(Data.get(<span class="number">4</span>)-Data.get(<span class="number">5</span>))>c) {</span><br><span class="line"> <span class="keyword">if</span>(Data.get(<span class="number">2</span>)<Data.get(<span class="number">3</span>)) {</span><br><span class="line"> Data.set(<span class="number">5</span>, Data.get(<span class="number">1</span>));<span class="comment">//b的值</span></span><br><span class="line"> Data.set(<span class="number">1</span>, Data.get(<span class="number">0</span>));</span><br><span class="line"> Data.set(<span class="number">3</span>, Data.get(<span class="number">2</span>));</span><br><span class="line"> Data.set(<span class="number">0</span>, Data.get(<span class="number">4</span>)+(<span class="number">1</span>-T)*(Data.get(<span class="number">5</span>)-Data.get(<span class="number">4</span>)));</span><br><span class="line"> <span class="keyword">double</span> tempt = Data.set(<span class="number">0</span>, Data.get(<span class="number">4</span>)+(<span class="number">1</span>-T)*(Data.get(<span class="number">5</span>)-Data.get(<span class="number">4</span>)));</span><br><span class="line"> Data.set(<span class="number">2</span>, function(tempt));</span><br><span class="line"> }<span class="keyword">else</span> {</span><br><span class="line"> Data.set(<span class="number">4</span>, Data.get(<span class="number">0</span>));<span class="comment">//a的值</span></span><br><span class="line"> Data.set(<span class="number">0</span>,Data.get(<span class="number">1</span>));</span><br><span class="line"> Data.set(<span class="number">2</span>,Data.get(<span class="number">3</span>));</span><br><span class="line"> Data.set(<span class="number">1</span>,Data.get(<span class="number">4</span>)+T*(Data.get(<span class="number">5</span>)-Data.get(<span class="number">4</span>)));</span><br><span class="line"> <span class="keyword">double</span> tempt = Data.set(<span class="number">1</span>,Data.get(<span class="number">4</span>)+T*(Data.get(<span class="number">5</span>)-Data.get(<span class="number">4</span>)));</span><br><span class="line"> Data.set(<span class="number">3</span>,function(tempt));</span><br><span class="line"> }</span><br><span class="line"> }<span class="keyword">else</span> {</span><br><span class="line"> DecimalFormat sc = <span class="keyword">new</span> DecimalFormat(<span class="string">"0.000"</span>);</span><br><span class="line"> System.out.println(<span class="string">"最优解x="</span>+sc.format((Data.get(<span class="number">4</span>)+Data.get(<span class="number">5</span>))/<span class="number">2</span>));</span><br><span class="line"> <span class="keyword">break</span>;</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line">}</span><br></pre></td></tr></table></figure>]]></content>
<summary type="html"><p> 一种经典的一维搜索方法</p></summary>
<category term="最优化方法" scheme="http://sgnxotsmicf.com/categories/%E6%9C%80%E4%BC%98%E5%8C%96%E6%96%B9%E6%B3%95/"/>
<category term="java" scheme="http://sgnxotsmicf.com/tags/java/"/>
<category term="最优化算法" scheme="http://sgnxotsmicf.com/tags/%E6%9C%80%E4%BC%98%E5%8C%96%E7%AE%97%E6%B3%95/"/>
</entry>
<entry>
<title>阻尼Newton法</title>
<link href="http://sgnxotsmicf.com/2021/12/25/%E9%98%BB%E5%B0%BCNewton%E6%B3%95/"/>
<id>http://sgnxotsmicf.com/2021/12/25/%E9%98%BB%E5%B0%BCNewton%E6%B3%95/</id>
<published>2021-12-25T15:50:55.000Z</published>
<updated>2021-12-26T04:49:32.168Z</updated>
<content type="html"><![CDATA[<link rel="stylesheet" class="aplayer-secondary-style-marker" href="\assets\css\APlayer.min.css"><script src="\assets\js\APlayer.min.js" class="aplayer-secondary-script-marker"></script><script class="meting-secondary-script-marker" src="\assets\js\Meting.min.js"></script><p> 一般用来寻找二元函数最优解</p><span id="more"></span><figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">function</span> <span class="title">ZN_Newton</span><span class="params">(fun,x0,epsilon)</span></span></span><br><span class="line"><span class="comment">% 阻尼Newton法</span></span><br><span class="line"><span class="comment">% fun = (1-x1)^2+2*(x2-x1^2)^2; x0 = [0;0]; epsilon = 0.1;</span></span><br><span class="line"><span class="comment">% fun = (x1-1)^4+(x1-x2)^2; x0 = [0;0]; epsilon = 10^-6;</span></span><br><span class="line"><span class="comment">% fun = x1^2+x2^2-3*x1-x1*x2; x0 =[0;0];epsilon = 0.1;</span></span><br><span class="line"><span class="comment">% fun = (x1-2)^2+(x1-2*x2)^2; x0 =[0;3];epsilon = 0.4;</span></span><br><span class="line"><span class="comment">% clear;clc</span></span><br><span class="line"><span class="comment">% by -- 2021.12.9 -- cn.sgnxotsmicf.Lx</span></span><br><span class="line">clc</span><br><span class="line"><span class="built_in">disp</span>(<span class="string">'==================阻尼Newton法=================='</span>)</span><br><span class="line">syms x1 x2</span><br><span class="line">n = <span class="built_in">length</span>(x0); <span class="comment">% 解的长度</span></span><br><span class="line">k = <span class="number">0</span>; <span class="comment">% 迭代次数</span></span><br><span class="line"><span class="comment">% 初始化值</span></span><br><span class="line">grad_f = [diff(fun,x1);diff(fun,x2)];</span><br><span class="line">Hesse = hessian(fun,[x1,x2]);</span><br><span class="line">x = <span class="built_in">zeros</span>(n,<span class="number">10</span>); </span><br><span class="line">y = <span class="built_in">zeros</span>(n,<span class="number">10</span>);</span><br><span class="line">index = <span class="number">1</span>; <span class="comment">% 索引</span></span><br><span class="line">x(:,index) = x0;</span><br><span class="line"><span class="comment">% 进入迭代逻辑</span></span><br><span class="line"><span class="keyword">while</span>(<span class="built_in">true</span>)</span><br><span class="line"> syms lamuda <span class="built_in">real</span></span><br><span class="line"> gk = subs(grad_f,[x1;x2],x(:,index));</span><br><span class="line"> gk = double(gk);</span><br><span class="line"> gk_Hesse = subs(Hesse,[x1;x2],x(:,index));</span><br><span class="line"> y(:,index) = gk;</span><br><span class="line"> <span class="keyword">if</span>(norm(gk) > epsilon) <span class="comment">% 精度判断</span></span><br><span class="line"> dk = - gk_Hesse^<span class="number">-1</span> * gk;</span><br><span class="line"> f1 = subs(fun,[x1;x2],x(:,index)+lamuda * dk);</span><br><span class="line"> f1 = diff(f1);</span><br><span class="line"> lamuda = solve(f1); </span><br><span class="line"> lamuda = <span class="built_in">max</span>(lamuda); <span class="comment">% 进入实数循环</span></span><br><span class="line"> <span class="keyword">if</span>(lamuda ~= <span class="number">0</span>)</span><br><span class="line"> x(:,index+<span class="number">1</span>) = x(:,index) + lamuda*dk;</span><br><span class="line"> <span class="keyword">else</span></span><br><span class="line"> <span class="keyword">break</span>;</span><br><span class="line"> <span class="keyword">end</span> </span><br><span class="line"> <span class="keyword">else</span></span><br><span class="line"> <span class="comment">% 迭代结束进行输出</span></span><br><span class="line"> val = x(:,index);</span><br><span class="line"> fprintf(<span class="string">'阻尼Newton所求的函数为:%s\n迭代次数为;%d次\n'</span>,fun,k)</span><br><span class="line"> str = [<span class="string">'最优解为:['</span>,num2str(val'),<span class="string">']'</span>];</span><br><span class="line"> <span class="built_in">disp</span>(str)</span><br><span class="line"> <span class="keyword">break</span>;</span><br><span class="line"> <span class="keyword">end</span></span><br><span class="line"> index = index + <span class="number">1</span>;</span><br><span class="line"> k = k + <span class="number">1</span>;</span><br><span class="line"><span class="keyword">end</span></span><br><span class="line"><span class="keyword">end</span></span><br></pre></td></tr></table></figure>]]></content>
<summary type="html"><p> 一般用来寻找二元函数最优解</p></summary>
<category term="最优化方法" scheme="http://sgnxotsmicf.com/categories/%E6%9C%80%E4%BC%98%E5%8C%96%E6%96%B9%E6%B3%95/"/>
<category term="最优化算法" scheme="http://sgnxotsmicf.com/tags/%E6%9C%80%E4%BC%98%E5%8C%96%E7%AE%97%E6%B3%95/"/>
<category term="matlab" scheme="http://sgnxotsmicf.com/tags/matlab/"/>
</entry>
<entry>
<title>单纯性表</title>
<link href="http://sgnxotsmicf.com/2021/12/25/%E5%8D%95%E7%BA%AF%E6%80%A7%E8%A1%A8/"/>
<id>http://sgnxotsmicf.com/2021/12/25/%E5%8D%95%E7%BA%AF%E6%80%A7%E8%A1%A8/</id>
<published>2021-12-25T15:26:55.000Z</published>
<updated>2021-12-25T15:44:08.138Z</updated>
<content type="html"><![CDATA[<link rel="stylesheet" class="aplayer-secondary-style-marker" href="\assets\css\APlayer.min.css"><script src="\assets\js\APlayer.min.js" class="aplayer-secondary-script-marker"></script><script class="meting-secondary-script-marker" src="\assets\js\Meting.min.js"></script><p>用来求解线性规划问题</p><span id="more"></span><figure class="highlight java"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">package</span> cn.sgnxotsmicf.Demo;</span><br><span class="line"><span class="keyword">import</span> java.text.DecimalFormat;</span><br><span class="line"><span class="keyword">import</span> java.util.Scanner;</span><br><span class="line"></span><br><span class="line"><span class="keyword">public</span> <span class="class"><span class="keyword">class</span> <span class="title">Simplex_Method_Data</span> </span>{</span><br><span class="line"> <span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">double</span>[] c; <span class="comment">// 目标函数中变量xj的价值系数cj</span></span><br><span class="line"> <span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">double</span>[] pi; <span class="comment">// 检验系数π</span></span><br><span class="line"> <span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">double</span>[] theta; <span class="comment">// 入基时对应的θ</span></span><br><span class="line"> <span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">double</span>[] b0; <span class="comment">// 基本可行解中基变量的值</span></span><br><span class="line"> <span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">int</span> m; <span class="comment">// 秩</span></span><br><span class="line"> <span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">int</span> n; <span class="comment">// 变量个数</span></span><br><span class="line"> <span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">int</span>[] basicVar; <span class="comment">//用来存储基变量的索引</span></span><br><span class="line"> <span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">double</span> <span class="keyword">var</span>;</span><br><span class="line"> <span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">int</span>[] MData;</span><br><span class="line"> <span class="keyword">public</span> <span class="keyword">static</span> String YN;</span><br><span class="line"> <span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">double</span>[][] centreMatrix_Creat(Scanner sc){</span><br><span class="line"> System.out.println(<span class="string">"请输入约束矩阵A的秩:"</span>);</span><br><span class="line"> m = sc.nextInt();</span><br><span class="line"> System.out.println(<span class="string">"请输入约束矩阵变量个数:"</span>); <span class="comment">// 同时也确定了基变量和非基变量的个数</span></span><br><span class="line"> n = sc.nextInt();</span><br><span class="line"> <span class="keyword">double</span>[][]A = <span class="keyword">new</span> <span class="keyword">double</span>[m][n];</span><br><span class="line"> inputA(A,sc);</span><br><span class="line"> <span class="keyword">return</span> A;</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> <span class="comment">/**</span></span><br><span class="line"><span class="comment"> * 用于初始化A</span></span><br><span class="line"><span class="comment"> * <span class="doctag">@param</span> A 为初始约束条件系数矩阵</span></span><br><span class="line"><span class="comment"> */</span></span><br><span class="line"> <span class="function"><span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">inputA</span><span class="params">(<span class="keyword">double</span>[][]A,Scanner sc)</span> </span>{</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i < m; i++) {</span><br><span class="line"> System.out.println(<span class="string">"请输入第"</span>+(i+<span class="number">1</span>)+<span class="string">"行元素:"</span>);</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> j = <span class="number">0</span>; j < n; j++) {</span><br><span class="line"> A[i][j] = sc.nextDouble();</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> <span class="comment">/**</span></span><br><span class="line"><span class="comment"> * 打印单纯性表</span></span><br><span class="line"><span class="comment"> */</span></span><br><span class="line"> <span class="function"><span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">printMatrix</span><span class="params">(<span class="keyword">double</span>[][] E)</span></span>{</span><br><span class="line"> DecimalFormat sc = <span class="keyword">new</span> DecimalFormat(<span class="string">"0.00"</span>);</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i < m; i++) {</span><br><span class="line"> System.out.print(sc.format(b0[i])+<span class="string">","</span>);</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> j = <span class="number">0</span>; j < n; j++) {</span><br><span class="line"> System.out.print(sc.format(E[i][j])+(j==n-<span class="number">1</span>?<span class="string">""</span>:<span class="string">","</span>));</span><br><span class="line"> }</span><br><span class="line"> System.out.println();</span><br><span class="line"> }</span><br><span class="line"> System.out.print(<span class="keyword">var</span>+<span class="string">","</span>);</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> j = <span class="number">0</span>; j < n; j++) {</span><br><span class="line"> System.out.print(sc.format(pi[j])+(j==n-<span class="number">1</span>?<span class="string">""</span>:<span class="string">","</span>));</span><br><span class="line"> }</span><br><span class="line"> System.out.println();</span><br><span class="line"> }</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line"></span><br></pre></td></tr></table></figure><figure class="highlight java"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br><span class="line">82</span><br><span class="line">83</span><br><span class="line">84</span><br><span class="line">85</span><br><span class="line">86</span><br><span class="line">87</span><br><span class="line">88</span><br><span class="line">89</span><br><span class="line">90</span><br><span class="line">91</span><br><span class="line">92</span><br><span class="line">93</span><br><span class="line">94</span><br><span class="line">95</span><br><span class="line">96</span><br><span class="line">97</span><br><span class="line">98</span><br><span class="line">99</span><br><span class="line">100</span><br><span class="line">101</span><br><span class="line">102</span><br><span class="line">103</span><br><span class="line">104</span><br><span class="line">105</span><br><span class="line">106</span><br><span class="line">107</span><br><span class="line">108</span><br><span class="line">109</span><br><span class="line">110</span><br><span class="line">111</span><br><span class="line">112</span><br><span class="line">113</span><br><span class="line">114</span><br><span class="line">115</span><br><span class="line">116</span><br><span class="line">117</span><br><span class="line">118</span><br><span class="line">119</span><br><span class="line">120</span><br><span class="line">121</span><br><span class="line">122</span><br><span class="line">123</span><br><span class="line">124</span><br><span class="line">125</span><br><span class="line">126</span><br><span class="line">127</span><br><span class="line">128</span><br><span class="line">129</span><br><span class="line">130</span><br><span class="line">131</span><br><span class="line">132</span><br><span class="line">133</span><br><span class="line">134</span><br><span class="line">135</span><br><span class="line">136</span><br><span class="line">137</span><br><span class="line">138</span><br><span class="line">139</span><br><span class="line">140</span><br><span class="line">141</span><br><span class="line">142</span><br><span class="line">143</span><br><span class="line">144</span><br><span class="line">145</span><br><span class="line">146</span><br><span class="line">147</span><br><span class="line">148</span><br><span class="line">149</span><br><span class="line">150</span><br><span class="line">151</span><br><span class="line">152</span><br><span class="line">153</span><br><span class="line">154</span><br><span class="line">155</span><br><span class="line">156</span><br><span class="line">157</span><br><span class="line">158</span><br><span class="line">159</span><br><span class="line">160</span><br><span class="line">161</span><br><span class="line">162</span><br><span class="line">163</span><br><span class="line">164</span><br><span class="line">165</span><br><span class="line">166</span><br><span class="line">167</span><br><span class="line">168</span><br><span class="line">169</span><br><span class="line">170</span><br><span class="line">171</span><br><span class="line">172</span><br><span class="line">173</span><br><span class="line">174</span><br><span class="line">175</span><br><span class="line">176</span><br><span class="line">177</span><br><span class="line">178</span><br><span class="line">179</span><br><span class="line">180</span><br><span class="line">181</span><br><span class="line">182</span><br><span class="line">183</span><br><span class="line">184</span><br><span class="line">185</span><br><span class="line">186</span><br><span class="line">187</span><br><span class="line">188</span><br><span class="line">189</span><br><span class="line">190</span><br><span class="line">191</span><br><span class="line">192</span><br><span class="line">193</span><br><span class="line">194</span><br><span class="line">195</span><br><span class="line">196</span><br><span class="line">197</span><br><span class="line">198</span><br><span class="line">199</span><br><span class="line">200</span><br><span class="line">201</span><br><span class="line">202</span><br><span class="line">203</span><br><span class="line">204</span><br><span class="line">205</span><br><span class="line">206</span><br><span class="line">207</span><br><span class="line">208</span><br><span class="line">209</span><br><span class="line">210</span><br><span class="line">211</span><br><span class="line">212</span><br><span class="line">213</span><br><span class="line">214</span><br><span class="line">215</span><br><span class="line">216</span><br><span class="line">217</span><br><span class="line">218</span><br><span class="line">219</span><br><span class="line">220</span><br><span class="line">221</span><br><span class="line">222</span><br><span class="line">223</span><br><span class="line">224</span><br><span class="line">225</span><br><span class="line">226</span><br><span class="line">227</span><br><span class="line">228</span><br><span class="line">229</span><br><span class="line">230</span><br><span class="line">231</span><br><span class="line">232</span><br><span class="line">233</span><br><span class="line">234</span><br><span class="line">235</span><br><span class="line">236</span><br><span class="line">237</span><br><span class="line">238</span><br><span class="line">239</span><br><span class="line">240</span><br><span class="line">241</span><br><span class="line">242</span><br><span class="line">243</span><br><span class="line">244</span><br><span class="line">245</span><br><span class="line">246</span><br><span class="line">247</span><br><span class="line">248</span><br><span class="line">249</span><br><span class="line">250</span><br><span class="line">251</span><br><span class="line">252</span><br><span class="line">253</span><br><span class="line">254</span><br><span class="line">255</span><br><span class="line">256</span><br><span class="line">257</span><br><span class="line">258</span><br><span class="line">259</span><br><span class="line">260</span><br><span class="line">261</span><br><span class="line">262</span><br><span class="line">263</span><br><span class="line">264</span><br><span class="line">265</span><br><span class="line">266</span><br><span class="line">267</span><br><span class="line">268</span><br><span class="line">269</span><br><span class="line">270</span><br><span class="line">271</span><br><span class="line">272</span><br><span class="line">273</span><br><span class="line">274</span><br><span class="line">275</span><br><span class="line">276</span><br><span class="line">277</span><br><span class="line">278</span><br><span class="line">279</span><br><span class="line">280</span><br><span class="line">281</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">package</span> cn.sgnxotsmicf.Demo;</span><br><span class="line"></span><br><span class="line"><span class="keyword">import</span> java.text.DecimalFormat;</span><br><span class="line"><span class="keyword">import</span> java.time.LocalDateTime;</span><br><span class="line"><span class="keyword">import</span> java.time.temporal.ChronoUnit;</span><br><span class="line"><span class="keyword">import</span> java.util.Arrays;</span><br><span class="line"><span class="keyword">import</span> java.util.Scanner;</span><br><span class="line"></span><br><span class="line"><span class="keyword">public</span> <span class="class"><span class="keyword">class</span> <span class="title">Simplex_Method</span> <span class="keyword">extends</span> <span class="title">Simplex_Method_Data</span></span>{</span><br><span class="line"> <span class="keyword">static</span> {</span><br><span class="line"> System.out.println(<span class="string">"==================单纯形法求解线性规划================="</span>);</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> <span class="comment">/**</span></span><br><span class="line"><span class="comment"> * 主方法</span></span><br><span class="line"><span class="comment"> */</span></span><br><span class="line"> <span class="function"><span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">main</span><span class="params">(String[] args)</span> </span>{</span><br><span class="line"> Scanner sc = <span class="keyword">new</span> Scanner(System.in);</span><br><span class="line"> <span class="keyword">double</span>[][] A = centreMatrix_Creat(sc);</span><br><span class="line"> Search_XB(sc);</span><br><span class="line"> Init(sc,A);</span><br><span class="line"> LocalDateTime startTime = LocalDateTime.now();</span><br><span class="line"> LogicLoopJudge(A);</span><br><span class="line"> LocalDateTime endTime = LocalDateTime.now();</span><br><span class="line"> IterationTime(startTime,endTime);</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> <span class="comment">/**</span></span><br><span class="line"><span class="comment"> *用于计算迭代时间</span></span><br><span class="line"><span class="comment"> * <span class="doctag">@param</span> startTime 迭代前时间</span></span><br><span class="line"><span class="comment"> * <span class="doctag">@param</span> endTime 迭代后时间</span></span><br><span class="line"><span class="comment"> *</span></span><br><span class="line"><span class="comment"> */</span></span><br><span class="line"> <span class="function"><span class="keyword">private</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">IterationTime</span><span class="params">(LocalDateTime startTime, LocalDateTime endTime)</span> </span>{</span><br><span class="line"> System.out.println(<span class="string">"迭代时间大约为:"</span>+ ChronoUnit.MILLIS.between(startTime, endTime)+<span class="string">"毫秒"</span>);</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> <span class="function"><span class="keyword">private</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">LogicLoopJudge</span><span class="params">(<span class="keyword">double</span>[][] A)</span> </span>{</span><br><span class="line"> <span class="keyword">int</span> number = <span class="number">0</span>;</span><br><span class="line"> <span class="keyword">while</span> (<span class="keyword">true</span>){</span><br><span class="line"> <span class="keyword">int</span> tempt = <span class="number">0</span>;</span><br><span class="line"> <span class="keyword">int</span> negativeIndex = <span class="number">0</span>;</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i < pi.length; i++) {</span><br><span class="line"> <span class="keyword">if</span> (pi[i]<=<span class="number">0</span>){</span><br><span class="line"> tempt++;</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">else</span> <span class="keyword">if</span> (pi[i]><span class="number">0</span>) {</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> j = <span class="number">0</span>;j < m;j++){</span><br><span class="line"> <span class="keyword">if</span> (A[j][i]<=<span class="number">0</span>){</span><br><span class="line"> negativeIndex++;</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">if</span> (negativeIndex ==m){</span><br><span class="line"> System.out.println(<span class="string">"此线性规划问题无最优解！"</span>);</span><br><span class="line"> <span class="keyword">return</span>;</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">if</span> (tempt == n){</span><br><span class="line"> printResult(A,number);</span><br><span class="line"> <span class="keyword">break</span>;</span><br><span class="line"> }</span><br><span class="line"> <span class="comment">// 开始操作；</span></span><br><span class="line"> System.out.println(<span class="string">"=================开始第"</span>+(number+<span class="number">1</span>)+<span class="string">"次迭代================"</span>);</span><br><span class="line"> <span class="keyword">int</span> index = IntoBaseVar_Judge();<span class="comment">//入基索引</span></span><br><span class="line"> UpDataTheta(A,index);</span><br><span class="line"> <span class="keyword">int</span> index2 = OutOfBaseVar_Judge(index);<span class="comment">//出基索引</span></span><br><span class="line"> RotationTransformation(A,index,index2);</span><br><span class="line"> number++;</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> <span class="comment">/**</span></span><br><span class="line"><span class="comment"> *用于A的旋转变化，更新A</span></span><br><span class="line"><span class="comment"> * <span class="doctag">@param</span> index 入基变量</span></span><br><span class="line"><span class="comment"> * <span class="doctag">@param</span> index2 出基变量</span></span><br><span class="line"><span class="comment"> */</span></span><br><span class="line"> <span class="function"><span class="keyword">private</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">RotationTransformation</span><span class="params">(<span class="keyword">double</span>[][] A,<span class="keyword">int</span> index,<span class="keyword">int</span> index2)</span> </span>{</span><br><span class="line"> <span class="keyword">if</span> (A[index2][index] != <span class="number">1</span>) {</span><br><span class="line"> <span class="keyword">double</span> positiveFactor = A[index2][index];</span><br><span class="line"> <span class="comment">//A[index2][index] = 1;</span></span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> j = <span class="number">0</span>; j < n; j++) {</span><br><span class="line"> A[index2][j] /= positiveFactor; <span class="comment">//初等行变化</span></span><br><span class="line"> }</span><br><span class="line"> b0[index2] /= positiveFactor;</span><br><span class="line"> }</span><br><span class="line"> RotatingCenter(A,index,index2);</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> <span class="function"><span class="keyword">private</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">RotatingCenter</span><span class="params">(<span class="keyword">double</span>[][] A, <span class="keyword">int</span> index, <span class="keyword">int</span> index2)</span> </span>{</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i < m; i++) {</span><br><span class="line"> <span class="keyword">if</span> (A[i][index]!=<span class="number">0</span> && i!=index2){</span><br><span class="line"> <span class="keyword">double</span> factor = -A[index2][index]*A[i][index];</span><br><span class="line"> <span class="keyword">double</span> b0_factor = factor*b0[index2];</span><br><span class="line"> b0[i] += b0_factor;</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> j = <span class="number">0</span>; j < n; j++) {</span><br><span class="line"> <span class="keyword">double</span> A_factor = A[index2][j]*factor;</span><br><span class="line"> A[i][j] += A_factor;</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">double</span> factor = -A[index2][index]*pi[index];</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> j = <span class="number">0</span>; j < n; j++) {</span><br><span class="line"> <span class="keyword">double</span> pi_factor = factor*A[index2][j];</span><br><span class="line"> pi[j] += pi_factor;</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> <span class="function"><span class="keyword">private</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">printResult</span><span class="params">(<span class="keyword">double</span>[][] A,<span class="keyword">int</span> number)</span> </span>{</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i < m; i++) {</span><br><span class="line"> <span class="keyword">var</span> += c[basicVar[i]]*b0[i];</span><br><span class="line"> }</span><br><span class="line"> System.out.println(<span class="string">"======================循环结束======================"</span>);</span><br><span class="line"> System.out.println(<span class="string">"最优值为:"</span>+<span class="keyword">var</span>);</span><br><span class="line"> System.out.println(<span class="string">"最优解为:"</span>);</span><br><span class="line"> <span class="keyword">boolean</span> logic = OptimalSolution();</span><br><span class="line"> <span class="keyword">if</span> (!logic){</span><br><span class="line"> <span class="keyword">return</span>;</span><br><span class="line"> }</span><br><span class="line"> System.out.println(<span class="string">"基向量为:"</span>);</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> j : basicVar) {</span><br><span class="line"> System.out.print(<span class="string">"x"</span> + (j + <span class="number">1</span>)+<span class="string">"\t"</span>);</span><br><span class="line"> }</span><br><span class="line"> System.out.println();</span><br><span class="line"> System.out.println(<span class="string">"迭代次数:"</span>+number);</span><br><span class="line"> System.out.println(<span class="string">"最后一步单纯形表为:"</span>);</span><br><span class="line"> printMatrix(A);</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> <span class="function"><span class="keyword">private</span> <span class="keyword">static</span> <span class="keyword">boolean</span> <span class="title">OptimalSolution</span><span class="params">()</span> </span>{</span><br><span class="line"> DecimalFormat sc = <span class="keyword">new</span> DecimalFormat(<span class="string">"0.00"</span>);</span><br><span class="line"> String[] OptimalSData = <span class="keyword">new</span> String[n];</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i < basicVar.length; i++) {</span><br><span class="line"> OptimalSData[basicVar[i]] = sc.format(b0[i]);</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i < OptimalSData.length; i++) {</span><br><span class="line"> <span class="keyword">if</span> (OptimalSData[i] == <span class="keyword">null</span>){</span><br><span class="line"> OptimalSData[i] = <span class="string">"0.00"</span>;</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"> System.out.println(Arrays.toString(OptimalSData));</span><br><span class="line"> <span class="comment">//0.25 0 0 1 0 -0.25 -2 0 -1 0 1 -3 1 1 0 0 0 1 1.5 6 10 2 3 0 0 10000 10000</span></span><br><span class="line"> <span class="keyword">if</span> (YN.equals(<span class="string">"yes"</span>)) {</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> mDatum : MData) {</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> j = <span class="number">0</span>; j < m; j++) {</span><br><span class="line"> <span class="keyword">if</span> (mDatum == basicVar[j] && b0[j] != <span class="number">0</span>) {</span><br><span class="line"> System.out.println(<span class="string">"由于最优解中存在正的人工变量，则原问题是不可行的"</span>);</span><br><span class="line"> <span class="keyword">return</span> <span class="keyword">false</span>;</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">return</span> <span class="keyword">true</span>;</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> <span class="comment">/**</span></span><br><span class="line"><span class="comment"> *</span></span><br><span class="line"><span class="comment"> * <span class="doctag">@param</span> index 为入基索引</span></span><br><span class="line"><span class="comment"> * <span class="doctag">@return</span> 返回出基索引</span></span><br><span class="line"><span class="comment"> * 使用的是Dantzig出基规则</span></span><br><span class="line"><span class="comment"> */</span></span><br><span class="line"> <span class="function"><span class="keyword">private</span> <span class="keyword">static</span> <span class="keyword">int</span> <span class="title">OutOfBaseVar_Judge</span><span class="params">(<span class="keyword">int</span> index)</span> </span>{</span><br><span class="line"> <span class="keyword">int</span> index2 = <span class="number">0</span>;</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">1</span>; i < m; i++) {</span><br><span class="line"> <span class="keyword">if</span> (theta[index2]>theta[i]){</span><br><span class="line"> index2 = i;</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"> System.out.println(<span class="string">"出基变量为:x"</span>+((basicVar[index2]+<span class="number">1</span>)));</span><br><span class="line"> basicVar[index2] = index;</span><br><span class="line"> <span class="keyword">return</span> index2;</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> <span class="comment">/**</span></span><br><span class="line"><span class="comment"> *</span></span><br><span class="line"><span class="comment"> * <span class="doctag">@return</span> 返回的是入基索引</span></span><br><span class="line"><span class="comment"> * 使用的是Dantzig入基规则</span></span><br><span class="line"><span class="comment"> */</span></span><br><span class="line"> <span class="function"><span class="keyword">private</span> <span class="keyword">static</span> <span class="keyword">int</span> <span class="title">IntoBaseVar_Judge</span><span class="params">()</span> </span>{</span><br><span class="line"> <span class="keyword">int</span> index = <span class="number">0</span>;</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">1</span>; i < n; i++) {</span><br><span class="line"> <span class="keyword">if</span> (pi[index]<pi[i]){</span><br><span class="line"> index = i;</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"> System.out.println(<span class="string">"入基变量为:x"</span>+(index+<span class="number">1</span>));</span><br><span class="line"> <span class="keyword">return</span> index;</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> <span class="comment">/**</span></span><br><span class="line"><span class="comment"> * 用于更新θ</span></span><br><span class="line"><span class="comment"> * <span class="doctag">@param</span> index 入基变量</span></span><br><span class="line"><span class="comment"> */</span></span><br><span class="line"> <span class="function"><span class="keyword">private</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">UpDataTheta</span><span class="params">(<span class="keyword">double</span>[][] A,<span class="keyword">int</span> index)</span> </span>{</span><br><span class="line"> theta = <span class="keyword">new</span> <span class="keyword">double</span>[m];</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i < m; i++) {</span><br><span class="line"> <span class="keyword">if</span> (A[i][index]<=<span class="number">0</span>){</span><br><span class="line"> theta[i] = <span class="number">1000</span>;</span><br><span class="line"> }<span class="keyword">else</span> {</span><br><span class="line"> theta[i] = b0[i]/A[i][index];</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> <span class="comment">/**</span></span><br><span class="line"><span class="comment"> * 搜索单位矩阵</span></span><br><span class="line"><span class="comment"> */</span></span><br><span class="line"> <span class="function"><span class="keyword">private</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">Search_XB</span><span class="params">(Scanner sc)</span> </span>{</span><br><span class="line"> basicVar = <span class="keyword">new</span> <span class="keyword">int</span>[m];</span><br><span class="line"> System.out.println(<span class="string">"请输入单位矩阵所对应的"</span>+m+<span class="string">"个初始基变量的下标:"</span>);</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i < m; i++) {</span><br><span class="line"> <span class="keyword">int</span> tempt = sc.nextInt();</span><br><span class="line"> basicVar[i] = (tempt-<span class="number">1</span>);</span><br><span class="line"> }</span><br><span class="line"> System.out.println(<span class="string">"基变量为:"</span>);</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> j : basicVar) {</span><br><span class="line"> System.out.print(<span class="string">"x"</span> + (j + <span class="number">1</span>) + <span class="string">" "</span>);</span><br><span class="line"> }</span><br><span class="line"> System.out.println();</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> <span class="comment">/**</span></span><br><span class="line"><span class="comment"> * 数据初始化</span></span><br><span class="line"><span class="comment"> */</span></span><br><span class="line"> <span class="function"><span class="keyword">private</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">Init</span><span class="params">(Scanner sc,<span class="keyword">double</span>[][] A)</span> </span>{</span><br><span class="line"> <span class="keyword">int</span> MJudge = TheBigMMethod(sc);</span><br><span class="line"> <span class="comment">// 初始化基变量的值</span></span><br><span class="line"> b0 = <span class="keyword">new</span> <span class="keyword">double</span>[m];</span><br><span class="line"> <span class="keyword">if</span> (MJudge == <span class="number">1</span>){</span><br><span class="line"> System.out.println(<span class="string">"请输入人工变量的个数:"</span>);</span><br><span class="line"> <span class="keyword">int</span> M = sc.nextInt();</span><br><span class="line"> MData= <span class="keyword">new</span> <span class="keyword">int</span>[M];</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i < M; i++) {</span><br><span class="line"> System.out.println(<span class="string">"请输入第"</span>+(i+<span class="number">1</span>)+<span class="string">"个人工变量下标:"</span>);</span><br><span class="line"> <span class="keyword">int</span> tempt = sc.nextInt();</span><br><span class="line"> MData[i] = (tempt-<span class="number">1</span>); <span class="comment">//人工变量索引</span></span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"> System.out.println(<span class="string">"请输入基本可行解中基变量的初始值:"</span>); <span class="comment">//等于B^-1*b ---基于初始单位阵I</span></span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i < m; i++) {</span><br><span class="line"> b0[i] = sc.nextDouble();</span><br><span class="line"> }</span><br><span class="line"> <span class="comment">// 初始化 价值系数c</span></span><br><span class="line"> c = <span class="keyword">new</span> <span class="keyword">double</span>[n];</span><br><span class="line"> System.out.println(<span class="string">"请输入目标函数中变量xj的价值系数cj:"</span>);</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i < n; i++) {</span><br><span class="line"> c[i] = sc.nextDouble();</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> <span class="comment">// 初始检验数</span></span><br><span class="line"> initPi(A);</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> <span class="comment">/**</span></span><br><span class="line"><span class="comment"> * 大M法判断</span></span><br><span class="line"><span class="comment"> * <span class="doctag">@return</span> 返回判断逻辑</span></span><br><span class="line"><span class="comment"> */</span></span><br><span class="line"> <span class="function"><span class="keyword">private</span> <span class="keyword">static</span> <span class="keyword">int</span> <span class="title">TheBigMMethod</span><span class="params">(Scanner sc)</span> </span>{</span><br><span class="line"> System.out.println(<span class="string">"是否有人工变量？(yes/no)"</span>);</span><br><span class="line"> String MJudge = sc.next();</span><br><span class="line"> YN = MJudge;</span><br><span class="line"> <span class="keyword">if</span> (MJudge.equals(<span class="string">"yes"</span>)){</span><br><span class="line"> System.out.println(<span class="string">"==================下面进行大M法求解线性最优化问题============="</span>);</span><br><span class="line"> <span class="keyword">return</span> <span class="number">1</span>;</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line"> }</span><br><span class="line"></span><br><span class="line"> <span class="comment">/**</span></span><br><span class="line"><span class="comment"> *初始化检验数π</span></span><br><span class="line"><span class="comment"> */</span></span><br><span class="line"> <span class="function"><span class="keyword">private</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">initPi</span><span class="params">(<span class="keyword">double</span>[][] A)</span> </span>{</span><br><span class="line"> pi = <span class="keyword">new</span> <span class="keyword">double</span>[n];</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i < n; i++) {</span><br><span class="line"> <span class="keyword">for</span> (<span class="keyword">int</span> j = <span class="number">0</span>; j < m; j++) {</span><br><span class="line"> pi[i] += c[basicVar[j]]*A[j][i];</span><br><span class="line"> }</span><br><span class="line"> pi[i] -= c[i];</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line">}</span><br></pre></td></tr></table></figure>]]></content>
<summary type="html"><p>用来求解线性规划问题</p></summary>
<category term="最优化方法" scheme="http://sgnxotsmicf.com/categories/%E6%9C%80%E4%BC%98%E5%8C%96%E6%96%B9%E6%B3%95/"/>
<category term="java" scheme="http://sgnxotsmicf.com/tags/java/"/>
<category term="最优化算法" scheme="http://sgnxotsmicf.com/tags/%E6%9C%80%E4%BC%98%E5%8C%96%E7%AE%97%E6%B3%95/"/>
<category term="线性规划" scheme="http://sgnxotsmicf.com/tags/%E7%BA%BF%E6%80%A7%E8%A7%84%E5%88%92/"/>
</entry>
<entry>
<title>markdown语法及数学公式使用规则</title>
<link href="http://sgnxotsmicf.com/2021/12/25/markdown%E6%95%B0%E5%AD%A6%E5%85%AC%E5%BC%8F%E4%BD%BF%E7%94%A8%E8%A7%84%E5%88%99/"/>
<id>http://sgnxotsmicf.com/2021/12/25/markdown%E6%95%B0%E5%AD%A6%E5%85%AC%E5%BC%8F%E4%BD%BF%E7%94%A8%E8%A7%84%E5%88%99/</id>
<published>2021-12-25T12:56:43.310Z</published>
<updated>2022-01-13T11:45:48.749Z</updated>
<content type="html"><![CDATA[<link rel="stylesheet" class="aplayer-secondary-style-marker" href="\assets\css\APlayer.min.css"><script src="\assets\js\APlayer.min.js" class="aplayer-secondary-script-marker"></script><script class="meting-secondary-script-marker" src="\assets\js\Meting.min.js"></script><h3 id="语法"><a href="#语法" class="headerlink" title="语法"></a>语法</h3><blockquote><p><a href="https://www.jianshu.com/p/191d1e21f7ed">这里</a></p></blockquote><h3 id="数学公式"><a href="#数学公式" class="headerlink" title="数学公式"></a>数学公式</h3><blockquote><p><a href="https://zhuanlan.zhihu.com/p/261750408?utm_source=wechat_session">这里1</a></p><p><a href="https://blog.csdn.net/c20182030/article/details/84840373?depth_1-utm_source=distribute.pc_relevant.none-task-blog-BlogCommendFromBaidu-1&utm_source=distribute.pc_relevant.none-task-blog-BlogCommendFromBaidu-1#LaTexKaTex_26">这里2</a></p></blockquote><span id="more"></span><blockquote><p><a href="https://www.cnblogs.com/VincentValentine/p/15419593.html">这里3</a></p><p><a href="https://www.jianshu.com/p/e74eb43960a1">这里4</a></p><p><a href="https://www.jianshu.com/p/d63887d0c706">箭头集合</a></p></blockquote><p>$a^{\prime}$</p><p>$a’$</p><p>…………..</p> <img src="/2021/12/25/markdown%E6%95%B0%E5%AD%A6%E5%85%AC%E5%BC%8F%E4%BD%BF%E7%94%A8%E8%A7%84%E5%88%99/1Q7L5JUV5Y7_%60%7B9%7DE_K3N4V.jpg" class="" title="1Q7L5JUV5Y7_`{9}E_K3N4V"><hr>]]></content>
<summary type="html"><h3 id="语法"><a href="#语法" class="headerlink" title="语法"></a>语法</h3><blockquote>
<p><a href="https://www.jianshu.com/p/191d1e21f7ed">这里</a></p>
</blockquote>
<h3 id="数学公式"><a href="#数学公式" class="headerlink" title="数学公式"></a>数学公式</h3><blockquote>
<p><a href="https://zhuanlan.zhihu.com/p/261750408?utm_source=wechat_session">这里1</a></p>
<p><a href="https://blog.csdn.net/c20182030/article/details/84840373?depth_1-utm_source=distribute.pc_relevant.none-task-blog-BlogCommendFromBaidu-1&amp;utm_source=distribute.pc_relevant.none-task-blog-BlogCommendFromBaidu-1#LaTexKaTex_26">这里2</a></p>
</blockquote></summary>
<category term="markdown" scheme="http://sgnxotsmicf.com/categories/markdown/"/>
</entry>
<entry>
<title>第二积分中值定理</title>
<link href="http://sgnxotsmicf.com/2021/12/22/%E7%AC%AC%E4%BA%8C%E7%A7%AF%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86/"/>
<id>http://sgnxotsmicf.com/2021/12/22/%E7%AC%AC%E4%BA%8C%E7%A7%AF%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86/</id>
<published>2021-12-22T07:29:15.000Z</published>
<updated>2021-12-30T04:50:08.078Z</updated>
<content type="html"><![CDATA[<link rel="stylesheet" class="aplayer-secondary-style-marker" href="\assets\css\APlayer.min.css"><script src="\assets\js\APlayer.min.js" class="aplayer-secondary-script-marker"></script><script class="meting-secondary-script-marker" src="\assets\js\Meting.min.js"></script><p>定理、证明、例题————复习</p><p>来源于网络！！！</p><span id="more"></span><h2 id="证明"><a href="#证明" class="headerlink" title="证明"></a>证明</h2><p>设 $f(x)$ 在 $[a, b]$ 上可积， $g(x)$ 在 $[a, b]$ 上单调，则存在 $\xi \in[a, b]$ ，使得</p><script type="math/tex; mode=display">\int_{a}^{b} f(x) g(x) d x=g(a) \int_{a}^{\xi} f(x) d x+g(b) \int_{\xi}^{b} f(x) d x .</script><p>这个定理的推导比较复杂，牵扯到积分上限函数: $\varphi(x)=\int f(t) d t$ 。以下用 $\int_{a}^{b} f(x) d x$ 表示从 $\mathrm{a}$ 到 $\mathrm{b}$ 的定积分。 首先需要证明，若函数 $f(x)$ 在 $[a, b]$ 内可积分，则 $\Phi(x)$ 在此区间内为一连续函数。</p><p>证明: 设 $F(x)=\int_{a}^{x} f(t) d t$, 则 $\int_{a}^{b} f(x) g(x) d x=\int_{a}^{b} g(x) d F(x)=\left.g(x) F(x)\right|_{a} ^{b}-\int_{a}^{b} F(x) g^{\prime}(x) d x$ $=g(b) \int_{a}^{b} f(x) d x-\int_{a}^{b} F(x) g^{\prime}(x) d x,\left(F(x)=\int_{a}^{x} f(t) d t\right) .$</p><p>因 $g^{\prime}(x)$ 在 $[a, b]$ 上不变号，则由积分第一中值定理知，在 $[a, b]$ 上至少存在一点 $\xi ，$ 使得 $\int_{a}^{b} F(x) g^{\prime}(x) d x=F(\xi) \int_{a}^{b} g^{\prime}(x) d x=F(\xi)[g(b)-g(a)] .$</p><p>于是，有</p><script type="math/tex; mode=display">\begin{gathered}\left.\int_{a}^{b} f(x) g(x) d x=g(b) \int_{a}^{b} f(x) d x-[g(b)]-g(a)\right] F(\xi) \\=g(b) \int_{a}^{b} f(x) d x-[g(b)-g(a)] \int_{a}^{\xi} f(x) d x \\=g(b)\left[\int_{a}^{b} f(x) d x-\int_{a}^{\xi} f(x) d x\right]+g(a) \int_{a}^{\xi} f(x) d x \\=g(b) \int_{\xi}^{b} f(x) d x+g(a) \int_{a}^{\xi} f(x) d x, \\\text { 即 } \int_{a}^{b} f(x) g(x) d x=g(a) \int_{a}^{\xi} f(x) d x+g(b) \int_{\xi}^{b} f(x) d x, \text { 得证。 }\end{gathered}</script><h2 id="例题"><a href="#例题" class="headerlink" title="例题"></a>例题</h2><ol><li>定理的直接应用<br>例1. 设 $f(x)$ 在 $[a, b]$ 上可积,$g(x)$ 在 $[a, b]$ 上单调递增且非负,在 $a, b$ 处连续,那么在 $[a, b]$ 上存在 $\xi$,使$\int_{a}^{b} f(x) g(x) d x=f(b) \int_{\xi}^{b} f(x) d x .$</li></ol><script type="math/tex; mode=display">\begin{aligned}&\text { 证 明 : 令 } \mathrm{x}=\mathrm{b}-\mathrm{t}, g(b-t)=h(t) \int_{0}^{b-a} f(b-t) h(t) d(t) \text {, 因为 } \mathrm{h}(\mathrm{t}) \text { 非负且单调递减 }(0<\mathrm{t}<\mathrm{b}-\mathrm{a}) \text { 利用公式有 } \\&\int_{0}^{b-a} f(b-t) h(t) d t=h(0) \int_{0}^{\gamma} f(b-t) d t=g(b) \int_{b-\gamma}^{b} f(x) d x(0<\gamma<b-a) \text {, 而 } \mathrm{a}<\xi<\mathrm{b}-\gamma \text { ，即 } \\&\int_{a}^{b} f(x) g(x) d x=f(b) \int_{\xi}^{b} f(x) d x .\end{aligned}</script><ol><li>积分第二中值定理在证明不等式中的应用<br>例2. 证明 $x>0$ 时， $\left|\int_{x}^{x+e} \sin t^{2} d t\right| \leq \frac{1}{x}$.<br>证明:取 $u=t^{2}, t=\sqrt{u}, d t=\frac{d u}{2 \sqrt{u}}$, 由积分中值定理和它的推论可得:</li></ol><script type="math/tex; mode=display">\left|\int_{x}^{x+\tau} \sin t^{2} d t\right|=\left|\int_{x}^{x+\tau} \frac{\sin u}{2 \sqrt{u}} d u\right|=\frac{1}{2 \sqrt{x^{2}}} \int_{x^{2}}^{\gamma} \sin u d u=\frac{1}{2 x}\left(\cos x^{2}-\cos \gamma\right) \leq \frac{2}{2 x}=\frac{1}{x} .</script><ol><li>积分中值定理在极限中的应用<br>例3.证明极限 $\lim _{n \rightarrow \infty} \frac{x^{n}}{1+x} d x=0$.<br>证明: 由积分中值定理和它的推论可得:令 $f(x)=\frac{1}{1+x}, g(x)=x^{n}$,<br>令可知 $g(x)$ 在 $[0,1]$ 上连续,而且不变号。所以存在 $\xi$ 使得 $\int_{0}^{1} f(x) g(x) d x=f(\xi) \int_{0}^{1} g(x) d x$, 因此有以下式子<br>$0 \leq \int_{0}^{1} \frac{x^{n}}{1+x} d x=\frac{1}{1+\xi} \int_{0}^{1} x^{n} d x=\frac{1}{(n+1)(1+\xi)} \leq \frac{1}{n+1} \rightarrow 0 \rightarrow \lim _{n \rightarrow \infty} \int_{0}^{1} \frac{x^{n}}{1+x} d x=0$</li></ol>]]></content>
<summary type="html"><p>定理、证明、例题————复习</p>
<p>来源于网络！！！</p></summary>
<category term="积分" scheme="http://sgnxotsmicf.com/categories/%E7%A7%AF%E5%88%86/"/>
<category term="第二积分中值定理" scheme="http://sgnxotsmicf.com/tags/%E7%AC%AC%E4%BA%8C%E7%A7%AF%E5%88%86%E4%B8%AD%E5%80%BC%E5%AE%9A%E7%90%86/"/>
</entry>
<entry>
<title>很叼的猜拳游戏</title>
<link href="http://sgnxotsmicf.com/2021/10/08/%E5%BE%88%E5%8F%BC%E7%9A%84%E7%8C%9C%E6%8B%B3%E6%B8%B8%E6%88%8F/"/>
<id>http://sgnxotsmicf.com/2021/10/08/%E5%BE%88%E5%8F%BC%E7%9A%84%E7%8C%9C%E6%8B%B3%E6%B8%B8%E6%88%8F/</id>
<published>2021-10-08T08:39:54.000Z</published>
<updated>2021-10-08T08:50:45.189Z</updated>
<content type="html"><![CDATA[<link rel="stylesheet" class="aplayer-secondary-style-marker" href="\assets\css\APlayer.min.css"><script src="\assets\js\APlayer.min.js" class="aplayer-secondary-script-marker"></script><script class="meting-secondary-script-marker" src="\assets\js\Meting.min.js"></script><p><img src="https://cdn.mathpix.com/snip/images/pYX6MuIpPsS7tB3CpbeeuHjI_hrr9M5UdpfYzmbGKdA.original.fullsize.png" alt=""><br><span id="more"></span></p><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">import</span> random</span><br><span class="line"><span class="built_in">input</span>(<span class="string">"游戏规则:输入1代表出<拳头> 输入2代表出<剪刀> 输入3代表出<布>"</span></span><br><span class="line"> <span class="string">"\n 按回车继续......"</span>)</span><br><span class="line"><span class="built_in">input</span>(<span class="string">"胜负关系为:1>2 2>3 3>1\n按回车开始游戏......"</span>)</span><br><span class="line"><span class="built_in">print</span>(<span class="string">"----------------------------------------------------------------------"</span>)</span><br><span class="line">number = <span class="built_in">int</span>(<span class="built_in">input</span>(<span class="string">"请输入比赛回合:"</span>))</span><br><span class="line">i = <span class="number">1</span></span><br><span class="line">win = <span class="number">0</span></span><br><span class="line">lose = <span class="number">0</span></span><br><span class="line">noun = <span class="number">0</span></span><br><span class="line"><span class="keyword">while</span> i <= number:</span><br><span class="line"> user = <span class="built_in">input</span>(<span class="string">"请输入需要出的拳:石头(1)/剪刀(2)/布(3):"</span>)</span><br><span class="line"> user = <span class="built_in">eval</span>(user)</span><br><span class="line"> computer = random.randint(<span class="number">1</span>, <span class="number">300</span>) <span class="comment"># 随机产生1-300之间的随机整数</span></span><br><span class="line"> h = number - i</span><br><span class="line"> <span class="keyword">if</span> user == <span class="number">1</span>:</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">" 你出石头:%d"</span> % user)</span><br><span class="line"> <span class="keyword">elif</span> user == <span class="number">2</span>:</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">" 你出剪刀:%d"</span> % user)</span><br><span class="line"> <span class="keyword">else</span>:</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">" 你出布: %d"</span> % user)</span><br><span class="line"> <span class="keyword">if</span> (computer >= <span class="number">1</span>) <span class="keyword">and</span> computer <= <span class="number">100</span>:</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">"电脑出石头:1 --(%d)"</span> % computer)</span><br><span class="line"> <span class="keyword">elif</span> (computer > <span class="number">100</span>) <span class="keyword">and</span> computer <= <span class="number">200</span>:</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">"电脑出剪刀:2 --(%d)"</span> % computer)</span><br><span class="line"> <span class="keyword">else</span>:</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">" 电脑出布:3 --(%d)"</span> % computer)</span><br><span class="line"> <span class="comment"># print("-----规则:1>2 2>3 3>1-----")</span></span><br><span class="line"> <span class="keyword">if</span> (user == <span class="number">1</span> <span class="keyword">and</span> ((computer >= <span class="number">1</span>) <span class="keyword">and</span> computer <= <span class="number">100</span>)) <span class="keyword">or</span> (</span><br><span class="line"> user == <span class="number">2</span> <span class="keyword">and</span> ((computer > <span class="number">100</span>) <span class="keyword">and</span> computer <= <span class="number">200</span>)) <span class="keyword">or</span> (</span><br><span class="line"> user == <span class="number">3</span> <span class="keyword">and</span> ((computer > <span class="number">200</span>) <span class="keyword">and</span> computer <= <span class="number">300</span>)):</span><br><span class="line"> noun = <span class="number">1</span></span><br><span class="line"> <span class="built_in">print</span>(<span class="string">"本回合平局!无胜负关系，重新再来!"</span>)</span><br><span class="line"> <span class="keyword">elif</span> user == <span class="number">1</span> <span class="keyword">and</span> (computer > <span class="number">100</span>) <span class="keyword">and</span> computer <= <span class="number">200</span> <span class="keyword">or</span> user == <span class="number">2</span> <span class="keyword">and</span> (</span><br><span class="line"> computer > <span class="number">200</span>) <span class="keyword">and</span> computer <= <span class="number">300</span> <span class="keyword">or</span> user == <span class="number">3</span> <span class="keyword">and</span> (computer >= <span class="number">1</span>) <span class="keyword">and</span> computer <= <span class="number">100</span>:</span><br><span class="line"> noun = <span class="number">0</span></span><br><span class="line"> win += <span class="number">1</span></span><br><span class="line"> <span class="keyword">if</span> h == <span class="number">0</span>:</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">f"本回合你赢了!回合结束,正在计算胜负关系......"</span>)</span><br><span class="line"> <span class="keyword">else</span>:</span><br><span class="line"> <span class="keyword">if</span> win == <span class="built_in">int</span>(number / <span class="number">2</span> + <span class="number">1</span>):</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">"本回合你赢了!回合结束,正在计算胜负关系......"</span>)</span><br><span class="line"> <span class="keyword">else</span>:</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">f"本回合你赢了!但别太高兴，你只赢了<span class="subst">{win}</span>个回合-----------------------还剩<span class="subst">{h}</span>个回合!"</span>)</span><br><span class="line"> <span class="keyword">if</span> win == <span class="built_in">int</span>(number / <span class="number">2</span> + <span class="number">1</span>):</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">"----------------------------------------------------------------------"</span>)</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">f"好吧，你赢了，<span class="subst">{number}</span>局游戏赢了<span class="subst">{win}</span>局!"</span>)</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">"恭喜你!成功赢了电脑!牛牛牛牛牛牛牛牛牛牛牛牛牛牛牛牛牛牛牛牛牛牛牛牛牛牛牛牛!"</span>)</span><br><span class="line"> <span class="keyword">break</span></span><br><span class="line"> <span class="keyword">else</span>:</span><br><span class="line"> noun = <span class="number">0</span></span><br><span class="line"> lose += <span class="number">1</span></span><br><span class="line"> <span class="keyword">if</span> h == <span class="number">0</span>:</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">"本回合你输了!回合结束，正在计算胜负关系......"</span>)</span><br><span class="line"> <span class="keyword">else</span>:</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">f"本回合你输了!菜狗，电脑都赢不了，你已经输了<span class="subst">{lose}</span>回合了-----------------------还剩<span class="subst">{h}</span>个回合"</span>)</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">"----------------------------------------------------------------------"</span>)</span><br><span class="line"> <span class="comment"># if (i == number) and (win >= 0 and win < number/2):</span></span><br><span class="line"> <span class="comment"># print(f"{number}次比赛你只赢了{win}个回合，你输了!")</span></span><br><span class="line"> <span class="keyword">if</span> lose == <span class="built_in">int</span>(number / <span class="number">2</span> + <span class="number">1</span>):</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">"很遗憾，你输了!"</span>)</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">f"<span class="subst">{number}</span>个回合你已经输了<span class="subst">{lose}</span>个回合了，我奶奶玩都比你强!"</span>)</span><br><span class="line"> <span class="keyword">break</span></span><br><span class="line"> <span class="keyword">elif</span> (i == number) <span class="keyword">and</span> win == number / <span class="number">2</span>:</span><br><span class="line"> <span class="built_in">print</span>(<span class="string">f"<span class="subst">{number}</span>个回合你只赢了<span class="subst">{win}</span>个回合:平局!"</span>)</span><br><span class="line"> <span class="keyword">if</span> noun == <span class="number">1</span>:</span><br><span class="line"> i = i</span><br><span class="line"> <span class="keyword">else</span>:</span><br><span class="line"> i += <span class="number">1</span></span><br></pre></td></tr></table></figure>]]></content>
<summary type="html"><p><img src="https://cdn.mathpix.com/snip/images/pYX6MuIpPsS7tB3CpbeeuHjI_hrr9M5UdpfYzmbGKdA.original.fullsize.png" alt=""><br></summary>
<category term="小游戏" scheme="http://sgnxotsmicf.com/categories/%E5%B0%8F%E6%B8%B8%E6%88%8F/"/>
<category term="while逻辑运算符" scheme="http://sgnxotsmicf.com/tags/while%E9%80%BB%E8%BE%91%E8%BF%90%E7%AE%97%E7%AC%A6/"/>
<category term="if逻辑运算符" scheme="http://sgnxotsmicf.com/tags/if%E9%80%BB%E8%BE%91%E8%BF%90%E7%AE%97%E7%AC%A6/"/>
<category term="python" scheme="http://sgnxotsmicf.com/tags/python/"/>
</entry>
<entry>
<title>努力学习</title>
<link href="http://sgnxotsmicf.com/2021/09/23/%E5%8A%AA%E5%8A%9B%E5%AD%A6%E4%B9%A0/"/>
<id>http://sgnxotsmicf.com/2021/09/23/%E5%8A%AA%E5%8A%9B%E5%AD%A6%E4%B9%A0/</id>
<published>2021-09-23T13:58:56.000Z</published>
<updated>2021-12-21T06:50:54.265Z</updated>
<content type="html"><![CDATA[<link rel="stylesheet" class="aplayer-secondary-style-marker" href="\assets\css\APlayer.min.css"><script src="\assets\js\APlayer.min.js" class="aplayer-secondary-script-marker"></script><script class="meting-secondary-script-marker" src="\assets\js\Meting.min.js"></script><p>想要把平时感兴趣但没有深入了解的数学知识研究透彻，要把这些”兴趣”，或者说学习中的疑难困惑解决清楚，因此特发此篇，以来总结。<br><span id="more"></span></p><blockquote><p>伯努利数</p><p>p级数求和</p><p>反常积分求解</p><p>留数定理深入学习</p><p>拉普拉斯变换</p><p>傅里叶变换</p><p>lebesgue积分深入</p></blockquote><p>…..</p><hr>]]></content>
<summary type="html"><p>想要把平时感兴趣但没有深入了解的数学知识研究透彻，要把这些”兴趣”，或者说学习中的疑难困惑解决清楚，因此特发此篇，以来总结。<br></summary>
<category term="兴趣" scheme="http://sgnxotsmicf.com/tags/%E5%85%B4%E8%B6%A3/"/>
</entry>
<entry>
<title>很简单的一道题</title>
<link href="http://sgnxotsmicf.com/2021/09/20/%E5%BE%88%E7%AE%80%E5%8D%95%E7%9A%84%E4%B8%80%E9%81%93%E9%A2%98/"/>
<id>http://sgnxotsmicf.com/2021/09/20/%E5%BE%88%E7%AE%80%E5%8D%95%E7%9A%84%E4%B8%80%E9%81%93%E9%A2%98/</id>
<published>2021-09-20T14:38:18.000Z</published>
<updated>2021-12-30T04:55:41.230Z</updated>
<content type="html"><![CDATA[<link rel="stylesheet" class="aplayer-secondary-style-marker" href="\assets\css\APlayer.min.css"><script src="\assets\js\APlayer.min.js" class="aplayer-secondary-script-marker"></script><script class="meting-secondary-script-marker" src="\assets\js\Meting.min.js"></script><p>预定31号题！非数竞赛一道题，第一次做没想到（+1）</p><span id="more"></span><h3 id="题目"><a href="#题目" class="headerlink" title="题目"></a>题目</h3><p>设区间$\big(0,+\infty)$上的函数$u(x)$定义为：$u(x)=\int^{+\infty}_{0}{e^{-xt^{2}}{dt}}$，则$u(x)$的初等函数表达式为$?$</p><h3 id="分析"><a href="#分析" class="headerlink" title="分析"></a>分析</h3><p>这道题的核心就是二重积分，直接求导(利用原函数存在定理)和求积分显然是不可取的。</p><p>我们可以人为的再建立一个$u(x)=\int^{+\infty}_{0}{e^{-xm^2}{dm}}$,这种形式是可以存在的，下来就简单了…</p><h3 id="解题"><a href="#解题" class="headerlink" title="解题"></a>解题</h3><p>通过运算使得$u^2(x)=\iint_{s}{e^{-x\big(t^2+m^2)}dtdm}$ , 其中$t\in \big(0,+\infty)$ $m\in \big(0,+\infty)$</p><p>因为被积函数中有平方和，因此利用极坐标解该二重积分：</p><p>$\begin{cases}t=rcos{\theta}\\m=rsin{\theta}\end{cases}$</p><p>$u^2(x)=\int^{\frac{\pi}{2}}_{0}d\theta\int^{+\infty}_{0}re^{-xr^2}dr=\frac{\pi}{2x}\int^{+\infty}_{0}{e^{-xr^2}}d\big(xr^2)={\frac{\pi}{2x}}{\int^{+\infty}_{0}e^{-p}dp}$ </p><p> $p=xr^{2}$</p><p>解得$u^{2}(x)=-\frac{\pi}{2x}$</p><p>则$u(x)=\sqrt{-\frac{\pi}{2x}}$</p><p>完毕！</p>]]></content>
<summary type="html"><p>预定31号题！非数竞赛一道题，第一次做没想到（+1）</p></summary>
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<category term="二重积分" scheme="http://sgnxotsmicf.com/tags/%E4%BA%8C%E9%87%8D%E7%A7%AF%E5%88%86/"/>
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