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search.xml
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<?xml version="1.0" encoding="utf-8"?>
<search>
<entry>
<title>民法关键名词解释</title>
<link href="/2022/09/22/%E6%B0%91%E6%B3%95%E5%85%B3%E9%94%AE%E5%90%8D%E8%AF%8D%E8%A7%A3%E9%87%8A/"/>
<url>/2022/09/22/%E6%B0%91%E6%B3%95%E5%85%B3%E9%94%AE%E5%90%8D%E8%AF%8D%E8%A7%A3%E9%87%8A/</url>
<content type="html"><![CDATA[<style>#web_bg{ background: url(https://gimg2.baidu.com/image_search/src=http%3A%2F%2Fpic1.win4000.com%2Fwallpaper%2F2020-07-16%2F5f1014f55de06.jpg&refer=http%3A%2F%2Fpic1.win4000.com&app=2002&size=f9999,10000&q=a80&n=0&g=0n&fmt=auto?sec=1666432279&t=d4d7271bee5ffc7fcc32a7c419707b2a)!important; /*重新定义background会导致原有定位属性失效，所以也需要再声明一次加权的定位属性*/ background-position: center !important; background-size: cover !important; background-repeat: no-repeat !important;}</style><h1 id="民法关键名词解释"><a class="markdownIt-Anchor" href="#民法关键名词解释"></a> 民法关键名词解释</h1><h2 id="民法"><a class="markdownIt-Anchor" href="#民法"></a> 民法</h2><h3 id="概念"><a class="markdownIt-Anchor" href="#概念"></a> 概念</h3><ul><li>调整平等主体的自然人、法人和非法人组织之间的人身关系和财产关系的法律规范的总和</li></ul><h3 id="任务"><a class="markdownIt-Anchor" href="#任务"></a> 任务</h3><ul><li>保障民事权益</li><li>调整民事关系</li><li>维护社会和经济秩序</li></ul><h2 id="民法的调整对象"><a class="markdownIt-Anchor" href="#民法的调整对象"></a> 民法的调整对象</h2><h3 id="平等性"><a class="markdownIt-Anchor" href="#平等性"></a> 平等性</h3><ul><li>法律地位平等</li><li>适用规则平等</li><li>权利保护平等</li></ul><h3 id="人身关系"><a class="markdownIt-Anchor" href="#人身关系"></a> 人身关系</h3><ul><li><p>人格权</p><ul><li>生命权、身体权、健康权<br />姓名权、肖像权</li></ul></li><li><p>特点</p><ul><li>非财产性</li><li>专属性</li></ul></li></ul><h3 id="一定身份"><a class="markdownIt-Anchor" href="#一定身份"></a> 一定身份</h3><ul><li>亲属关系（继承权、监护权）</li><li>知识产权（商标权、专利权）</li></ul><h3 id="财产关系重心交易关系"><a class="markdownIt-Anchor" href="#财产关系重心交易关系"></a> 财产关系（重心：交易关系）</h3><ul><li>财产归属关系</li><li>财产流转关系</li></ul><h2 id="民法的渊源表现形式"><a class="markdownIt-Anchor" href="#民法的渊源表现形式"></a> 民法的渊源（表现形式）</h2><h3 id="宪法"><a class="markdownIt-Anchor" href="#宪法"></a> 宪法</h3><h3 id="民事法律"><a class="markdownIt-Anchor" href="#民事法律"></a> 民事法律</h3><h3 id="行政法规"><a class="markdownIt-Anchor" href="#行政法规"></a> 行政法规</h3><h3 id="司法解释"><a class="markdownIt-Anchor" href="#司法解释"></a> 司法解释</h3><h3 id="国际条约和国际惯例"><a class="markdownIt-Anchor" href="#国际条约和国际惯例"></a> 国际条约和国际惯例</h3><h3 id="不违背公序良俗的习惯"><a class="markdownIt-Anchor" href="#不违背公序良俗的习惯"></a> 不违背公序良俗的习惯</h3><h2 id="民法的基本原则"><a class="markdownIt-Anchor" href="#民法的基本原则"></a> 民法的基本原则</h2><h3 id="平等原则"><a class="markdownIt-Anchor" href="#平等原则"></a> 平等原则</h3><ul><li>人格平等</li><li>法律地位平等</li><li>补救方法平等</li></ul><h3 id="意思自治原则"><a class="markdownIt-Anchor" href="#意思自治原则"></a> 意思自治原则</h3><ul><li>民事主体享有在法定范围内广泛的行为自由，并可以根据自己的意志产生、变更和消灭民事法律关系</li><li>私法任意性特点的集中体现</li><li>从事某种活动与否、选择行为内容和相对人、选择行为方式、选择救济方式</li></ul><h3 id="公平原则"><a class="markdownIt-Anchor" href="#公平原则"></a> 公平原则</h3><ul><li>民事法律内容应当合乎公平</li></ul><h3 id="诚实信用原则"><a class="markdownIt-Anchor" href="#诚实信用原则"></a> 诚实信用原则</h3><ul><li>不损害他人利益来获利</li></ul><h3 id="公序良俗原则"><a class="markdownIt-Anchor" href="#公序良俗原则"></a> 公序良俗原则</h3><h2 id="民事法律关系"><a class="markdownIt-Anchor" href="#民事法律关系"></a> 民事法律关系</h2><h3 id="核心"><a class="markdownIt-Anchor" href="#核心"></a> 核心</h3><ul><li>民事权利与义务</li></ul><h3 id="绝对法律关系"><a class="markdownIt-Anchor" href="#绝对法律关系"></a> 绝对法律关系</h3><ul><li>基于绝对权成立的法律关系，其他一切人都必须尊重</li></ul><h3 id="相对法律关系"><a class="markdownIt-Anchor" href="#相对法律关系"></a> 相对法律关系</h3><ul><li>以特定主体为限，相对主体旅行积极义务</li></ul><h3 id="主体"><a class="markdownIt-Anchor" href="#主体"></a> 主体</h3><ul><li>人（自然人和法人）</li></ul><h3 id="客体"><a class="markdownIt-Anchor" href="#客体"></a> 客体</h3><ul><li><p>财产</p><ul><li>有体财产</li><li>无体财产</li></ul></li><li><p>行为</p><ul><li>债权法律关系中，债务人的给付</li></ul></li><li><p>人身利益</p><ul><li><p>人格利益</p><ul><li>名誉、隐私、健康等</li></ul></li><li><p>身份利益</p><ul><li>特定的身份地位（亲属关系、非亲属的社会关系）（例如抚养权）</li></ul></li></ul></li></ul><h3 id="民事法律事实"><a class="markdownIt-Anchor" href="#民事法律事实"></a> 民事法律事实</h3><ul><li><p>能够引起民事法律《产生、变更或消灭》的《客观》情况</p></li><li><p>分类</p><ul><li><p>自然事实</p></li><li><p>人的行为</p><ul><li>表示行为（以人的意思表示为要素）</li><li>非表示行为（人主观上没有表示，客观上引起法律效果变化）</li></ul></li></ul></li></ul><h2 id="民事权利"><a class="markdownIt-Anchor" href="#民事权利"></a> 民事权利</h2><h3 id="核心-2"><a class="markdownIt-Anchor" href="#核心-2"></a> 核心</h3><ul><li>民事主体的利益</li></ul><h3 id="分类"><a class="markdownIt-Anchor" href="#分类"></a> 分类</h3><ul><li><p>1</p><ul><li><p>绝对权</p><ul><li>任何人都必须尊重的一个权利</li></ul></li><li><p>相对权</p><ul><li>请求特定人为一定行为或不为一定行为的权力</li></ul></li></ul></li><li><p>2</p><ul><li><p>支配权</p><ul><li>对权利客体直接支配并享受其利益</li></ul></li><li><p>请求权</p><ul><li>请求他人满足自己请求的利益</li></ul></li></ul></li><li><p>3</p><ul><li><p>形成权</p><ul><li>完全凭借单方意思表示就可以改变民事法律关系</li></ul></li><li><p>抗辩权</p><ul><li>用于对抗请求权或其他权利效力的权利</li></ul></li></ul></li><li><p>4</p><ul><li><p>既得权</p><ul><li>完整权，已经取得的权利</li></ul></li><li><p>期待权</p><ul><li>将未来可能取得的权利作为一种现实的合法权益加以保护</li></ul></li></ul></li><li><p>5</p><ul><li><p>原权利</p><ul><li>基于法律或当事人意思产生的权利</li></ul></li><li><p>救济权</p><ul><li>对原权利产生救援性作用的权利</li></ul></li></ul></li><li><p>6</p><ul><li><p>主权利</p><ul><li>能够独立存在</li></ul></li><li><p>从权利</p><ul><li>不能够独立存在</li></ul></li></ul></li><li><p>7</p><ul><li><p>专属权</p><ul><li>特定的民事主体</li></ul></li><li><p>非专属权</p><ul><li>不依附于民事主体</li></ul></li></ul></li></ul><h3 id="取得"><a class="markdownIt-Anchor" href="#取得"></a> 取得</h3><ul><li><p>原始取得</p><ul><li>独立的、不以他人既存权利为前提而取得的权利</li></ul></li><li><p>继受取得</p><ul><li>基于他人既存权利而取得的权利</li></ul></li></ul><h3 id="救济"><a class="markdownIt-Anchor" href="#救济"></a> 救济</h3><ul><li><p>自卫</p><ul><li>正当防卫</li><li>紧急避险</li></ul></li><li><p>自助</p><ul><li>情势紧急</li></ul></li></ul><h2 id="民事责任"><a class="markdownIt-Anchor" href="#民事责任"></a> 民事责任</h2><h3 id="按份责任"><a class="markdownIt-Anchor" href="#按份责任"></a> 按份责任</h3><ul><li>按照各自出资份额承担责任</li></ul><h3 id="连带责任"><a class="markdownIt-Anchor" href="#连带责任"></a> 连带责任</h3><ul><li>各义务人均有义务就共同责任向权利主体全部承担</li></ul><h3 id="违约责任"><a class="markdownIt-Anchor" href="#违约责任"></a> 违约责任</h3><ul><li>只能是合同关系当中</li></ul>]]></content>
<categories>
<category> 法学 </category>
</categories>
<tags>
<tag> 法学 </tag>
<tag> 名词解释 </tag>
<tag> 民法学 </tag>
</tags>
</entry>
<entry>
<title>刑法学关键名词解释</title>
<link href="/2022/09/22/%E5%88%91%E6%B3%95%E5%85%B3%E9%94%AE%E5%90%8D%E8%AF%8D%E8%A7%A3%E9%87%8A/"/>
<url>/2022/09/22/%E5%88%91%E6%B3%95%E5%85%B3%E9%94%AE%E5%90%8D%E8%AF%8D%E8%A7%A3%E9%87%8A/</url>
<content type="html"><![CDATA[<style>#web_bg{ background: url(https://gimg2.baidu.com/image_search/src=http%3A%2F%2Fup.enterdesk.com%2Fphoto%2F2010-5-11%2Fenterdesk.com-53B6DAC62FDECA76A436FAAA545415EC.png&refer=http%3A%2F%2Fup.enterdesk.com&app=2002&size=f9999,10000&q=a80&n=0&g=0n&fmt=auto?sec=1666418521&t=4d6f51b9a1bd7789f963b56ada040138)!important; /*重新定义background会导致原有定位属性失效，所以也需要再声明一次加权的定位属性*/ background-position: center !important; background-size: cover !important; background-repeat: no-repeat !important;}</style><h1 id="刑法关键名词解释"><a class="markdownIt-Anchor" href="#刑法关键名词解释"></a> 刑法关键名词解释</h1><h2 id="刑法的基本原则"><a class="markdownIt-Anchor" href="#刑法的基本原则"></a> 刑法的基本原则</h2><h3 id="罪刑法定"><a class="markdownIt-Anchor" href="#罪刑法定"></a> 罪刑法定</h3><h3 id="罪责刑相适应"><a class="markdownIt-Anchor" href="#罪责刑相适应"></a> 罪责刑相适应</h3><h3 id="适用刑法人人平等"><a class="markdownIt-Anchor" href="#适用刑法人人平等"></a> 适用刑法人人平等</h3><h2 id="刑法的空间效力"><a class="markdownIt-Anchor" href="#刑法的空间效力"></a> 刑法的空间效力</h2><h3 id="在中国领域内犯罪的一律适用本法"><a class="markdownIt-Anchor" href="#在中国领域内犯罪的一律适用本法"></a> 在中国领域内犯罪的一律适用本法</h3><h3 id="领域"><a class="markdownIt-Anchor" href="#领域"></a> 领域</h3><ul><li><p>国土</p></li><li><p>拟制领土</p><ul><li>船舶</li><li>航空器</li><li>不包含国际列车</li></ul></li></ul><h3 id="犯罪地遍在地说"><a class="markdownIt-Anchor" href="#犯罪地遍在地说"></a> 犯罪地：遍在地说</h3><ul><li>有一项在领域内即在我国犯罪</li></ul><h3 id="普遍管辖"><a class="markdownIt-Anchor" href="#普遍管辖"></a> 普遍管辖</h3><ul><li>对于外国人：或引渡或起诉</li></ul><h3 id="属人属地遍在地"><a class="markdownIt-Anchor" href="#属人属地遍在地"></a> 属人+属地+遍在地</h3><h2 id="从旧兼从轻"><a class="markdownIt-Anchor" href="#从旧兼从轻"></a> 从旧兼从轻</h2><h3 id="已经生效的判决不得适用此原则被推翻"><a class="markdownIt-Anchor" href="#已经生效的判决不得适用此原则被推翻"></a> 已经生效的判决不得适用此原则被推翻</h3><h3 id="仅适用于未决案"><a class="markdownIt-Anchor" href="#仅适用于未决案"></a> 仅适用于未决案</h3><h2 id="犯罪构成"><a class="markdownIt-Anchor" href="#犯罪构成"></a> 犯罪构成</h2><h3 id="内容"><a class="markdownIt-Anchor" href="#内容"></a> 内容</h3><ul><li>犯罪客体</li><li>犯罪主体</li><li>犯罪客观方面</li><li>犯罪主观方面</li></ul><h3 id="构成"><a class="markdownIt-Anchor" href="#构成"></a> 构成</h3><ul><li><p>单一的犯罪构成</p><ul><li>各要件都为单一要素：例如非法侵入住宅，主体（能承担刑事责任，客体是住宅不受侵犯的权利，客观方面是非法侵入的行为，主观方面为故意</li></ul></li><li><p>混合的犯罪构成</p><ul><li><p>复合的犯罪构成</p><ul><li>包含数个危害行为</li><li>包含数个犯罪对象</li></ul></li><li><p>择一的犯罪构成</p><ul><li>刑法条文中规定的犯罪构成诸要件要素中有可供选择余地的犯罪构成。例如，走私淫秽物品的犯罪目的，可以是牟利也可以是传播。</li></ul></li></ul></li><li><p>基本的犯罪构成</p><ul><li>刑法对某一犯罪的《单独》犯的《既遂》状态</li></ul></li><li><p>修正的犯罪构成</p><ul><li>故意犯罪的《未完成》形态、《主犯共犯》《胁从犯教唆犯》</li></ul></li><li><p>普通、加重的、减轻的犯罪构成</p></li></ul><h2 id="犯罪客体侵犯的社会关系"><a class="markdownIt-Anchor" href="#犯罪客体侵犯的社会关系"></a> 犯罪客体（侵犯的社会关系）</h2><h3 id="定义"><a class="markdownIt-Anchor" href="#定义"></a> 定义</h3><ul><li>被刑法保护的、被犯罪行为侵犯的社会关系</li></ul><h3 id="犯罪对象"><a class="markdownIt-Anchor" href="#犯罪对象"></a> 犯罪对象</h3><ul><li>被犯罪行为直接侵害的、被刑法所保护的、具体的人或物</li></ul><h3 id="一般客体共性"><a class="markdownIt-Anchor" href="#一般客体共性"></a> 一般客体（共性）</h3><ul><li>一切犯罪所共同侵犯的社会关系，该客体揭示了犯罪的本质</li></ul><h3 id="同类客体刑法制定的依据"><a class="markdownIt-Anchor" href="#同类客体刑法制定的依据"></a> 同类客体（刑法制定的依据）</h3><ul><li>某一类犯罪所共同侵犯的、我国刑法保护的社会关系</li></ul><h3 id="直接客体"><a class="markdownIt-Anchor" href="#直接客体"></a> 直接客体</h3><ul><li><p>定义</p><ul><li>受到犯罪行为直接侵犯的客体</li></ul></li><li><p>分类</p><ul><li><p>简单客体（单一客体）</p><ul><li>犯罪行为仅仅侵犯一种具体的社会关系</li></ul></li><li><p>复杂客体</p><ul><li>犯罪行为直接侵犯两种及以上的客体</li><li>主要客体、次要客体</li></ul></li></ul></li></ul><h2 id="犯罪客观方面犯罪事实"><a class="markdownIt-Anchor" href="#犯罪客观方面犯罪事实"></a> 犯罪客观方面（犯罪事实）</h2><h3 id="定义-2"><a class="markdownIt-Anchor" href="#定义-2"></a> 定义</h3><ul><li>能够说明犯罪行为对刑法所保护的客体造成了侵害的客观事实特征</li></ul><h3 id="包含"><a class="markdownIt-Anchor" href="#包含"></a> 包含</h3><ul><li><p>危害行为</p><ul><li><p>定义</p><ul><li>客观上危害了社会</li><li>主观上是人的意志或意识的行为（主动性）</li></ul></li><li><p>分类</p><ul><li><p>作为（不应为而为）</p></li><li><p>不作为（应为而不为）</p><ul><li><p>特征</p><ul><li>负有实施某种特定积极行为的法律义务，非道德义务</li><li>消极的身体活动</li><li>触犯刑法</li></ul></li><li><p>纯正的不作为犯</p><ul><li>只能由不作为形式完成的犯罪（例如遗弃罪）</li></ul></li><li><p>不纯正的不作为犯</p><ul><li>由不作为方式实施通常由作为方式实施的犯罪（例如医生通过不救人来杀人）</li></ul></li></ul></li><li><p>持有</p><ul><li><p>对（非法）物品的控制状态</p></li><li><p>注意</p><ul><li>能管理、控制即可，不是拿在手中</li><li>不一定直接持有，即使有第三者介入，仍然可能成立持有</li></ul></li></ul></li><li><p>实行行为</p><ul><li>构成要件行为，即刑法规定的要完成某个犯罪必须具有的行为</li></ul></li><li><p>非实行行为</p><ul><li>对犯罪起补充、唆使、帮助的行为（教唆犯、帮助犯）</li></ul></li></ul></li></ul></li><li><p>危害结果</p><ul><li><p>非构成要件结果</p><ul><li>行为犯、预备犯、未遂犯、中止犯</li><li>间接结果</li><li>精神损害</li></ul></li></ul></li><li><p>行为与结果的因果关系</p><ul><li><p>中断的因果关系</p><ul><li><p>介入因素的异常性大小</p></li><li><p>介入因素对结果的影响力是否超过最初行为的结果范围</p><ul><li>（甲想杀乙，只砍了一刀，丙开了一枪->丙故意杀人既遂，甲未遂）</li><li>（甲杀乙，已经杀成重伤，丙路过对乙进行辱骂，乙气急攻心死亡->甲故意杀人既遂）</li></ul></li></ul></li></ul></li><li><p>时间地点方法</p></li></ul><h2 id="犯罪主体"><a class="markdownIt-Anchor" href="#犯罪主体"></a> 犯罪主体</h2><h3 id="12-14岁犯故意杀人-伤害罪造成死亡结果的或者以特别残忍手段-致人重伤-造成残疾-情节恶劣的最高院核准追诉的应当负刑事责任是指具体的罪名"><a class="markdownIt-Anchor" href="#12-14岁犯故意杀人-伤害罪造成死亡结果的或者以特别残忍手段-致人重伤-造成残疾-情节恶劣的最高院核准追诉的应当负刑事责任是指具体的罪名"></a> 12-14岁：犯故意杀人、伤害罪，造成死亡结果的或者以特别残忍手段-致人重伤-造成残疾-情节恶劣的，最高院核准追诉的，应当负刑事责任（是指具体的罪名）</h3><h3 id="14岁-16岁负刑事责任故意杀人故意伤害致人重伤或死亡-强奸-抢劫-贩卖毒品-放火-爆炸-投毒仅仅指犯罪行为不是具体罪名"><a class="markdownIt-Anchor" href="#14岁-16岁负刑事责任故意杀人故意伤害致人重伤或死亡-强奸-抢劫-贩卖毒品-放火-爆炸-投毒仅仅指犯罪行为不是具体罪名"></a> 14岁-16岁负刑事责任：故意杀人，故意伤害致人重伤或死亡、强奸、抢劫、贩卖毒品、放火、爆炸、投毒（仅仅指犯罪行为，不是具体罪名）</h3><h3 id="16岁以上完全刑事责任能力"><a class="markdownIt-Anchor" href="#16岁以上完全刑事责任能力"></a> 16岁以上：完全刑事责任能力</h3><h3 id="岁数的计算"><a class="markdownIt-Anchor" href="#岁数的计算"></a> 岁数的计算</h3><ul><li>生日当天不算满岁，第二天才算满周岁</li></ul><h3 id="特殊身份"><a class="markdownIt-Anchor" href="#特殊身份"></a> 特殊身份</h3><ul><li><p>真正身份犯</p><ul><li>只能由某主体实施</li></ul></li><li><p>不真正身份犯</p><ul><li>某种特殊身份成立加重情节（国家公职人员、军警人员…）</li></ul></li></ul><h3 id="单位犯罪"><a class="markdownIt-Anchor" href="#单位犯罪"></a> 单位犯罪</h3><ul><li><p>必要条件：具有法人资格</p></li><li><p>一人公司视为刑法上的单位</p></li><li><p>犯罪体现单位意志（事前就要体现，事后追认不算单位意志）</p></li><li><p>采用双罚制</p><ul><li>罚单位罚金</li><li>罚直接责任人</li></ul></li></ul><h2 id="犯罪主观方面"><a class="markdownIt-Anchor" href="#犯罪主观方面"></a> 犯罪主观方面</h2><h3 id="犯罪故意"><a class="markdownIt-Anchor" href="#犯罪故意"></a> 犯罪故意</h3><ul><li><p>直接故意</p><ul><li>对危害结果持期望态度，积极追求犯罪结果的发生</li><li>认识到危害结果可能发生或者必然发生</li></ul></li><li><p>间接故意</p><ul><li>对危害结果持有放任、容忍态度</li><li>只能认识到危害结果可能发生，不能认识到危害结果必然发生</li></ul></li><li><p>犯意转化</p><ul><li>罪数=1，对象相同</li></ul></li><li><p>另起犯意</p><ul><li>罪数≥2，前一罪可既遂、中止、未遂，对象可以不同</li></ul></li></ul><h3 id="犯罪过失"><a class="markdownIt-Anchor" href="#犯罪过失"></a> 犯罪过失</h3><ul><li><p>疏忽大意</p></li><li><p>过于自信</p><ul><li>区分过于自信和间接故意：结果发生是否出乎行为人意料</li></ul></li></ul><h3 id="无罪过事件"><a class="markdownIt-Anchor" href="#无罪过事件"></a> 无罪过事件</h3><h3 id="刑法上的认识错误"><a class="markdownIt-Anchor" href="#刑法上的认识错误"></a> 刑法上的认识错误</h3><ul><li><p>法律认识错误</p><ul><li>假想无罪：实际有罪</li></ul></li><li><p>事实认识错误</p><ul><li><p>客体错误</p><ul><li>按照意图侵犯的客体定罪（想偷钱却发现里面是枪，按盗窃罪而不是盗窃枪支罪）</li></ul></li><li><p>对象错误</p><ul><li><p>想要侵犯的对象不存在</p><ul><li>想杀人却发现想杀的人其实是一只猫：犯罪未遂</li></ul></li><li><p>同类客体，不影响定罪</p><ul><li>想偷手机，发现包里是手表</li></ul></li></ul></li><li><p>工具错误</p><ul><li><p>成立未遂犯（工具不能犯）</p><ul><li>把白糖当成砒霜</li></ul></li></ul></li><li><p>因果关系错误</p><ul><li><p>坚持主客观相统一</p><ul><li>在饭菜中下毒，但被害人是噎死->犯罪未遂</li></ul></li><li><p>故意伤害（轻伤），被害失血过多而死->故意伤害既遂，过失致人死亡</p></li></ul></li><li><p>打击错误</p><ul><li>同类客体：想杀张三杀成李四：既遂</li><li>不同类客体：想杀张三杀成狗：未遂+另一罪（如果成立犯罪），数罪并罚</li></ul></li></ul></li></ul><h2 id="犯罪预备形态可以从轻-减轻或者免除处罚"><a class="markdownIt-Anchor" href="#犯罪预备形态可以从轻-减轻或者免除处罚"></a> 犯罪预备形态（”可以“从轻、减轻或者免除处罚）</h2><h3 id="为犯罪准备工具"><a class="markdownIt-Anchor" href="#为犯罪准备工具"></a> 为犯罪准备工具</h3><h3 id="为犯罪制造条件"><a class="markdownIt-Anchor" href="#为犯罪制造条件"></a> 为犯罪制造条件</h3><ul><li>打探被害人行踪、跟踪、埋伏、引诱被害人前往犯罪现场</li></ul><h3 id="可以从轻-减轻或者免除"><a class="markdownIt-Anchor" href="#可以从轻-减轻或者免除"></a> ”可以“从轻、减轻或者免除</h3><h2 id="犯罪未遂形态可以比照既遂犯从轻或减轻处罚"><a class="markdownIt-Anchor" href="#犯罪未遂形态可以比照既遂犯从轻或减轻处罚"></a> 犯罪未遂形态（”可以“比照既遂犯从轻或减轻处罚）</h2><h3 id="已经着手实施犯罪但因犯罪人意志以外的原因使得犯罪未得逞"><a class="markdownIt-Anchor" href="#已经着手实施犯罪但因犯罪人意志以外的原因使得犯罪未得逞"></a> 已经着手实施犯罪，但因犯罪人意志以外的原因使得犯罪未得逞</h3><h3 id="原因"><a class="markdownIt-Anchor" href="#原因"></a> 原因</h3><ul><li><p>犯罪人自身以外的原因</p></li><li><p>犯罪人自身能力的原因</p><ul><li>自己体能低下、作案手段拙劣、犯罪技巧欠缺</li></ul></li><li><p>主观认识错误</p></li></ul><h3 id="实行终了的未遂"><a class="markdownIt-Anchor" href="#实行终了的未遂"></a> 实行终了的未遂</h3><ul><li>行为人完成所有自以为能实现犯罪的行为</li></ul><h3 id="能犯未遂"><a class="markdownIt-Anchor" href="#能犯未遂"></a> 能犯未遂</h3><ul><li>可以既遂但是因为罪犯意志以外的原因没能既遂</li></ul><h3 id="不能犯未遂"><a class="markdownIt-Anchor" href="#不能犯未遂"></a> 不能犯未遂</h3><ul><li><p>对象不能犯</p><ul><li>犯罪对象不具有犯罪行为想要侵犯的客体（想强奸女性发现是男性）</li></ul></li><li><p>工具不能犯</p><ul><li>犯罪工具性能问题使得犯罪不能既遂（白糖当砒霜、劣质盗窃钥匙）</li></ul></li></ul><h2 id="犯罪中止必须从宽处罚"><a class="markdownIt-Anchor" href="#犯罪中止必须从宽处罚"></a> 犯罪中止（“必须”从宽处罚）</h2><h3 id="主动停止犯罪并积极避免危害结果发生二者必须同时满足"><a class="markdownIt-Anchor" href="#主动停止犯罪并积极避免危害结果发生二者必须同时满足"></a> 主动停止犯罪并积极避免危害结果发生（二者必须同时满足）</h3><h2 id="共同犯罪"><a class="markdownIt-Anchor" href="#共同犯罪"></a> 共同犯罪</h2><h3 id="成立条件缺一不可"><a class="markdownIt-Anchor" href="#成立条件缺一不可"></a> 成立条件（缺一不可）</h3><ul><li><p>两人及以上共同的犯罪故意（注意：只能是故意犯罪才能成立共同犯罪）</p></li><li><p>有意思联络</p></li><li><p>共同行为</p><ul><li>作为和不作为可随意搭配</li></ul></li></ul><h3 id="存在共同犯罪的预备或中止共谋而未实行或主动放弃"><a class="markdownIt-Anchor" href="#存在共同犯罪的预备或中止共谋而未实行或主动放弃"></a> 存在共同犯罪的预备或中止（共谋而未实行或主动放弃）</h3><h3 id="共同过失犯罪"><a class="markdownIt-Anchor" href="#共同过失犯罪"></a> 共同过失犯罪</h3><ul><li>存在共同过失犯罪，但不成立共犯</li></ul><h3 id="片面共犯"><a class="markdownIt-Anchor" href="#片面共犯"></a> 片面共犯</h3><ul><li>两人共同完成某个罪行，但一方存在犯罪故意，另一方无犯意</li></ul><h3 id="实行过限的共同犯罪"><a class="markdownIt-Anchor" href="#实行过限的共同犯罪"></a> 实行过限的共同犯罪</h3><ul><li><p>过限部分不成立共同犯罪</p></li><li><p>教唆犯</p><ul><li>被教唆者犯的罪是教唆者所教唆的罪的加重结果：教唆者负责，若不是则教唆者不负责</li><li>看行为是否符合教唆者意图（教唆人抢劫，结果被教唆人强奸了房主，则教唆犯不对强奸行为负责）</li></ul></li></ul><h3 id="共同犯罪的分类"><a class="markdownIt-Anchor" href="#共同犯罪的分类"></a> 共同犯罪的分类</h3><ul><li><p>任意的共同犯罪</p><ul><li>共同犯下的罪是一个人也可以实施的犯罪</li></ul></li><li><p>必要的共同犯罪</p><ul><li>只能由多人完成这个罪名（例如：聚众xxx）</li></ul></li><li><p>简单的共同犯罪</p><ul><li>犯罪成员实施单一犯罪行为</li></ul></li><li><p>复杂的共同犯罪</p><ul><li>有分工的犯罪行为</li></ul></li><li><p>一般的共同犯罪</p><ul><li>犯罪完成即解散团伙</li></ul></li><li><p>有组织的共同犯罪（犯罪集团）</p><ul><li>团伙性、组织性</li></ul></li></ul><h3 id="共犯的刑事责任"><a class="markdownIt-Anchor" href="#共犯的刑事责任"></a> 共犯的刑事责任</h3><ul><li><p>主犯</p><ul><li>犯罪集团的首要分子</li><li>所有共犯成员都可以是主犯</li></ul></li><li><p>教唆犯</p><ul><li>通常起主要作用，负主要责任</li><li>被教唆人没有犯所教唆的罪，“可以”从轻或者减轻处罚</li><li>教唆未遂与未遂的教唆犯：教唆未遂的被教唆人没有实行罪行；未遂的教唆犯的被教唆人犯罪未遂</li></ul></li></ul><h2 id="刑罚裁量"><a class="markdownIt-Anchor" href="#刑罚裁量"></a> 刑罚裁量</h2><h3 id="服刑期间发现漏罪"><a class="markdownIt-Anchor" href="#服刑期间发现漏罪"></a> 服刑期间发现漏罪</h3><ul><li>先并后减</li></ul><h3 id="服刑期间又犯新罪"><a class="markdownIt-Anchor" href="#服刑期间又犯新罪"></a> 服刑期间又犯新罪</h3><ul><li>先减后并</li></ul><h3 id="自首"><a class="markdownIt-Anchor" href="#自首"></a> 自首</h3><ul><li><p>一般自首</p><ul><li>犯罪嫌疑人自动投案</li><li>投案行为基于本人意志</li><li>亲属扭送也算自首</li><li>必须如实供述自己犯罪事实（是犯罪的，且是自己犯下的罪的事实而不是其他什么事实）</li></ul></li><li><p>准自首</p><ul><li>如实供述司法机关《还没有掌握》的本人的其他的犯罪事实</li><li>司法机关没有证据证实其他犯罪行为</li></ul></li><li><p>特别自首</p><ul><li>主体：行贿那一类的罪</li></ul></li><li><p>单位自首</p><ul><li>单位直接负责的主管人员主动投案</li><li>单位没有自首，但直接责任人员主动投案的，该直接责任人员认定为自首</li></ul></li></ul><h3 id="坦白"><a class="markdownIt-Anchor" href="#坦白"></a> 坦白</h3><ul><li><p>时间</p><ul><li>罪犯《被动》归案以后，被提起公诉之前</li></ul></li><li><p>《主动》《如实》供述自己所犯罪行</p></li><li><p>可以供述机关已经掌握的事实，也可以供述机关没有掌握的事实（以自首论）</p></li></ul><h3 id="立功"><a class="markdownIt-Anchor" href="#立功"></a> 立功</h3><ul><li>时间：到案后、判决前</li><li>揭发他人犯罪事实+查证属实/提供重要线索</li><li>性质：有利于国家和社会</li></ul><h3 id="数罪并罚"><a class="markdownIt-Anchor" href="#数罪并罚"></a> 数罪并罚</h3><ul><li><p>规律</p><ul><li>有期+拘役=有期</li><li>没收全部财产+罚金=没收全部财产</li><li>无期+无期+…+无期=无期</li><li>罚金+罚金=罚金x1，加重</li><li>有期/拘役+管制=不变</li></ul></li><li><p>漏罪</p><ul><li><p>犯一罪，漏数罪</p><ul><li>先并再判</li></ul></li><li><p>犯数罪，漏一罪</p><ul><li>先判再并</li></ul></li><li><p>犯数罪，漏数罪</p><ul><li>先判再并</li></ul></li></ul></li></ul><h3 id="缓刑与死缓"><a class="markdownIt-Anchor" href="#缓刑与死缓"></a> 缓刑与死缓</h3><ul><li><p>不同</p><ul><li>缓刑不予关押且不能减刑，死缓必须关押</li></ul></li><li><p>缓刑的适用</p><ul><li>三年有期以下、罪行较轻</li><li>有悔罪表现、没有再犯罪的危险</li><li>犯罪分子不是累犯或者犯罪集团的首要分子</li></ul></li></ul><h3 id="减刑"><a class="markdownIt-Anchor" href="#减刑"></a> 减刑</h3><ul><li><p>条件</p><ul><li>被判处管制、拘役、有期、无期的犯罪分子</li></ul></li><li><p>可以减刑</p><ul><li>有立功表现或者确有悔改表现</li></ul></li><li><p>应当减刑</p><ul><li>重大立功表现</li></ul></li><li><p>限度</p><ul><li>不应少于原判的一半</li><li>无期不少于13年</li></ul></li><li><p>时间</p><ul><li><p>有期</p><ul><li>不满5年：执行一年以上，每次间隔1年</li><li>5<=x<10：执行一年6个月以上，每次间隔1年</li><li>x>=10：执行2年以上，每次间隔超过1年6个月</li></ul></li><li><p>无期</p><ul><li>执行2年以上</li></ul></li></ul></li><li><p>幅度</p><ul><li><p>有期、拘役、管制</p><ul><li>悔改或立功：减不超9个月</li><li>悔改+立功=不超1年</li><li>重大立功=不超1年6个月</li><li>重大立功+悔改=不超2年</li></ul></li><li><p>无期</p><ul><li>悔改/立功=22年</li><li>悔改+立功=21<x<22</li><li>重大立功= 20<x<21</li><li>重大立功+悔改=19<x<20</li></ul></li></ul></li></ul><h3 id="假释"><a class="markdownIt-Anchor" href="#假释"></a> 假释</h3><ul><li>有期执行一半以上，无期13年以上，死缓15年以上</li><li>考验期为未执行完毕的刑期</li><li>暴力犯罪、累犯、有组织的暴力性犯罪罪犯不得假释</li><li>死缓减为有期不得假释</li><li>无期假释考验期：10年</li></ul><h3 id="追诉时效"><a class="markdownIt-Anchor" href="#追诉时效"></a> 追诉时效</h3><ul><li><p>法定最高刑</p><ul><li>5年以下（不含）：5年</li><li>5年（含）-10年以下（不含）：10年</li><li>10年以上（含）：15年</li><li>无期、死刑：20年（可以经过核准后追诉）</li></ul></li><li><p>起算</p><ul><li>不持续的犯罪：犯罪时</li><li>持续的犯罪：犯罪终止时</li></ul></li><li><p>停止计算</p><ul><li>立案时刻停止计算追诉期</li></ul></li><li><p>罪犯逃匿的躲避侦察的，不受追诉时效限制</p></li></ul><h2 id="刑罚的体系和种类"><a class="markdownIt-Anchor" href="#刑罚的体系和种类"></a> 刑罚的体系和种类</h2><h3 id="体系"><a class="markdownIt-Anchor" href="#体系"></a> 体系</h3><ul><li><p>主刑</p><ul><li>管制、拘役、有期徒刑、无期徒刑、死刑</li></ul></li><li><p>附加刑</p><ul><li>罚金、剥夺政治权利、没收财产、驱逐出境</li></ul></li></ul><h3 id="种类"><a class="markdownIt-Anchor" href="#种类"></a> 种类</h3><ul><li><p>主刑</p><ul><li><p>管制</p><ul><li>羁押一天抵两天刑期</li><li>3个月以上两年以下，数罪并罚不超3年</li><li>机关：公安机关</li></ul></li><li><p>拘役</p><ul><li>1-6个月，数罪并罚不超1年</li></ul></li><li><p>有期徒刑</p><ul><li>6个月以上15年以下，数罪并罚总刑期<35的，最高不超20年；≥35的，不超过25年；</li><li>死缓减为有期的，定为25年</li></ul></li><li><p>无期徒刑（不能独立适用）</p><ul><li>必须附带剥夺政治权利终身</li><li>减刑后实际服刑不能少于13年</li></ul></li><li><p>死刑</p><ul><li>《犯罪》时不满18的、《审判》时（羁押时）怀孕的（或者人工流产或者自然流产的）不适用死刑</li><li>审判时（不是犯罪时）满75岁的，不是”以特别残忍手段致人死亡“的，不适用死刑</li><li>全部应当报请最高人民法院批准</li><li>死缓期间《故意》（不是过失）犯罪、不核准死刑的，死缓从罪行查实那天起重新计算</li><li>必须附加剥夺政治权利终身</li></ul></li></ul></li><li><p>附加刑（可以独立适用）</p><ul><li><p>罚金</p><ul><li>民事赔偿优先（被害人所遭受的物质损失（不含精神损失费））</li></ul></li><li><p>剥夺政治权利</p><ul><li><p>担任国家机关职务、担任国有企业职务</p></li><li><p>期限与执行</p><ul><li>死刑、无期徒刑：主刑执行之日起计算</li><li>死缓减为无期和无期减为有期时，剥夺政治权利的年限应该修改为3-10年，从减刑之后的有期徒刑执行完毕或者假释之日起算</li><li>独立使用：1-5年</li><li>有期徒刑、拘役，执行完毕或者假释之日开始计算；效力当然适用于主刑执行期间</li></ul></li></ul></li><li><p>没收财产</p><ul><li><p>”应当“（不是可以）为犯罪人本人及其抚养的家属（可以是成年家属）保留必需的生活费用</p></li><li><p>不得没收属于或者应当属于其家属的部分</p></li><li><p>债务清偿</p><ul><li>判处没收财产之前所负的债务</li><li>正当合法债务</li><li>经过债权人请求</li></ul></li></ul></li></ul></li></ul><h2 id="刑罚目的-功能"><a class="markdownIt-Anchor" href="#刑罚目的-功能"></a> 刑罚目的、功能</h2><h3 id="目的"><a class="markdownIt-Anchor" href="#目的"></a> 目的</h3><ul><li><p>一般预防论</p><ul><li>预防主体：犯罪人以外的一般人</li></ul></li><li><p>特殊预防论</p><ul><li>预防主体：使已经犯罪的人不再犯罪的效果</li></ul></li></ul><h3 id="功能"><a class="markdownIt-Anchor" href="#功能"></a> 功能</h3><ul><li><p>威慑功能</p><ul><li>意图实施犯罪的人</li></ul></li><li><p>教育鼓励功能</p><ul><li>公民</li></ul></li></ul><h2 id="刑事责任"><a class="markdownIt-Anchor" href="#刑事责任"></a> 刑事责任</h2><h3 id="产生阶段"><a class="markdownIt-Anchor" href="#产生阶段"></a> 产生阶段</h3><ul><li>从犯罪行为实施起，到立案止</li></ul><h3 id="确认阶段"><a class="markdownIt-Anchor" href="#确认阶段"></a> 确认阶段</h3><ul><li>立案时起，到审判生效止</li></ul><h3 id="实现阶段"><a class="markdownIt-Anchor" href="#实现阶段"></a> 实现阶段</h3><ul><li>审判生效起，到刑罚执行完毕止</li></ul><h2 id="罪数形态"><a class="markdownIt-Anchor" href="#罪数形态"></a> 罪数形态</h2><h3 id="实质的一罪"><a class="markdownIt-Anchor" href="#实质的一罪"></a> 实质的一罪</h3><ul><li><p>继续犯</p><ul><li>基于一个犯罪故意实施《一种》危害行为</li><li>持续侵犯《同一》直接客体</li><li>不法状态与犯罪行为同时存在</li></ul></li><li><p>想象竞合犯</p><ul><li>想犯几个罪、实施《仅1个》危害行为、触犯数个罪名</li><li>从一重罪</li><li>若该行为同时触犯的几个法条之间存在重叠，则称这种犯罪为法条竞合犯，而不是想象竞合犯</li></ul></li><li><p>结果加重犯</p></li></ul><h3 id="法定的一罪"><a class="markdownIt-Anchor" href="#法定的一罪"></a> 法定的一罪</h3><ul><li><p>结合犯</p><ul><li>行为人犯的几个不同罪加起来等于一个另外的罪</li><li>处断：按照结合的罪名处罚</li><li>我国刑法没有规定结合犯</li></ul></li><li><p>集合犯（惯犯）</p><ul><li>以实施多次犯罪行为为目的</li><li>实施了数个同种犯罪行为</li><li>例如：“以赌博为业”，“非法行医”</li></ul></li></ul><h3 id="处断的一罪其实罪犯犯了不止一个罪只是将他们综合起来按一个罪处理"><a class="markdownIt-Anchor" href="#处断的一罪其实罪犯犯了不止一个罪只是将他们综合起来按一个罪处理"></a> 处断的一罪（其实罪犯犯了不止一个罪，只是将他们综合起来按一个罪处理）</h3><ul><li><p>连续犯（按照该罪从重处罚）</p><ul><li>同一犯罪故意</li><li>想要连续实施这种犯罪</li><li>连续实施同一罪名的罪</li><li>是数个行为，数个独立的罪</li></ul></li><li><p>牵连犯（刑法无规定的，从一重罪）</p><ul><li>基于最终犯罪目的</li><li>前面犯的罪为最终之罪提供条件</li><li>触犯不同罪名</li></ul></li><li><p>吸收犯（按照吸收之罪处罚）</p><ul><li>一罪被另一罪吸收</li><li>重罪吸收轻罪</li><li>实行了数个基本性质相同的危害行为</li></ul></li></ul>]]></content>
<categories>
<category> 法学 </category>
</categories>
<tags>
<tag> 法学 </tag>
<tag> 刑法学 </tag>
<tag> 大纲 </tag>
<tag> 名词解释 </tag>
</tags>
</entry>
<entry>
<title>宪法学</title>
<link href="/2022/09/22/%E5%AE%AA%E6%B3%95%E5%AD%A6/"/>
<url>/2022/09/22/%E5%AE%AA%E6%B3%95%E5%AD%A6/</url>
<content type="html"><![CDATA[<style>#web_bg{ background: url(https://gimg2.baidu.com/image_search/src=http%3A%2F%2Fimg2.niutuku.com%2Fdesk%2F1208%2F1436%2Fntk-1436-14765.jpg&refer=http%3A%2F%2Fimg2.niutuku.com&app=2002&size=f9999,10000&q=a80&n=0&g=0n&fmt=auto?sec=1666418255&t=f4c37bb612ba86469eb25d3e03cb9574)!important; /*重新定义background会导致原有定位属性失效，所以也需要再声明一次加权的定位属性*/ background-position: center !important; background-size: cover !important; background-repeat: no-repeat !important;}</style><h1 id="宪法学"><a class="markdownIt-Anchor" href="#宪法学"></a> 宪法学</h1><h2 id="第一章总论"><a class="markdownIt-Anchor" href="#第一章总论"></a> 第一章：总论</h2><h3 id="第一节"><a class="markdownIt-Anchor" href="#第一节"></a> 第一节</h3><ul><li><p>宪法的特征</p><ul><li><p>规定了国家最根本的问题</p></li><li><p>制定修改更严格</p></li><li><p>最高法律效力</p><ul><li>体现1:宪法是制定其他法律的依据任何法律违宪则无效</li><li>体现2:宪法是一切国家机关、社会团体和全体公民的最高行为准则</li></ul></li></ul></li><li><p>宪法的本质</p><ul><li><p>1.民主制度化、法律化的基本形式</p></li><li><p>2.各种政治力量对比关系的集中体现</p><ul><li>阶级斗争的产物</li><li>规定社会各阶级在国家中的地位及相互关系</li><li>宪法随着阶级力量对比关系的变化而变化</li></ul></li></ul></li></ul><h3 id="第二节"><a class="markdownIt-Anchor" href="#第二节"></a> 第二节</h3><ul><li><p>宪法的分类</p><ul><li>成文/不成文</li><li>刚性/柔性</li><li>钦定/民定/协定</li><li>近代/现代</li></ul></li><li><p>宪法的渊源（表现形式）</p><ul><li><p>成文法国家</p><ul><li>宪法典</li><li>宪法解释</li></ul></li><li><p>不成文法国家</p><ul><li>宪法性法律</li><li>宪法判例</li><li>宪法惯例</li></ul></li></ul></li></ul><h3 id="第三节"><a class="markdownIt-Anchor" href="#第三节"></a> 第三节</h3><ul><li><p>宪法的制定</p><ul><li><p>制宪机关</p><ul><li>我国：一届全国人大一次会议</li></ul></li><li><p>制宪程序</p><ul><li>设立制宪机关</li><li>提出宪法草案</li><li>通过宪法草案</li></ul></li></ul></li><li><p>宪法的解释</p><ul><li><p>有权解释/学理解释（解释有无法律效力）</p><ul><li>只有全国人大常委会有权解释宪法，全国人大无权解释宪法</li></ul></li><li><p>语法（文义）/逻辑/历史/系统（体系）/目的解释</p></li><li><p>合宪解释与补充解释</p></li><li><p>原则</p><ul><li>符合宪法的基本原则和精神</li><li>符合宪法规定的国家的根本任务和目的</li><li>协调宪法的基本原则和内容</li><li>协调宪法规范与社会的关系</li></ul></li></ul></li><li><p>宪法的修改</p><ul><li><p>必要性</p><ul><li>1.使宪法的规定适应社会实际的发展和变化</li><li>2.弥补宪法规范在实施过程中出现的缺漏</li></ul></li><li><p>程序</p><ul><li><p>提案-先决投票-公告-议决-公布</p></li><li><p>我国修宪提案主体</p><ul><li>全国人大常委会</li><li>1/5以上全国人大代表（600人以上）</li></ul></li></ul></li></ul></li></ul><h3 id="第四节"><a class="markdownIt-Anchor" href="#第四节"></a> 第四节</h3><ul><li><p>宪法关系</p><ul><li>国家与公民：最核心的权利义务关系</li><li>国家与国内各民族</li><li>国家与社会团体、企业、事业组织以及其他组织</li><li>国家机关之间</li><li>国家与政党</li><li>国家与外国人、无国籍人</li></ul></li><li><p>宪法规范</p><ul><li><p>特点</p><ul><li>政治性、最高性、原则性、组织性和限制性</li></ul></li><li><p>种类</p><ul><li>1.授权性/义务性规范</li><li>2.宣告性/确认性规范</li><li>3.倡导性/任意性/强制性规范</li><li>4.保护性/奖励性/制裁性规范</li></ul></li></ul></li></ul><h3 id="第五节"><a class="markdownIt-Anchor" href="#第五节"></a> 第五节</h3><ul><li><p>宪法的效力</p><ul><li><p>空间效力</p><ul><li>一个国家的领土</li></ul></li><li><p>时间效力</p></li><li><p>对人效力</p><ul><li>公权力主体</li><li>自然人主体</li></ul></li><li><p>对事效力</p></li><li><p>宪法序言：与宪法法律效力相同</p></li></ul></li><li><p>宪法的作用：确认和巩固、限制和规范</p><ul><li>确认和规范国家权力</li><li>保障公民基本权利：最主要，最核心的价值</li><li>维护国家法制统一</li><li>确认经济制度、促进经济发展</li><li>维护国家统一和世界和平</li></ul></li></ul><h2 id="第二章宪法的历史发展"><a class="markdownIt-Anchor" href="#第二章宪法的历史发展"></a> 第二章：宪法的历史发展</h2><h3 id="资本主义宪法"><a class="markdownIt-Anchor" href="#资本主义宪法"></a> 资本主义宪法</h3><ul><li><p>国家</p><ul><li><p>1.英国（宪法之母）</p><ul><li><p>不成文宪法</p><ul><li>1215.自由大宪章</li><li>权利法案</li><li>王位继承法</li></ul></li></ul></li><li><p>2.美国（1787美国宪法是世界上第一部成文宪法）</p><ul><li>1776《独立宣言》（马克思评价：第一个人权宣言）</li><li>1777《邦联条例》</li><li>1787《美利坚合众国宪法》（美国宪法）</li><li>1791《权利法案》</li></ul></li><li><p>3.法国</p><ul><li>1789《人权宣言》（保障权利，权力分立）</li><li>1791 欧洲第一部成文宪法</li><li>1958《第五共和国宪法》（沿用至今）</li></ul></li><li><p>4.日本</p><ul><li><p>1889《大日本帝国宪法》（明治宪法）</p></li><li><p>1946《日本国宪法》</p><ul><li>确立了和平主义原则（放弃战争力量，不承认交战权）</li></ul></li></ul></li><li><p>5.德国</p><ul><li>1919《德意志国宪法》（魏玛宪法）（标志着近代宪法向现代宪法的转型）</li><li>1949《德国基本法》</li></ul></li></ul></li></ul><h3 id="社会主义宪法"><a class="markdownIt-Anchor" href="#社会主义宪法"></a> 社会主义宪法</h3><ul><li><p>国家</p><ul><li><p>苏联</p><ul><li>1918《苏俄宪法》（世界上第一部社会主义类型的宪法）</li></ul></li></ul></li></ul><h3 id="中华人民共和国成立前的宪法"><a class="markdownIt-Anchor" href="#中华人民共和国成立前的宪法"></a> 中华人民共和国成立前的宪法</h3><ul><li><p>清末预备立宪</p><ul><li>1908《钦定宪法大纲》</li><li>1911《十九信条》</li></ul></li><li><p>1912.《中华民国临时约法》（中国宪法史上第一部，也是唯一一部资产阶级宪法性质的文件）</p><ul><li>1.确立资产阶级民主共和国的国家制度</li><li>2.规定人民享有较为广泛的权利和自由</li></ul></li><li><p>北洋军阀</p><ul><li><p>1913《中华民国宪法（草案）》</p></li><li><p>1914《中华民国约法》</p><ul><li>规定了“三权鼎立”的政权组织形式</li><li>规人民享有广泛的权利和自由</li></ul></li><li><p>1923《中华民国宪法》（贿选宪法）</p></li><li><p>1924《中华民国宪法草案》</p></li></ul></li><li><p>南京国民政府</p><ul><li>1931《中华民国训政时期约法》</li><li>1936 ”五五宪草“</li><li>1946《中华民国宪法》</li></ul></li><li><p>革命根据地宪法文件性文件</p><ul><li>1931《宪法大纲》（人民政权制定并公布的第一部宪法性文件）</li><li>1941《施政纲领》</li><li>1946《陕甘宁边区宪法原则》</li></ul></li></ul><h3 id="我国现行宪法的产生和发展"><a class="markdownIt-Anchor" href="#我国现行宪法的产生和发展"></a> 我国现行宪法的产生和发展</h3><ul><li><p>1949《中国人民政治协商会议共同纲领》（由政协通过）</p></li><li><p>1954年宪法</p></li><li><p>1975年宪法</p></li><li><p>1978年宪法</p><ul><li>1979年第一次修改</li><li>1980年第二次修改（取消了“大鸣、大放、大辩论、大字报）</li></ul></li><li><p>1982年宪法</p><ul><li><p>内容</p><ul><li><p>1.确立指导思想和根本任务</p></li><li><p>2.完善了对公民权利的保障</p></li><li><p>3.总结历史经验，加强社会主义民主法治建设</p></li><li><p>4.维护国家统一和民族团结，保障“一国两制”</p></li><li><p>5.完善国家机构体系</p><ul><li>废除领导干部任职终身制</li><li>恢复国家主席设置</li></ul></li></ul></li></ul></li><li><p>1988年-1次修正</p><ul><li>1.确认了私营经济的宪法地位（注意是私营经济，不是个体经济）</li><li>2.修改了土地政策，明确了土地使用权可以依法转让</li></ul></li><li><p>1993年-2次修正</p><ul><li>1.明确我国处于社会主义初级阶段</li><li>2.明确规定逐步实现现代化，把我国建设成富强、民主、文明的社会主义国家</li><li>3.确认多党合作和政协制度将长期存在（即将政协制度写入宪法）</li><li>4.规定实施社会主义市场经济</li><li>5.将县、不设区的市、市辖区的人民代表大会每届任期由3年改为5年</li></ul></li><li><p>1999年-3次修正</p><ul><li>1.确立邓小平理论的指导地位</li><li>2.确认实行“依法治国，建设社会主义法治国家”的治国方略</li><li>3.规定基本经济制度，分配制度</li><li>4.将宪法第二十八条”反革命活动“修改为”危害国家安全的犯罪活动“</li></ul></li><li><p>2004年-4次修正</p><ul><li>1.确认三个代表重要思想的指导地位</li><li>2.规定国家可以征用公民私有财产并给予补偿</li><li>3.规定国家主席可以进行国事活动</li><li>4.将乡、镇的人民代表大会的任期延长为5年</li><li>5.建立社保制度</li><li>6.国家尊重和保障人权</li><li>7.国家鼓励支持引导非公有制经济发展</li><li>8.爱国统一战线中添加“社会主义事业的建设者”</li><li>9.明确我国国歌是”义勇军进行曲“</li></ul></li><li><p>2018年-5次修正</p><ul><li>1.确立科学发展观与习思的指导地位</li><li>2.调整第二个百年奋斗目标</li><li>3.监察委员会</li><li>4.增加倡导社会主义核心价值观相关内容</li><li>5.修改主席任期</li><li>6.将”全国人大法律委员会“更名为”全国人大宪法和法律委员会“</li><li>7.充实和平外交政策方面的内容</li><li>8.充实完善我国革命和建设发展历程的内容</li></ul></li></ul><h2 id="第三章宪法的指导思想和基本原则"><a class="markdownIt-Anchor" href="#第三章宪法的指导思想和基本原则"></a> 第三章：宪法的指导思想和基本原则</h2><h3 id="指导思想"><a class="markdownIt-Anchor" href="#指导思想"></a> 指导思想</h3><ul><li><p>内容</p><ul><li>马克思列宁主义</li><li>毛泽东思想</li><li>邓小平理论，三个代表重要思想（由邓提出）</li><li>科学发展观（胡锦涛）</li><li>习近平新时代中国特色社会主义思想</li></ul></li><li><p>作用</p><ul><li>1.是全国各族人民团结奋斗的共同思想基础</li><li>2.党的主张和人民意志的有机统一</li><li>3.宪法制定、修改和实施的根本依据</li></ul></li></ul><h3 id="基本原则"><a class="markdownIt-Anchor" href="#基本原则"></a> 基本原则</h3><ul><li><p>坚持中国共产党的领导</p></li><li><p>人民主权</p></li><li><p>社会主义法治</p></li><li><p>尊重和保障人权</p></li><li><p>权利的监督与制约</p><ul><li>1.人民对国家权力的监督</li><li>2.公民对国家机关和国家工作人员的监督</li><li>3.国家机关之间的制约和监督</li></ul></li><li><p>民主集中制</p></li></ul><h2 id="第四章国家性质和国家形式"><a class="markdownIt-Anchor" href="#第四章国家性质和国家形式"></a> 第四章：国家性质和国家形式</h2><h3 id="国家性质国体"><a class="markdownIt-Anchor" href="#国家性质国体"></a> 国家性质：国体；</h3><p>我国国体：人民民主专政<br />（国家制度的核心）<br />（国体不能脱离政体而单独存在）</p><ul><li><p>有关内容</p><ul><li>我国的国体：人民民主专政</li><li>根本制度：社会主义制度</li></ul></li><li><p>人民民主专政——主要内容</p><ul><li><p>必然要求和根本标志：工人阶级对社会主义国家的领导</p></li><li><p>阶级基础：工农联盟</p></li><li><p>中国共产党的领导是中国特色社会主义最本质的特征</p></li><li><p>新型民主与新型专政的结合</p></li><li><p>爱国统一战线</p><ul><li>2004年宪法修正案：在关于爱国统一战线的表述中增加了“社会主义事业的建设者”</li></ul></li></ul></li></ul><h3 id="中国特色社会主义最本质的特征中国共产党的领导"><a class="markdownIt-Anchor" href="#中国特色社会主义最本质的特征中国共产党的领导"></a> 中国特色社会主义最本质的特征：中国共产党的领导</h3><h3 id="国家形式"><a class="markdownIt-Anchor" href="#国家形式"></a> 国家形式</h3><ul><li><p>政权组织形式（政体）<br />我国政体：人民代表大会制度（也是根本政治制度）</p><ul><li><p>内容</p><ul><li>政体是国家最重要的外在表现形态</li><li>政体是国家政权机关组织和活动的系统体制</li><li>国体决定政体，政体对国体具有反作用</li></ul></li><li><p>分类</p><ul><li><p>资本主义国家的政体</p><ul><li><p>1.君主立宪制（英国（议会君主立宪制）、比利时、瑞典、荷兰、日本）</p></li><li><p>2.共和制</p><ul><li>议会内阁制（德日意）</li><li>总统制（美）</li><li>半总统制（法）</li><li>委员会制 （瑞士）</li></ul></li></ul></li><li><p>社会主义国家的政体</p><ul><li>1.公社制</li><li>2.苏维埃制</li><li>3.人民代表会议制</li></ul></li></ul></li><li><p>人民代表大会制度：人民当家做主的根本途径和最高实现形式</p></li></ul></li><li><p>结构形式</p><ul><li><p>单一制</p><ul><li>我国为统一、多民族的单一制国家</li></ul></li><li><p>联邦制</p></li></ul></li><li><p>行政区划</p><ul><li><p>我国的行政区划原则</p><ul><li>1.便于人民群众行使国家权力</li><li>2.有利于各民族之间团结</li><li>3.有利于经济社会发展和国防建设</li><li>4.尊重历史沿革和文化传统</li></ul></li><li><p>批准关系</p><ul><li>全国人大：批准省、自治区和直辖市的建制</li><li>国务院：除全国人大部分的其他部分</li></ul></li><li><p>民族区域自治地方：区、州、县</p></li></ul></li><li><p>国家标志</p><ul><li>国旗</li><li>国歌</li><li>国徽</li><li>首都</li></ul></li></ul><h2 id="第六章公民的基本权利和义务"><a class="markdownIt-Anchor" href="#第六章公民的基本权利和义务"></a> 第六章：公民的基本权利和义务</h2><h3 id="基本权利的一般原理"><a class="markdownIt-Anchor" href="#基本权利的一般原理"></a> 基本权利的一般原理</h3><ul><li><p>概念</p><ul><li>作为人所应享有的固有权利，即由人性所派生的或为维护“人的尊严”而应享有的、不可或缺的、具有重要意义的权利</li></ul></li><li><p>特点</p><ul><li>1.是固有权利，也是法定权利</li><li>2.不受侵犯，但是在特定条件下也受到限制和制约</li><li>3.具有普遍性，也具有特殊性</li></ul></li></ul><h3 id="基本权利的保障和限制"><a class="markdownIt-Anchor" href="#基本权利的保障和限制"></a> 基本权利的保障和限制</h3><ul><li><p>保障：绝对保障、相对保障、折中型保障</p></li><li><p>限制：内在限制、外在限制</p><ul><li>限制方式：法律或宪法</li></ul></li></ul><h3 id="公民的基本权利"><a class="markdownIt-Anchor" href="#公民的基本权利"></a> 公民的基本权利</h3><ul><li><p>1.平等权</p><ul><li>法律面前人人平等</li><li>禁止不合理的差别对待</li></ul></li><li><p>2.政治权利</p><ul><li><p>选举权与被选举权</p></li><li><p>言论自由</p></li><li><p>出版自由</p></li><li><p>结社自由</p><ul><li>社会团体的登记管理机关：国务院民政部门和县级以上地方各级民政部门</li></ul></li><li><p>集会、游行、示威自由</p><ul><li>主管部门：市、县公安局、城市公安分局</li><li>许可与否：2日前告知</li></ul></li></ul></li><li><p>3.宗教信仰自由</p><ul><li>合法性原则</li><li>政教分离原则</li><li>各宗教一律平等</li><li>独立办教原则（独立自主自办的方针）</li></ul></li><li><p>4.人身自由</p><ul><li><p>生命权的保障</p></li><li><p>人身自由不受侵犯</p></li><li><p>人格尊严不受侵犯</p><ul><li>姓名权</li><li>肖像权</li><li>名誉权</li><li>荣誉权</li><li>隐私权</li></ul></li><li><p>住宅不受侵犯</p></li><li><p>通信自由和通信秘密受宪法保护</p></li></ul></li><li><p>5.社会经济权利</p><ul><li><p>1）财产权</p></li><li><p>2）劳动权</p><ul><li>既是权利也是义务</li></ul></li><li><p>3）休息权</p><ul><li>主体：仅限于劳动者</li></ul></li><li><p>4）社会保障权</p></li><li><p>5）物质帮助权</p><ul><li>主体要求：年老、疾病或者丧失劳动能力（不包括自然灾害）</li></ul></li></ul></li><li><p>6.文化教育权利</p><ul><li><p>受教育权</p><ul><li>义务教育阶段：是义务</li><li>非义务教育阶段：是权利</li></ul></li><li><p>科学研究、文学艺术创作、进行其他文化活动的自由</p></li></ul></li><li><p>7.监督权与请求权</p><ul><li><p>监督权</p><ul><li>1.批评、建议权</li><li>2.控告、检举权</li><li>3.申诉权</li></ul></li><li><p>国家赔偿请求权</p></li></ul></li></ul><h3 id="公民的基本义务"><a class="markdownIt-Anchor" href="#公民的基本义务"></a> 公民的基本义务</h3><ul><li><p>1.维护国家统一和民族团结</p></li><li><p>2.遵守宪法和法律</p><ul><li>1）保守国家秘密</li><li>2）保护公共财产</li><li>3）遵守劳动纪律</li><li>4）遵守公共秩序</li><li>5）尊重社会公德</li></ul></li><li><p>3.维护祖国安全、荣誉和利益</p></li><li><p>4.依法服兵役</p><ul><li>注意，被剥夺政治权利的人无服兵役资格</li></ul></li><li><p>5.依法纳税</p></li><li><p>6.宪法规定的其他义务</p></li></ul><h2 id="第七章国家机构"><a class="markdownIt-Anchor" href="#第七章国家机构"></a> 第七章：国家机构</h2><h3 id="我国国家机构的组织活动原则"><a class="markdownIt-Anchor" href="#我国国家机构的组织活动原则"></a> 我国国家机构的组织活动原则</h3><ul><li>1.坚持党的领导</li><li>2.民主集中制</li><li>3.为人民服务原则</li><li>4.权责统一原则</li><li>5.精简和效率原则</li><li>6.法治原则</li></ul><h3 id="全国人大及其常委会"><a class="markdownIt-Anchor" href="#全国人大及其常委会"></a> 全国人大及其常委会</h3><ul><li><p>全国人大</p><ul><li><p>性质和地位</p><ul><li>1.全国人大会作为全国人民的代表机关，具有广泛的代表性</li><li>2.全国人大统一行使国家权力</li><li>3.全国人大在整个国家机构体系中居于最高地位</li></ul></li><li><p>职权</p><ul><li>1.修改宪法和监督宪法实施的权力</li><li>2.制定和修改基本法律的权力</li><li>3.对中央国家机关组成人员选举、决定任选和罢免的权力</li><li>4.决定重大国家事项的权力</li><li>5.监督其他中央国家机关的权力</li><li>6.应当由最高国家权力机关行使的其他权利</li></ul></li><li><p>主要工作程序</p><ul><li>1.提出议案</li><li>2.审议工作报告，审查国家计划和国家预算</li><li>3.选举</li><li>4.决定人选</li><li>5.罢免</li><li>6.询问和质询</li></ul></li></ul></li><li><p>全国人大常委会</p><ul><li><p>职权</p><ul><li>有权解释宪法，监督宪法的实施</li><li>立法权</li><li>法律解释权</li><li>监督权</li><li>重大国家事项决定权</li><li>人事任免权</li></ul></li><li><p>工作程序</p><ul><li>1.提出议案</li><li>2.听取和审议专项工作报告</li><li>3.审查和批准决算，听取和审议国民经济和社会发展计划、预算的执行情况报告</li><li>4.法律实施情况的检查</li><li>5.规范性文件备案审查</li><li>6.质询和询问</li><li>7.特定问题调查</li></ul></li></ul></li></ul><h3 id="国家主席"><a class="markdownIt-Anchor" href="#国家主席"></a> 国家主席</h3><ul><li>公布法律、发布命令</li><li>任免权</li><li>外事权</li><li>授予荣誉权</li></ul><h3 id="国务院"><a class="markdownIt-Anchor" href="#国务院"></a> 国务院</h3><ul><li><p>性质和地位</p><ul><li>国务院是中央人民政府</li><li>国务院是最高国家权力机关的执行机关</li><li>国务院是最高国家行政机关</li></ul></li><li><p>职权</p><ul><li>1.根据宪法和法律，规定行政措施，制定行政法规，发布和决定命令</li><li>2.向全国人民代表大会或全国人大常委会提出议案</li><li>3.规定各部和各委员会的任务和职责</li><li>4.领导各类有关国家建设的事务</li><li>5.保障少数民族的平等权利和民族自治地方的自治权利，保护华侨正当的权利和利益，保护归侨和侨眷的合法权利和利益</li></ul></li><li><p>结构</p><ul><li><p>总理</p><ul><li><p>副总理|国务委员|秘书长</p><ul><li>办公厅</li><li>组成部门</li><li>直属特设机构</li><li>直属机构</li><li>办事机构</li><li>直属事业单位</li><li>组成部门管理的国家局</li><li>议事协调机构</li></ul></li></ul></li></ul></li></ul><h3 id="中央军委"><a class="markdownIt-Anchor" href="#中央军委"></a> 中央军委</h3><ul><li><p>领导体制</p><ul><li>其组成人选由中央军委主席提名，报全国人大决定</li></ul></li><li><p>领导部队</p><ul><li>解放军现役+预备役</li><li>武警部队</li><li>民兵</li></ul></li></ul><h3 id="地方各级人大会和地方各级人民政府"><a class="markdownIt-Anchor" href="#地方各级人大会和地方各级人民政府"></a> 地方各级人大会和地方各级人民政府</h3><ul><li><p>地方各级人大会</p><ul><li><p>性质</p><ul><li>各级人大会有权任免政府官员，各级人大常委会有权任免副职领导人员</li><li>各级人大代表代表10人以上联名，可对本级人民政府及其工作部门提出质询案（人大会召开期间）</li><li>省、自治区、直辖市、自治州、设区的市，各级人大常委会组成人员5人以上，县级常委会组成人员3人以上，可向常委会书面提出质询案</li><li>地方政府规章报备：国务院+本级人大常委会</li></ul></li><li><p>职权</p><ul><li>选举本级常委会，本级检察长，法院院长（须报上一级人民检察院检察长提请该级人大常委会批准）</li><li>改变或撤销本级人大常委会不当决议，撤销本级人民政府不适当的决定和命令</li></ul></li></ul></li><li><p>专门委员会和调查委员会</p><ul><li>县级以上的地方各级人大或其常委会可以组织关于特定问题的调查委员会</li></ul></li><li><p>地方各级常委会</p><ul><li><p>职权</p><ul><li>1.领导或主持本级人大会代表的选举；召集本级人大会</li><li>2.任免副职政府领导，任免法院、检察院除正职干部外的其他司法人员（如检察员、审判员）</li><li>3.制定和颁布地方性法规</li></ul></li></ul></li><li><p>地方各级人民政府</p><ul><li><p>派出机关</p><ul><li>区公所</li><li>街道办事处</li></ul></li></ul></li></ul><h3 id="民族自治地方的自治机关"><a class="markdownIt-Anchor" href="#民族自治地方的自治机关"></a> 民族自治地方的自治机关</h3><ul><li><p>组成</p><ul><li>由该自治区民族公民担任主任或副主任（有一个即可）</li><li>自治区主席、州长、县长由该少数民族公民担任</li></ul></li><li><p>自治权</p><ul><li>若上级指示不适合实际情况，可以变通或停止执行</li><li>自治区的自治条例和单行条例须报全国人大常委会批准生效，自治州、县的条例报省、自治区和直辖市的人大常委会批准生效，报全国人大常委会和国务院备案</li></ul></li></ul><h3 id="监察委员会"><a class="markdownIt-Anchor" href="#监察委员会"></a> 监察委员会</h3><ul><li><p>职能</p><ul><li>对所有行使公权力的公职人员进行监察</li><li>调查职务违法和职务犯罪</li><li>开展廉政建设和反腐败工作</li></ul></li><li><p>职责</p><ul><li>监督职责</li><li>调查职责</li><li>处置职责</li></ul></li><li><p>范围</p><ul><li>公务员</li><li>管理公共事务的事业单位的工作人员</li><li>国企管理人员</li><li>公办教育、科研、文化等单位的管理人员</li><li>基层群众自治组织中从事管理的人员</li></ul></li><li><p>原则和方针</p><ul><li>依法独立行使监察权</li><li>与审判机关、检察机关和执法部门互相配合、互相制约</li><li>有关机关和单位对检察机关的协助义务</li><li>严格遵守宪法和法律，以事实为根据，以法律为准绳</li><li>在使用法律上一律平等，保障当事人的合法权益</li><li>权责对等，严格监督</li><li>惩戒与教育相结合，宽严相济</li></ul></li></ul><h3 id="人民法院"><a class="markdownIt-Anchor" href="#人民法院"></a> 人民法院</h3><ul><li><p>含义</p><ul><li>1.人民法院是审判机关</li><li>2.各级人民法院都是国家审判机关，而不是地方审判机关</li></ul></li><li><p>职权</p><ul><li><p>最高人民法院</p><ul><li>1.审判案件权</li><li>2.制定司法解释和发布指导性案例权</li><li>3.审判监督权</li></ul></li><li><p>高级人民法院</p><ul><li>1.审判案件权</li><li>2.审判监督权</li></ul></li><li><p>中级人民法院</p><ul><li>1.审判案件权</li><li>2.审判监督权</li></ul></li><li><p>基层人民法院</p><ul><li><p>1.审判案件权</p></li><li><p>2.业务指导权</p><ul><li>对人民调解委员会的调解工作进行业务指导</li></ul></li></ul></li></ul></li><li><p>审判工作原则</p><ul><li>1.依法独立行使审判权</li><li>2.平等适用法律原则</li><li>3.司法公正原则</li><li>4.司法民主原则</li><li>5.公开审判原则</li><li>6.司法责任制原则</li><li>7.使用本民族语言文字进行诉讼原则</li><li>8.当事人有权获得辩护原则</li></ul></li></ul><h3 id="人民检察院"><a class="markdownIt-Anchor" href="#人民检察院"></a> 人民检察院</h3><ul><li><p>工作原则</p><ul><li>依法独立行使检察权原则</li><li>平等适用法律原则</li><li>司法公正原则</li><li>司法民主原则</li><li>检务公开原则</li><li>司法责任制原则</li><li>使用本民族语言文字进行诉讼原则</li></ul></li></ul><h2 id="基本权利的主体只能是公民不包括法人及非法人组织"><a class="markdownIt-Anchor" href="#基本权利的主体只能是公民不包括法人及非法人组织"></a> 基本权利的主体只能是公民！不包括法人及非法人组织</h2><h2 id="易考"><a class="markdownIt-Anchor" href="#易考"></a> 易考</h2><h3 id="特别行政区制度"><a class="markdownIt-Anchor" href="#特别行政区制度"></a> 特别行政区制度</h3><ul><li><p>基本内容</p><ul><li>1.维护国家统一、主权和领土完整</li><li>2.实行”一国两制“</li><li>3.实行高度自治</li><li>4.保持原有的资本主义制度和生活方式不变</li><li>5.实行当地人治理</li></ul></li><li><p>法律地位</p><ul><li>特别行政区是我国不可分离的部分</li><li>特别行政区是直辖于中央人民政府的地方行政区域</li><li>特别行政区是实行高度自治的地方行政区域</li></ul></li><li><p>中央对特别行政区直接行使的权利</p><ul><li>1.管理与特别行政区有关的外交事务</li><li>2.管理特别行政区的防务</li><li>3.任命行政长官和政府主要官员</li><li>4.审查和发回特别行政区制定的法律</li><li>5.对于列入附件三的在特别行政区实施的全国性法律做出增减</li><li>6.决定特别行政区进入非常状态</li><li>7.解释基本法</li><li>8.修改基本法</li></ul></li><li><p>特别行政区行使的高度自治权</p><ul><li>1.行政管理权</li><li>2.立法权</li><li>3.独立的司法权和终审权</li><li>4.自行处理对外事务的权力（对外事务不是外交事务）</li></ul></li><li><p>行政长官的任免</p><ul><li><p>香港</p><ul><li>年满40周岁、在港居住满20年、在外国无居留权、特区永久性居民、中国国籍</li></ul></li><li><p>澳门</p><ul><li>年满40周岁、在澳居住满20年、任职期间无外国居留权、特区永久性居民、中国国籍</li></ul></li></ul></li><li><p>行政机关</p><ul><li>特别行政区政府</li></ul></li><li><p>立法机关</p><ul><li>立法会</li></ul></li><li><p>司法机关</p><ul><li><p>香港</p><ul><li>终审法院</li><li>高等法院</li><li>区域法院</li><li>裁判署法庭和其他专门法庭</li><li>香港的司法机关中没有检察院。其主管刑事检察工作的律政司属于行政机关</li></ul></li><li><p>澳门</p><ul><li><p>法院</p><ul><li>初级法院</li><li>中级法院</li><li>终审法院</li></ul></li><li><p>检察院（司法机关）</p></li></ul></li></ul></li></ul><h3 id="合宪性审查"><a class="markdownIt-Anchor" href="#合宪性审查"></a> 合宪性审查</h3><ul><li><p>主体</p><ul><li>全国人大及其常委会</li></ul></li><li><p>程序</p><ul><li><p>依职权审查</p><ul><li>备案审查</li><li>批准审查</li><li>移送审查</li></ul></li><li><p>依申请审查</p><ul><li>经国家机关要求启动审查</li><li>经组织、公民建议启动审查</li></ul></li><li><p>专项审查</p></li></ul></li><li><p>期限</p><ul><li>询问：一个月内给答复</li><li>审查：三个月内完成审查工作</li></ul></li></ul><h3 id="十三届全国人大"><a class="markdownIt-Anchor" href="#十三届全国人大"></a> 十三届全国人大</h3><ul><li>将”内务司法委员会“更名为”监察和司法委员会“</li><li>增设”社会建设委员会“</li></ul><h3 id="党领导的多党合作和政治协商制度"><a class="markdownIt-Anchor" href="#党领导的多党合作和政治协商制度"></a> 党领导的多党合作和政治协商制度</h3><ul><li><p>方针</p><ul><li>长期共存、互相监督、肝胆相照、荣辱与共</li></ul></li><li><p>显著特征</p><ul><li>在政党关系上，坚持共产党领导、多党派合作</li><li>在政权运作方式上，坚持共产党执政、多党派参政</li><li>在协调利益关系上，坚持维护国家和人民的根本利益，照顾同盟者的具体利益</li><li>在民主形式上，坚持充分协商、广泛参与</li></ul></li><li><p>重要作用</p><ul><li>服从大局、广泛凝聚力量，围绕和国家事业发展提供强有力的支持</li><li>充分发扬民主、扩大有序参与，推动社会主义民主政治发展</li><li>积极协调关系、努力化解矛盾，维护社会和谐稳定</li><li>高举爱国主义、社会主义两面旗帜，加强团结联谊，促进祖国和平统一大业</li></ul></li></ul><h3 id="民族区域自治制度"><a class="markdownIt-Anchor" href="#民族区域自治制度"></a> 民族区域自治制度</h3><ul><li><p>概念</p><ul><li>指在国家的统一领导之下，以少数民族聚居区为基础，建立相应的民族自治地方，设立民族自治机关，行使宪法和法律规定的自治权的制度</li></ul></li><li><p>核心内容</p><ul><li>自治权</li></ul></li><li><p>必然性</p><ul><li>历史依据：我国是有着悠久历史的统一的多民族国家</li><li>现实情况：”大杂居、小聚居“的少数民族的分布格局</li><li>政治基础：在长期的革命改革发展的过程中，各民族形成了合作共存的政治认同</li><li>理论依据：马克思列宁主义认为民族区域自治是建立现代真正民主国家的条件，是解决民族问题的重要途径</li></ul></li><li><p>建立原则</p><ul><li>1.以少数民族聚居区为基础</li><li>2.尊重历史传统</li><li>3.各民族共同协商</li></ul></li><li><p>意义</p><ul><li>有利于保障各少数民族人民当家作主</li><li>有利于促进民族关系的巩固和发展</li><li>有利于维护国家统一</li><li>有利于促进少数民族地区政治、经济、文化和社会事业的发展</li></ul></li><li><p>自治权</p><ul><li>”上级国家机关的决议如不适合民族自治地方实际情况的，经过上级国家机关批准可以变通执行或者停止执行“</li></ul></li></ul><h3 id="选举制度-p150"><a class="markdownIt-Anchor" href="#选举制度-p150"></a> 选举制度-P150</h3><ul><li><p>原则</p><ul><li>普遍性原则</li><li>平等性原则</li><li>直接选举与间接选举并用原则</li><li>无记名投票原则</li></ul></li><li><p>组织和程序</p><ul><li>组织机构</li><li>选区划分</li><li>选民登记</li><li>代表候选人的提名与介绍</li><li>选举投票与结果确认</li><li>代表的辞职、罢免以及补选</li></ul></li></ul><h3 id="如何发展全过程民主"><a class="markdownIt-Anchor" href="#如何发展全过程民主"></a> 如何发展全过程民主</h3><ul><li>坚持党对民主政治建设工作的集中统一领导</li><li>坚持和完善人民代表大会制度（人大代表具有广泛的代表性）</li><li>进一步健全社会主义协商民主制度</li></ul><h3 id="人民代表大会制度的优越性"><a class="markdownIt-Anchor" href="#人民代表大会制度的优越性"></a> 人民代表大会制度的优越性</h3><ul><li>使国家一切权力属于人民的原则得到充分落实</li><li>保证了人民权利的统一性</li><li>适合我国国情</li><li>既能保证中央的统一领导，又能充分发挥地方的主动性和积极性</li></ul><h3 id="国体和政体"><a class="markdownIt-Anchor" href="#国体和政体"></a> 国体和政体</h3><ul><li><p>国体：即国家的阶级本质，他集中反映了社会各阶级、阶层在社会中的不同地位</p><ul><li>我国国体：工人阶级领导的、以工农联盟为基础的人民民主专政的社会主义国家</li></ul></li><li><p>政体：政权组织形式，是指特定社会的统治阶级依据一定原则建立的行使国家权力、实现国家统治和管理职能的政权机关的组织与活动体制</p><ul><li>我国政体：人民代表大会制度</li></ul></li></ul><h3 id="新时代如何发展人大制"><a class="markdownIt-Anchor" href="#新时代如何发展人大制"></a> 新时代如何发展人大制</h3><ul><li>要全面贯彻实施宪法，维护宪法权威和尊严</li><li>要加快完善中国特色社会主义法律体系，以良法促进发展，保障善治</li><li>要用好宪法赋予人大的监督权，实行正确监督、有效监督、依法监督</li><li>要充分发挥人大代表作用，做到民有所呼，我有所应</li><li>要强化政治机关意识，加强人大自身建设</li><li>要加强党对人大工作的全面领导</li></ul><h3 id="基层群众自治制度"><a class="markdownIt-Anchor" href="#基层群众自治制度"></a> 基层群众自治制度</h3><ul><li><p>概念</p><ul><li>由居民（村民）选举的成员组成居民（村民）委员会，实行自我管理、自我教育、自我服务、自我监督的制度</li></ul></li><li><p>分支</p><ul><li><p>居民委员会</p><ul><li>召集和主持居民会议</li><li>涉及全体居民利益的重要问题，必须提请居民会议讨论决定</li><li>采用少数服从多数的原则</li></ul></li><li><p>村民委员会</p><ul><li>实行少数服从多数的民主决策机制，公开透明的工作原则</li><li>实行村务公开制度</li><li>村民会议由村民委员会依法召集</li></ul></li></ul></li></ul><h2 id="基本政治制度"><a class="markdownIt-Anchor" href="#基本政治制度"></a> 基本政治制度</h2><h2 id="民主集中制的优势"><a class="markdownIt-Anchor" href="#民主集中制的优势"></a> 民主集中制的优势</h2><h2 id="不对任何部门负责"><a class="markdownIt-Anchor" href="#不对任何部门负责"></a> 不对任何部门负责</h2><h2 id="变通或者停止"><a class="markdownIt-Anchor" href="#变通或者停止"></a> 变通或者停止</h2><h2 id="我国的法律监督机关检察院"><a class="markdownIt-Anchor" href="#我国的法律监督机关检察院"></a> 我国的法律监督机关：检察院</h2>]]></content>
<categories>
<category> 法学 </category>
</categories>
<tags>
<tag> 法学 </tag>
<tag> 大纲 </tag>
<tag> 宪法学 </tag>
</tags>
</entry>
<entry>
<title>思想道德与法治</title>
<link href="/2022/09/22/%E6%80%9D%E4%BF%AE/"/>
<url>/2022/09/22/%E6%80%9D%E4%BF%AE/</url>
<content type="html"><![CDATA[<style>#web_bg{ background: url(https://gimg2.baidu.com/image_search/src=http%3A%2F%2Fimg9.51tietu.net%2Fpic%2F2019-091116%2Fbtyk3glwn5xbtyk3glwn5x.jpg&refer=http%3A%2F%2Fimg9.51tietu.net&app=2002&size=f9999,10000&q=a80&n=0&g=0n&fmt=auto?sec=1666432323&t=aa3540f2f10bfc986bd95d7dd8c89660)!important; /*重新定义background会导致原有定位属性失效，所以也需要再声明一次加权的定位属性*/ background-position: center !important; background-size: cover !important; background-repeat: no-repeat !important;}</style><h1 id="思修"><a class="markdownIt-Anchor" href="#思修"></a> 思修</h1><h2 id="绪论"><a class="markdownIt-Anchor" href="#绪论"></a> 绪论</h2><h3 id="中特社新时代论断"><a class="markdownIt-Anchor" href="#中特社新时代论断"></a> 中特社新时代论断</h3><ul><li><p>意义</p><ul><li>中国人民站、富、强，迎来了实现伟大复兴的光明前景</li><li>中特社具有生机活力</li><li>中特社制度体系发展->推动现代化</li></ul></li><li><p>特征</p><ul><li>承前启后，继往开来+继续夺取新胜利</li><li>决胜全面小康+建成社会主义现代化强国</li><li>各族人民团结奋斗->共同富裕</li><li>伟大复兴中国梦</li></ul></li></ul><h3 id="新时代对大学生的要求"><a class="markdownIt-Anchor" href="#新时代对大学生的要求"></a> 新时代对大学生的要求</h3><ul><li><p>立大志</p><ul><li>坚定理想信念+中国梦内化于心外化于行</li></ul></li><li><p>明大德</p><ul><li>用真善美雕琢自己，感化他人</li></ul></li><li><p>成大才</p><ul><li>危机感、紧迫感，加强学习</li></ul></li><li><p>担大任</p><ul><li>知行合一求真务实，勇敢面对困难</li></ul></li></ul><h3 id="思想道德素质与法治素养的关系"><a class="markdownIt-Anchor" href="#思想道德素质与法治素养的关系"></a> 思想道德素质与法治素养的关系</h3><ul><li><p>同</p><ul><li>都是调节思想行为，协调人际关系，维护社会秩序的重要手段</li></ul></li><li><p>一方面，思德为法提供思想指引和价值基础</p></li><li><p>另一方面，法为思德提供制度保障+强制力</p></li></ul><h2 id="第一章"><a class="markdownIt-Anchor" href="#第一章"></a> 第一章</h2><h3 id="马克思关于人的本质"><a class="markdownIt-Anchor" href="#马克思关于人的本质"></a> 马克思关于人的本质</h3><ul><li><p>原话：“人的本质不是单个人固有的抽象物，在其现实性上，它是一切社会关系的总和”</p></li><li><p>理解</p><ul><li>人处于社会之中，社会属性是人的本质属性</li></ul></li></ul><h3 id="个人与社会的辩证关系"><a class="markdownIt-Anchor" href="#个人与社会的辩证关系"></a> 个人与社会的辩证关系</h3><ul><li>个人与社会是对立统一的</li><li>社会由人组成，离开了人就没有社会</li><li>人是社会中的人，社会是人的存在形式</li></ul><h3 id="人生观"><a class="markdownIt-Anchor" href="#人生观"></a> 人生观</h3><ul><li><p>人生目的（人生观的核心）</p><ul><li>人们在社会实践中关于自身行为的根本指向和人生追求</li></ul></li><li><p>人生态度</p><ul><li>人们通过生活实践形成的对人生问题的一种相对稳定的心里倾向和精神状态</li></ul></li><li><p>人生价值</p><ul><li><p>定义</p><ul><li>人的生命及其实践活动对于社会和个人所具有的作用和意义</li></ul></li><li><p>构成</p><ul><li><p>自我价值</p><ul><li>对于自我生存和发展所具有的价值</li></ul></li><li><p>社会价值</p><ul><li>个体的实践活动对于社会、他人所具有的价值</li></ul></li></ul></li><li><p>评价</p><ul><li>其实践是否符合社会发展的客观规律？促进历史进步？</li><li>既要看贡献大小，也要看尽力程度</li><li>尊重无论是物质还是精神的贡献</li><li>既注重社会贡献，也注重自身完善</li></ul></li><li><p>实现条件</p><ul><li>从社会客观条件出发</li><li>从个体自身条件出发</li><li>发挥主观能动性</li></ul></li></ul></li></ul><h3 id="高尚的人生观"><a class="markdownIt-Anchor" href="#高尚的人生观"></a> 高尚的人生观</h3><ul><li>高尚的人生追求</li><li>积极进取的人生态度</li></ul><h2 id="第二章"><a class="markdownIt-Anchor" href="#第二章"></a> 第二章</h2><h3 id="理想信念"><a class="markdownIt-Anchor" href="#理想信念"></a> 理想信念</h3><ul><li><p>理想</p><ul><li><p>内涵</p><ul><li>在实践中形成的、有实现可能性的、对未来社会和自身发展目标的向往和追求</li></ul></li><li><p>特征</p><ul><li>超越性</li><li>实践性</li><li>时代性</li></ul></li></ul></li><li><p>信念</p><ul><li><p>内涵</p><ul><li>人们在一定认识基础上确立的对某种思想或事物坚信不疑并身体力行的精神状态</li></ul></li><li><p>特征</p><ul><li>执着性</li><li>支撑性</li><li>多样性</li></ul></li></ul></li><li><p>理想信念共同作用</p><ul><li>昭示奋斗目标</li><li>催生前进动力</li><li>提供精神支柱</li><li>提高精神境界</li></ul></li></ul><h3 id="坚定信仰信念信心"><a class="markdownIt-Anchor" href="#坚定信仰信念信心"></a> 坚定信仰信念信心</h3><ul><li><p>信仰</p><ul><li><p>增强对马克思主义、共产主义的信仰</p><ul><li><p>原因</p><ul><li>科学的</li><li>人民的</li><li>实践的</li><li>发展的</li></ul></li></ul></li></ul></li><li><p>信念</p><ul><li>增强对中特社的信念</li></ul></li><li><p>信心</p><ul><li>增强实现中华民族伟大复兴的信心</li></ul></li></ul><h3 id="理想与现实的关系"><a class="markdownIt-Anchor" href="#理想与现实的关系"></a> 理想与现实的关系</h3><ul><li>辩证统一</li><li>现实中包含着理想的因素，孕育着理想的发展</li><li>理想受现实的限制</li></ul><h3 id="个人理想与社会理想的关系"><a class="markdownIt-Anchor" href="#个人理想与社会理想的关系"></a> 个人理想与社会理想的关系</h3><ul><li>二者有机地联系在一起</li><li>个人理想以社会理想为指引</li><li>社会理想是个人理想的汇聚和升华</li></ul><h2 id="第三章"><a class="markdownIt-Anchor" href="#第三章"></a> 第三章</h2><h3 id="第一节"><a class="markdownIt-Anchor" href="#第一节"></a> 第一节</h3><ul><li><p>中华民族崇尚精神</p><ul><li><p>表现</p><ul><li>对物质生活与精神生活相互关系的独到理解</li><li>对理想的不懈追求</li><li>对品格养成的重视</li></ul></li></ul></li><li><p>中国精神的丰富内涵</p><ul><li>团结、奋斗、创造、梦想</li></ul></li><li><p>中国精神</p><ul><li><p>以爱国主义为核心的民族精神</p><ul><li><p>爱国主义</p><ul><li><p>人们对自己家园、文化的认同感、归属感、尊严感和荣誉感的统一</p></li><li><p>基本内涵</p><ul><li>爱江山、爱同胞、爱文化、爱国家本身</li></ul></li></ul></li></ul></li><li><p>以改革创新为核心的时代精神</p></li></ul></li></ul><h3 id="第二节"><a class="markdownIt-Anchor" href="#第二节"></a> 第二节</h3><ul><li><p>做爱国者</p><ul><li>坚持爱国爱党爱社会相统一</li><li>维护祖国统一和民族团结</li><li>尊重和传承中华民族历史文化</li><li>坚持立足中国同时面向世界</li></ul></li><li><p>总体国家安全观</p><ul><li>国家利益至上、以人民安全为宗旨、以政治安全为根本、以经济安全为基础、以军事文化社会安全为保障、以促进国际安全为依托</li></ul></li><li><p>改革创新的时代要求</p><ul><li>把创新作为引领发展的第一动力</li><li>把人才作为支撑发展的第一资源</li><li>增强自主创新能力</li><li>破除体制机制障碍</li><li>解放和激发科技作为第一生产力的潜能</li></ul></li><li><p>做改革创新生力军</p><ul><li><p>树立改革创新意识</p><ul><li>增强改革创新责任感</li><li>树立敢于突破常规的意识</li><li>敢于探索未知领域</li></ul></li><li><p>增强改革创新的能力本领</p><ul><li>夯实创新基础</li><li>投身创新实践</li></ul></li></ul></li></ul><h2 id="第四章"><a class="markdownIt-Anchor" href="#第四章"></a> 第四章</h2><h3 id="第一节-2"><a class="markdownIt-Anchor" href="#第一节-2"></a> 第一节</h3><ul><li><p>价值</p><ul><li><p>价值</p><ul><li><p>定义</p><ul><li>实践基础上形成的主体和客体之间的意义关系</li></ul></li></ul></li><li><p>价值观</p><ul><li><p>定义</p><ul><li>主体对客体有无价值、价值大小的立场和态度</li></ul></li><li><p>特点</p><ul><li>反映时代精神</li><li>体现民族特色</li><li>蕴含阶级立场</li></ul></li></ul></li><li><p>核心价值观</p><ul><li>一定社会形态、社会性质的集中体现，在一个社会的思想观念体系中处于主导地位</li></ul></li><li><p>社会主义核心价值观</p><ul><li><p>基本内容</p><ul><li><p>国家</p><ul><li>富强民主文明和谐</li></ul></li><li><p>社会</p><ul><li>自由平等公正法治</li></ul></li><li><p>个人</p><ul><li>爱国敬业诚信友善</li></ul></li></ul></li></ul></li><li><p>当代中国发展进步的精神指引（如何培育践行社会主义核心价值观）</p><ul><li>坚持和发展中国特色社会主义的价值遵循</li><li>提高国家文化软实力的迫切要求</li><li>推进社会团结奋进的“最大公约数”</li></ul></li></ul></li></ul><h3 id="第二节-2"><a class="markdownIt-Anchor" href="#第二节-2"></a> 第二节</h3><ul><li><p>社会主义核心价值观的显著特征</p><ul><li><p>反映人类社会发展进步的价值理念</p><ul><li>体现社会主义的本质属性</li><li>扎根中华优秀传统文化土壤</li><li>吸纳世界文明有益成果</li></ul></li><li><p>彰显人民至上的价值立场</p><ul><li>尊重人民群众的历史主体地位</li><li>体现以人民为中心的价值导向</li></ul></li><li><p>因真实可信而具有强大的道义力量</p></li></ul></li><li><p>积极践行社会主义核心价值观</p><ul><li><p>扣好人生的扣子</p><ul><li>坚持正确价值观的引领</li></ul></li><li><p>把社会主义核心价值观落细落小落实</p><ul><li>勤学</li><li>修德</li><li>明辨</li><li>笃实</li></ul></li></ul></li></ul><h2 id="第六章"><a class="markdownIt-Anchor" href="#第六章"></a> 第六章</h2><h3 id="第一节-3"><a class="markdownIt-Anchor" href="#第一节-3"></a> 第一节</h3><ul><li><p>法律的含义</p><ul><li>由国家创制和实施的行为规范</li></ul></li><li><p>社会主义法律的本质特征和运行</p><ul><li><p>特征</p><ul><li>体现了党的主张和人民意志的统一</li><li>具有科学性和先进性</li><li>是中特社建设的重要保障</li></ul></li><li><p>运行</p><ul><li>法律制定</li><li>法律执行</li><li>法律适用</li><li>法律遵守</li></ul></li></ul></li></ul><h3 id="第二节-3"><a class="markdownIt-Anchor" href="#第二节-3"></a> 第二节</h3><ul><li><p>习思的主要内容</p><ul><li><p>政治方向</p></li><li><p>战略地位</p><ul><li>全面依法治国是新时代坚持和发展中特社的基本方略</li></ul></li><li><p>工作布局</p><ul><li>明确全面依法治国的总目标</li></ul></li><li><p>主要任务</p></li><li><p>重大关系</p><ul><li>政治与法治、改革与法治、德治与法治</li></ul></li><li><p>重要保障</p><ul><li>全面依法治国的人才支撑和“关键少数”</li></ul></li></ul></li><li><p>坚持走中特社法治道路的原则</p><ul><li>坚持党的领导</li><li>坚持人民主体地位</li><li>坚持法律面前人人平等</li><li>坚持依法治国和以德治国相结合</li><li>坚持从中国实际出发</li></ul></li><li><p>建设中特社法治体系</p><ul><li><p>内容</p><ul><li>完备的法律规范体系</li><li>高效的法治实施体系</li><li>严密的法治监督体系</li><li>有力的法治保障体系</li><li>完善的党内法规体系</li></ul></li></ul></li><li><p>坚持全面依法治国的要求</p><ul><li>加强法治政府、法治社会建设</li><li>坚持全面推进科学立法、严格执法、公正司法、全民守法</li></ul></li><li><p>宪法</p><ul><li><p>形成和发展</p><ul><li>P207</li></ul></li><li><p>基本原则</p><ul><li>党的领导</li><li>人民当家作主</li><li>尊重和保障人权</li><li>社会主义法治</li><li>民主集中制</li></ul></li><li><p>公民的基本权利和义务</p><ul><li><p>权利</p><ul><li><p>政治权利</p><ul><li>选举权与被选举权</li><li>表达权</li><li>民主管理权</li></ul></li><li><p>人身权</p><ul><li>生命健康权</li><li>人身自由权</li><li>人格尊严权</li><li>住宅安全权</li></ul></li><li><p>财产权</p><ul><li>私有财产权</li><li>继承权</li></ul></li><li><p>社会经济权</p><ul><li>劳动权</li><li>休息权</li><li>社会保障权</li></ul></li><li><p>宗教信仰自由</p></li><li><p>文化教育权利</p><ul><li>受教育权</li><li>创作自由</li></ul></li></ul></li><li><p>义务</p><ul><li>维护国家统一和民族团结</li><li>遵守宪法和法律</li><li>维护国家安全、荣誉和利益</li><li>依法服兵役</li><li>依法纳税</li></ul></li></ul></li></ul></li><li><p>法治思维的基本内容</p><ul><li>法律之上</li><li>权力制约</li><li>公平正义</li><li>权利保障</li><li>程序正当</li></ul></li></ul><h2 id="第五章"><a class="markdownIt-Anchor" href="#第五章"></a> 第五章</h2><h3 id="第一节-4"><a class="markdownIt-Anchor" href="#第一节-4"></a> 第一节</h3><ul><li><p>道德</p><ul><li><p>起源</p><ul><li>劳动是道德起源的首要前提</li><li>社会关系是道德赖以产生的客观条件</li><li>人的自我意识是道德产生的主观条件</li></ul></li><li><p>本质</p><ul><li>反映社会经济关系的特殊意识形态</li><li>社会利益关系的特殊调节方式</li><li>一种实践精神</li></ul></li><li><p>功能</p><ul><li><p>规范功能</p><ul><li>正确善恶观的指引下->规范社会成员的行为</li></ul></li><li><p>调节功能</p><ul><li>通过评价等方式指导和纠正人们的行为</li></ul></li></ul></li><li><p>作用</p><ul><li>道德的功能所发挥和实现所产生的社会影响及实际效果</li></ul></li><li><p>基本类型</p><ul><li><p>原始社会</p></li><li><p>奴隶社会</p></li><li><p>封建社会</p></li><li><p>资本主义社会</p></li><li><p>社会主义社会</p><ul><li>共产主义社会</li></ul></li></ul></li><li><p>规律</p><ul><li>人类道德发展的历史过程与社会生产方式的发展进程大体一致</li></ul></li></ul></li><li><p>社会主义道德</p><ul><li><p>核心</p><ul><li><p>为人民服务</p><ul><li>为人民服务是社会主义经济基础和人际关系的客观要求</li><li>为人民服务是社会主义市场经济健康发展的要求</li></ul></li></ul></li><li><p>原则</p><ul><li><p>集体主义</p><ul><li><p>调节社会利益关系的基本原则</p><ul><li>强调国家利益、社会整体利益和个人利益的辩证统一</li><li>强调国家利益、社会利益高于个人利益</li><li>集体主义重视和保障个人的正当（不是不正当）利益</li></ul></li></ul></li></ul></li></ul></li></ul><h3 id="第二节-4"><a class="markdownIt-Anchor" href="#第二节-4"></a> 第二节</h3><ul><li><p>中华传统美德的基本精神</p><ul><li>重视整体利益，强调责任奉献</li><li>推崇仁爱原则，注重以和为贵</li><li>注重人伦关系，重视道德义务</li><li>追求精神境界，向往理想人格</li><li>强调道德修养，注重道德实践</li></ul></li><li><p>中国革命道德</p><ul><li><p>形成与发展</p><ul><li>形成于五四运动前后、发端于中国共产党成立以后的蓬勃发展的伟大工人运动和农民运动</li></ul></li><li><p>内容</p><ul><li>始终把革命利益放在首位</li><li>树立社会新风，建立新型人际关系</li><li>修身自律，保持节操</li></ul></li></ul></li><li><p>公共生活</p><ul><li><p>特征</p><ul><li>活动范围的广泛性</li><li>活动内容的开放性</li><li>交往对象的复杂性</li><li>活动方式的多样性</li></ul></li></ul></li><li><p>社会公德</p><ul><li>文明礼貌、助人为乐、爱护公物、保护环境、遵纪守法</li></ul></li><li><p>网络道德</p><ul><li>正确使用网络工具</li><li>加强网络文明自律</li><li>营造良好网络道德环境</li></ul></li><li><p>职业道德的内容</p><ul><li>从事一定职业的人在职业生活中应当遵循的具有职业特征的道德要求和行为准则</li></ul></li><li><p>家庭美德的内容</p><ul><li>尊老爱幼、男女平等、夫妻和谐、勤俭持家、邻里互助</li></ul></li><li><p>涵养高尚道德品格</p><ul><li>形成正确的道德认知和判断</li><li>激发正向的道德认同和情感</li><li>强化坚定的道德意志和道德信念</li></ul></li></ul>]]></content>
<categories>
<category> 思政课程 </category>
</categories>
<tags>
<tag> 大纲 </tag>
<tag> 思政课程 </tag>
<tag> 思维导图 </tag>
</tags>
</entry>
<entry>
<title>TOPSIS法</title>
<link href="/2022/09/05/TOPSIS%E6%B3%95/"/>
<url>/2022/09/05/TOPSIS%E6%B3%95/</url>
<content type="html"><![CDATA[<style>#web_bg{ background: url(https://img-baofun.zhhainiao.com/pcwallpaper_ugc/static/8e52a8e1fdf8f5578f7abc6728154cbb.jpg?x-oss-process=image%2fresize%2cm_lfit%2cw_1920%2ch_1080)!important; /*重新定义background会导致原有定位属性失效，所以也需要再声明一次加权的定位属性*/ background-position: center !important; background-size: cover !important; background-repeat: no-repeat !important;}</style><h1 id="topsis法"><a class="markdownIt-Anchor" href="#topsis法"></a> TOPSIS法</h1><h2 id="回顾层次分析法的局限性"><a class="markdownIt-Anchor" href="#回顾层次分析法的局限性"></a> 回顾层次分析法的局限性</h2><p>1.评价的决策层不能太多，太多的话n会很大，导致判断矩阵和一致矩阵的差异很大</p><p>2.如果决策层中数据的指标是已知的，会导致评价不准确</p><h2 id="两种指标及其互化"><a class="markdownIt-Anchor" href="#两种指标及其互化"></a> 两种指标及其互化</h2><p>1）极大型指标（收益型指标）</p><p>2）极小型指标（成本型指标）</p><p><em>指标正向化：<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>M</mi><mi>a</mi><mi>x</mi><mo>−</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">Max - x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">M</span><span class="mord mathdefault">a</span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span></span></span></span>（极小型指标转化为极大型指标）</em></p><h2 id="标准化处理消除不同量纲的影响"><a class="markdownIt-Anchor" href="#标准化处理消除不同量纲的影响"></a> 标准化处理：消除不同量纲的影响</h2><p>标准化处理的计算公式：加起来平方再开方</p><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">假</mi><mi mathvariant="normal">设</mi><mi mathvariant="normal">有</mi><mi>n</mi><mi mathvariant="normal">个</mi><mi mathvariant="normal">要</mi><mi mathvariant="normal">评</mi><mi mathvariant="normal">价</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">对</mi><mi mathvariant="normal">象</mi><mo separator="true">,</mo><mi>m</mi><mi mathvariant="normal">个</mi><mi mathvariant="normal">正</mi><mi mathvariant="normal">向</mi><mi mathvariant="normal">化</mi><mi mathvariant="normal">评</mi><mi mathvariant="normal">价</mi><mi mathvariant="normal">指</mi><mi mathvariant="normal">标</mi><mi mathvariant="normal">构</mi><mi mathvariant="normal">成</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">正</mi><mi mathvariant="normal">向</mi><mi mathvariant="normal">化</mi><mi mathvariant="normal">矩</mi><mi mathvariant="normal">阵</mi><mi mathvariant="normal">如</mi><mi mathvariant="normal">下</mi><mo>:</mo><mspace linebreak="newline"></mspace><mi>A</mi><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mn>11</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mn>12</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo>⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mrow><mn>1</mn><mi>n</mi></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mn>21</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mn>22</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo>⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mrow><mn>2</mn><mi>n</mi></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo separator="true">,</mo><mspace linebreak="newline"></mspace><mi mathvariant="normal">那</mi><mi mathvariant="normal">么</mi><mo separator="true">,</mo><mi mathvariant="normal">对</mi><mi mathvariant="normal">其</mi><mi mathvariant="normal">标</mi><mi mathvariant="normal">准</mi><mi mathvariant="normal">化</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">矩</mi><mi mathvariant="normal">阵</mi><mi mathvariant="normal">记</mi><mi mathvariant="normal">为</mi><mi>Z</mi><mo separator="true">,</mo><mi>Z</mi><mi mathvariant="normal">中</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">每</mi><mi mathvariant="normal">一</mi><mi mathvariant="normal">个</mi><mi mathvariant="normal">元</mi><mi mathvariant="normal">素</mi><mi mathvariant="normal">：</mi><msub><mi>z</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mfrac><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msqrt><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>x</mi><mrow><mi>i</mi><mi>j</mi></mrow><mn>2</mn></msubsup></mrow></msqrt></mfrac></mrow><annotation encoding="application/x-tex">假设有n个要评价的对象,m个正向化评价指标构成的正向化矩阵如下:\\A = \left[ \begin{matrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & & \vdots \\ \end{matrix} \right],\\ 那么,对其标准化的矩阵记为Z,Z中的每一个元素：z_{ij}=\frac{a_{ij}}{\sqrt{\sum^n_{i=1}x^2_{ij}}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord cjk_fallback">假</span><span class="mord cjk_fallback">设</span><span class="mord cjk_fallback">有</span><span class="mord mathdefault">n</span><span class="mord cjk_fallback">个</span><span class="mord cjk_fallback">要</span><span class="mord cjk_fallback">评</span><span class="mord cjk_fallback">价</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">对</span><span class="mord cjk_fallback">象</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">m</span><span class="mord cjk_fallback">个</span><span class="mord cjk_fallback">正</span><span class="mord cjk_fallback">向</span><span class="mord cjk_fallback">化</span><span class="mord cjk_fallback">评</span><span class="mord cjk_fallback">价</span><span class="mord cjk_fallback">指</span><span class="mord cjk_fallback">标</span><span class="mord cjk_fallback">构</span><span class="mord cjk_fallback">成</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">正</span><span class="mord cjk_fallback">向</span><span class="mord cjk_fallback">化</span><span class="mord cjk_fallback">矩</span><span class="mord cjk_fallback">阵</span><span class="mord cjk_fallback">如</span><span class="mord cjk_fallback">下</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault">A</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:4.26em;vertical-align:-1.8800000000000003em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.351015em;"><span style="top:-1.9499950000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-3.104995em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.351015em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.850025em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.38em;"><span style="top:-5.2275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.0275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.1674999999999995em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.8800000000000003em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.38em;"><span style="top:-5.2275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.0275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.1674999999999995em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.8800000000000003em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.38em;"><span style="top:-5.04em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-3.8399999999999994em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-1.9799999999999998em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.8800000000000003em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.38em;"><span style="top:-5.2275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.0275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.1674999999999995em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.8800000000000003em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.351015em;"><span style="top:-1.9499950000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-3.104995em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.351015em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.850025em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord cjk_fallback">那</span><span class="mord cjk_fallback">么</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord cjk_fallback">对</span><span class="mord cjk_fallback">其</span><span class="mord cjk_fallback">标</span><span class="mord cjk_fallback">准</span><span class="mord cjk_fallback">化</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">矩</span><span class="mord cjk_fallback">阵</span><span class="mord cjk_fallback">记</span><span class="mord cjk_fallback">为</span><span class="mord mathdefault" style="margin-right:0.07153em;">Z</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">Z</span><span class="mord cjk_fallback">中</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">每</span><span class="mord cjk_fallback">一</span><span class="mord cjk_fallback">个</span><span class="mord cjk_fallback">元</span><span class="mord cjk_fallback">素</span><span class="mord cjk_fallback">：</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.04398em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.83756em;vertical-align:-1.7300000000000002em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1075599999999999em;"><span style="top:-2.11em;"><span class="pstrut" style="height:3.18066em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.18066em;"><span class="svg-align" style="top:-3.8em;"><span class="pstrut" style="height:3.8em;"></span><span class="mord" style="padding-left:1em;"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.804292em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.795908em;"><span style="top:-2.4231360000000004em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4129719999999999em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.1406599999999996em;"><span class="pstrut" style="height:3.8em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.8800000000000001em;"><svg width='400em' height='1.8800000000000001em' viewBox='0 0 400000 1944' preserveAspectRatio='xMinYMin slice'><path d='M1001,80H400000v40H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5c53.7,-170.3,84.5,-266.8,92.5,-289.5c4,-6.7,10,-10,18,-10zM1001 80H400000v40H1013z'/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.65934em;"><span></span></span></span></span></span></span></span><span style="top:-3.41066em;"><span class="pstrut" style="height:3.18066em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.85766em;"><span class="pstrut" style="height:3.18066em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.7300000000000002em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><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></pre></td><td class="code"><pre><span class="line"><span class="comment">%Matlab代码实现</span></span><br><span class="line">X = input(<span class="string">'请输入一个正向化矩阵：'</span>)</span><br><span class="line">[n,m] = <span class="built_in">size</span>(x)</span><br><span class="line">x ./ <span class="built_in">repmat</span>(sum(A .* A) .^ <span class="number">0.5</span>, n, <span class="number">1</span>)</span><br></pre></td></tr></table></figure><h2 id="计算得分"><a class="markdownIt-Anchor" href="#计算得分"></a> 计算得分</h2><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">得</mi><mi mathvariant="normal">分</mi><mi>x</mi><mo>=</mo><mfrac><mrow><mi>x</mi><mi mathvariant="normal">与</mi><mi mathvariant="normal">最</mi><mi mathvariant="normal">小</mi><mi mathvariant="normal">值</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">距</mi><mi mathvariant="normal">离</mi></mrow><mrow><mi>x</mi><mi mathvariant="normal">与</mi><mi mathvariant="normal">最</mi><mi mathvariant="normal">大</mi><mi mathvariant="normal">值</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">距</mi><mi mathvariant="normal">离</mi><mo>+</mo><mi>x</mi><mi mathvariant="normal">与</mi><mi mathvariant="normal">最</mi><mi mathvariant="normal">小</mi><mi mathvariant="normal">值</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">距</mi><mi mathvariant="normal">离</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mi>min</mi><mo></mo></mrow><mrow><mi>max</mi><mo></mo><mo>−</mo><mi>min</mi><mo></mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mi>min</mi><mo></mo></mrow><mrow><mo stretchy="false">(</mo><mi>max</mi><mo></mo><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>min</mi><mo></mo><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">得分x=\frac{x与最小值的距离}{x与最大值的距离+x与最小值的距离}=\frac{x-\min}{\max-\min}=\frac{x-\min}{(\max-x)+(x-\min)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord cjk_fallback">得</span><span class="mord cjk_fallback">分</span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0987230000000001em;vertical-align:-0.403331em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.695392em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">x</span><span class="mord cjk_fallback mtight">与</span><span class="mord cjk_fallback mtight">最</span><span class="mord cjk_fallback mtight">大</span><span class="mord cjk_fallback mtight">值</span><span class="mord cjk_fallback mtight">的</span><span class="mord cjk_fallback mtight">距</span><span class="mord cjk_fallback mtight">离</span><span class="mbin mtight">+</span><span class="mord mathdefault mtight">x</span><span class="mord cjk_fallback mtight">与</span><span class="mord cjk_fallback mtight">最</span><span class="mord cjk_fallback mtight">小</span><span class="mord cjk_fallback mtight">值</span><span class="mord cjk_fallback mtight">的</span><span class="mord cjk_fallback mtight">距</span><span class="mord cjk_fallback mtight">离</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">x</span><span class="mord cjk_fallback mtight">与</span><span class="mord cjk_fallback mtight">最</span><span class="mord cjk_fallback mtight">小</span><span class="mord cjk_fallback mtight">值</span><span class="mord cjk_fallback mtight">的</span><span class="mord cjk_fallback mtight">距</span><span class="mord cjk_fallback mtight">离</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.403331em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.2648329999999999em;vertical-align:-0.403331em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.861502em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop mtight">max</span><span class="mspace mtight" style="margin-right:0.19516666666666668em;"></span><span class="mord mtight">−</span><span class="mspace mtight" style="margin-right:0.19516666666666668em;"></span><span class="mop mtight">min</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">x</span><span class="mbin mtight">−</span><span class="mop mtight">min</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.403331em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.381502em;vertical-align:-0.52em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.861502em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mop mtight">max</span><span class="mspace mtight" style="margin-right:0.19516666666666668em;"></span><span class="mord mtight">−</span><span class="mord mathdefault mtight">x</span><span class="mclose mtight">)</span><span class="mbin mtight">+</span><span class="mopen mtight">(</span><span class="mord mathdefault mtight">x</span><span class="mbin mtight">−</span><span class="mop mtight">min</span><span class="mclose mtight">)</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">x</span><span class="mbin mtight">−</span><span class="mop mtight">min</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.52em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></p><h2 id="类比只有一个指标计算得分"><a class="markdownIt-Anchor" href="#类比只有一个指标计算得分"></a> 类比只有一个指标计算得分</h2><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">假</mi><mi mathvariant="normal">设</mi><mi mathvariant="normal">有</mi><mi>n</mi><mi mathvariant="normal">个</mi><mi mathvariant="normal">要</mi><mi mathvariant="normal">评</mi><mi mathvariant="normal">价</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">对</mi><mi mathvariant="normal">象</mi><mo separator="true">,</mo><mi>m</mi><mi mathvariant="normal">个</mi><mi mathvariant="normal">正</mi><mi mathvariant="normal">向</mi><mi mathvariant="normal">化</mi><mi mathvariant="normal">评</mi><mi mathvariant="normal">价</mi><mi mathvariant="normal">指</mi><mi mathvariant="normal">标</mi><mi mathvariant="normal">构</mi><mi mathvariant="normal">成</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">正</mi><mi mathvariant="normal">向</mi><mi mathvariant="normal">化</mi><mi mathvariant="normal">矩</mi><mi mathvariant="normal">阵</mi><mi mathvariant="normal">如</mi><mi mathvariant="normal">下</mi><mo>:</mo><mspace linebreak="newline"></mspace><mi>Z</mi><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>z</mi><mn>11</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>z</mi><mn>12</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo>⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>z</mi><mrow><mn>1</mn><mi>n</mi></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>z</mi><mn>21</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>z</mi><mn>22</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo>⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>z</mi><mrow><mn>2</mn><mi>n</mi></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>z</mi><mrow><mi>n</mi><mn>1</mn></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>z</mi><mrow><mi>n</mi><mn>2</mn></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo>⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>z</mi><mrow><mi>n</mi><mi>n</mi></mrow></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo separator="true">,</mo></mrow><annotation encoding="application/x-tex">假设有n个要评价的对象,m个正向化评价指标构成的正向化矩阵如下:\\Z = \left[ \begin{matrix} z_{11} & z_{12} & \cdots & z_{1n} \\ z_{21} & z_{22} & \cdots & z_{2n} \\ \vdots & \vdots & & \vdots \\ z_{n1} & z_{n2} & \cdots & z_{nn} \end{matrix} \right],</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord cjk_fallback">假</span><span class="mord cjk_fallback">设</span><span class="mord cjk_fallback">有</span><span class="mord mathdefault">n</span><span class="mord cjk_fallback">个</span><span class="mord cjk_fallback">要</span><span class="mord cjk_fallback">评</span><span class="mord cjk_fallback">价</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">对</span><span class="mord cjk_fallback">象</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">m</span><span class="mord cjk_fallback">个</span><span class="mord cjk_fallback">正</span><span class="mord cjk_fallback">向</span><span class="mord cjk_fallback">化</span><span class="mord cjk_fallback">评</span><span class="mord cjk_fallback">价</span><span class="mord cjk_fallback">指</span><span class="mord cjk_fallback">标</span><span class="mord cjk_fallback">构</span><span class="mord cjk_fallback">成</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">正</span><span class="mord cjk_fallback">向</span><span class="mord cjk_fallback">化</span><span class="mord cjk_fallback">矩</span><span class="mord cjk_fallback">阵</span><span class="mord cjk_fallback">如</span><span class="mord cjk_fallback">下</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">Z</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:5.459999999999999em;vertical-align:-2.4799999999999995em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.953005em;"><span style="top:-1.3499850000000007em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.5049850000000005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.1059850000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.7069850000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.953005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4500349999999997em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9799999999999995em;"><span style="top:-5.8275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.04398em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.6275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.04398em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.7674999999999996em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-1.5675000000000006em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.04398em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4799999999999995em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9799999999999995em;"><span style="top:-5.8275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.04398em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.6275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.04398em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.7674999999999996em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-1.5675000000000006em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.04398em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4799999999999995em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9799999999999995em;"><span style="top:-5.64em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-4.44em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-2.5799999999999996em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"></span></span><span style="top:-1.3800000000000006em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4799999999999995em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9799999999999995em;"><span style="top:-5.8275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.04398em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.6275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.04398em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.7674999999999996em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-1.5675000000000006em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.04398em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4799999999999995em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.953005em;"><span style="top:-1.3499850000000007em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.5049850000000005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.1059850000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.7069850000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.953005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4500349999999997em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span></span></span></span></span></p>\begin{align}定义最大值Z^+ &= (Z_1^+,Z_2^+,\cdots,Z_m^+)\\&=(\max \{z_{11},z_{21},\cdots,z_{n1}\},\max\{z_{12},z_{22},\cdots,z_{n2}\},\cdots, \max\{z_{1m},z_{2m},\cdots,z_{nm}\})\\定义最小值Z^- &= (Z_1^-,Z_2^-,\cdots,Z_m^-)\\&=(\min\{z_{11},z_{21},\cdots,z_{n1}\},\min\{z_{12},z_{22},\cdots,z_{n2}\},\cdots,\min\{z_{1m},z_{2m},\cdots,z_{nm}\})\\\end{align}<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">定</mi><mi mathvariant="normal">义</mi><mi mathvariant="normal">第</mi><mi>i</mi><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">)</mo><mi mathvariant="normal">个</mi><mi mathvariant="normal">评</mi><mi mathvariant="normal">价</mi><mi mathvariant="normal">对</mi><mi mathvariant="normal">象</mi><mi mathvariant="normal">与</mi><mi mathvariant="normal">对</mi><mi mathvariant="normal">象</mi><mi mathvariant="normal">最</mi><mi mathvariant="normal">大</mi><mi mathvariant="normal">值</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">距</mi><mi mathvariant="normal">离</mi><msubsup><mi>D</mi><mi>i</mi><mo>+</mo></msubsup><mo>=</mo><msqrt><mrow><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><mo stretchy="false">(</mo><msubsup><mi>Z</mi><mi>j</mi><mo>+</mo></msubsup><mo>−</mo><msub><mi>z</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></msqrt><mspace linebreak="newline"></mspace><mi mathvariant="normal">定</mi><mi mathvariant="normal">义</mi><mi mathvariant="normal">第</mi><mi>i</mi><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">)</mo><mi mathvariant="normal">个</mi><mi mathvariant="normal">评</mi><mi mathvariant="normal">价</mi><mi mathvariant="normal">对</mi><mi mathvariant="normal">象</mi><mi mathvariant="normal">与</mi><mi mathvariant="normal">对</mi><mi mathvariant="normal">象</mi><mi mathvariant="normal">最</mi><mi mathvariant="normal">小</mi><mi mathvariant="normal">值</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">距</mi><mi mathvariant="normal">离</mi><msubsup><mi>D</mi><mi>i</mi><mo>−</mo></msubsup><mo>=</mo><msqrt><mrow><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><mo stretchy="false">(</mo><msubsup><mi>Z</mi><mi>j</mi><mo>−</mo></msubsup><mo>−</mo><msub><mi>z</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></msqrt><mspace linebreak="newline"></mspace><mi mathvariant="normal">那</mi><mi mathvariant="normal">么</mi><mi mathvariant="normal">，</mi><mi mathvariant="normal">我</mi><mi mathvariant="normal">们</mi><mi mathvariant="normal">可</mi><mi mathvariant="normal">以</mi><mi mathvariant="normal">计</mi><mi mathvariant="normal">算</mi><mi mathvariant="normal">得</mi><mi mathvariant="normal">出</mi><mi mathvariant="normal">第</mi><mi>i</mi><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">)</mo><mi mathvariant="normal">个</mi><mi mathvariant="normal">评</mi><mi mathvariant="normal">价</mi><mi mathvariant="normal">对</mi><mi mathvariant="normal">象</mi><mi mathvariant="normal">未</mi><mi mathvariant="normal">归</mi><mi mathvariant="normal">一</mi><mi mathvariant="normal">化</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">得</mi><mi mathvariant="normal">分</mi><mi mathvariant="normal">：</mi><msub><mi>S</mi><mi>i</mi></msub><mo>=</mo><mfrac><msubsup><mi>D</mi><mi>i</mi><mo>−</mo></msubsup><mrow><msubsup><mi>D</mi><mi>i</mi><mo>+</mo></msubsup><mo>+</mo><msubsup><mi>D</mi><mi>i</mi><mo>−</mo></msubsup></mrow></mfrac><mspace linebreak="newline"></mspace><mi mathvariant="normal">很</mi><mi mathvariant="normal">明</mi><mi mathvariant="normal">显</mi><mn>0</mn><mo>≤</mo><msub><mi>S</mi><mi>i</mi></msub><mo>≤</mo><mn>1</mn><mo separator="true">,</mo><mi mathvariant="normal">且</mi><msub><mi>S</mi><mi>i</mi></msub><mi mathvariant="normal">越</mi><mi mathvariant="normal">大</mi><msubsup><mi>D</mi><mi>i</mi><mo>+</mo></msubsup><mi mathvariant="normal">越</mi><mi mathvariant="normal">小</mi><mi mathvariant="normal">，</mi><mi mathvariant="normal">即</mi><mi mathvariant="normal">越</mi><mi mathvariant="normal">接</mi><mi mathvariant="normal">近</mi><mi mathvariant="normal">最</mi><mi mathvariant="normal">大</mi><mi mathvariant="normal">值</mi><mi mathvariant="normal">。</mi></mrow><annotation encoding="application/x-tex">定义第i(i=1,2,\cdots,n)个评价对象与对象最大值的距离D_i^+ = \sqrt{\sum^m_{j=1}(Z^+_j-z_{ij})^2}\\定义第i(i=1,2,\cdots,n)个评价对象与对象最小值的距离D_i^- = \sqrt{\sum^m_{j=1}(Z^-_j-z_{ij})^2}\\那么，我们可以计算得出第i(i=1,2,\cdots,n)个评价对象未归一化的得分：S_i=\frac{D_i^-}{D_i^++D_i^-}\\很明显0\leq S_i \leq 1,且S_i越大D_i^+越小，即越接近最大值。</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord cjk_fallback">定</span><span class="mord cjk_fallback">义</span><span class="mord cjk_fallback">第</span><span class="mord mathdefault">i</span><span class="mopen">(</span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.088326em;vertical-align:-0.266995em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">n</span><span class="mclose">)</span><span class="mord cjk_fallback">个</span><span class="mord cjk_fallback">评</span><span class="mord cjk_fallback">价</span><span class="mord cjk_fallback">对</span><span class="mord cjk_fallback">象</span><span class="mord cjk_fallback">与</span><span class="mord cjk_fallback">对</span><span class="mord cjk_fallback">象</span><span class="mord cjk_fallback">最</span><span class="mord cjk_fallback">大</span><span class="mord cjk_fallback">值</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">距</span><span class="mord cjk_fallback">离</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8213309999999999em;"><span style="top:-2.433005em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.266995em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.2929240000000006em;vertical-align:-1.4137769999999998em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8791470000000008em;"><span class="svg-align" style="top:-5.252924em;"><span class="pstrut" style="height:5.252924em;"></span><span class="mord" style="padding-left:1.056em;"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6513970000000007em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.4137769999999998em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.811462em;"><span style="top:-2.4231360000000004em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1031310000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.412972em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.04398em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.8391470000000005em;"><span class="pstrut" style="height:5.252924em;"></span><span class="hide-tail" style="min-width:0.742em;height:3.3329240000000007em;"><svg width='400em' height='3.3329240000000007em' viewBox='0 0 400000 3332' preserveAspectRatio='xMinYMin slice'><path d='M702 80H400000v40H742v3198l-4 4-4 4c-.667.7-2 1.5-4 2.5s-4.167 1.833-6.5 2.5-5.5 1-9.5 1h-12l-28-84c-16.667-52-96.667-294.333-240-727l-212 -643 -85 170c-4-3.333-8.333-7.667-13 -13l-13-13l77-155 77-156c66 199.333 139 419.667 219 661 l218 661zM702 80H400000v40H742z'/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.4137769999999998em;"><span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord cjk_fallback">定</span><span class="mord cjk_fallback">义</span><span class="mord cjk_fallback">第</span><span class="mord mathdefault">i</span><span class="mopen">(</span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.088326em;vertical-align:-0.266995em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">n</span><span class="mclose">)</span><span class="mord cjk_fallback">个</span><span class="mord cjk_fallback">评</span><span class="mord cjk_fallback">价</span><span class="mord cjk_fallback">对</span><span class="mord cjk_fallback">象</span><span class="mord cjk_fallback">与</span><span class="mord cjk_fallback">对</span><span class="mord cjk_fallback">象</span><span class="mord cjk_fallback">最</span><span class="mord cjk_fallback">小</span><span class="mord cjk_fallback">值</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">距</span><span class="mord cjk_fallback">离</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8213309999999999em;"><span style="top:-2.433005em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">−</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.266995em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.2929240000000006em;vertical-align:-1.4137769999999998em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8791470000000008em;"><span class="svg-align" style="top:-5.252924em;"><span class="pstrut" style="height:5.252924em;"></span><span class="mord" style="padding-left:1.056em;"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6513970000000007em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.4137769999999998em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.811462em;"><span style="top:-2.4231360000000004em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1031310000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">−</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.412972em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.04398em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.8391470000000005em;"><span class="pstrut" style="height:5.252924em;"></span><span class="hide-tail" style="min-width:0.742em;height:3.3329240000000007em;"><svg width='400em' height='3.3329240000000007em' viewBox='0 0 400000 3332' preserveAspectRatio='xMinYMin slice'><path d='M702 80H400000v40H742v3198l-4 4-4 4c-.667.7-2 1.5-4 2.5s-4.167 1.833-6.5 2.5-5.5 1-9.5 1h-12l-28-84c-16.667-52-96.667-294.333-240-727l-212 -643 -85 170c-4-3.333-8.333-7.667-13 -13l-13-13l77-155 77-156c66 199.333 139 419.667 219 661 l218 661zM702 80H400000v40H742z'/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.4137769999999998em;"><span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord cjk_fallback">那</span><span class="mord cjk_fallback">么</span><span class="mord cjk_fallback">，</span><span class="mord cjk_fallback">我</span><span class="mord cjk_fallback">们</span><span class="mord cjk_fallback">可</span><span class="mord cjk_fallback">以</span><span class="mord cjk_fallback">计</span><span class="mord cjk_fallback">算</span><span class="mord cjk_fallback">得</span><span class="mord cjk_fallback">出</span><span class="mord cjk_fallback">第</span><span class="mord mathdefault">i</span><span class="mopen">(</span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">n</span><span class="mclose">)</span><span class="mord cjk_fallback">个</span><span class="mord cjk_fallback">评</span><span class="mord cjk_fallback">价</span><span class="mord cjk_fallback">对</span><span class="mord cjk_fallback">象</span><span class="mord cjk_fallback">未</span><span class="mord cjk_fallback">归</span><span class="mord cjk_fallback">一</span><span class="mord cjk_fallback">化</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">得</span><span class="mord cjk_fallback">分</span><span class="mord cjk_fallback">：</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.466788em;vertical-align:-0.978326em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.488462em;"><span style="top:-2.2985379999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.811462em;"><span style="top:-2.4231360000000004em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1031310000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.811462em;"><span style="top:-2.4231360000000004em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1031310000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">−</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.6769999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.811462em;"><span style="top:-2.4231360000000004em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1031310000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">−</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.978326em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.78041em;vertical-align:-0.13597em;"></span><span class="mord cjk_fallback">很</span><span class="mord cjk_fallback">明</span><span class="mord cjk_fallback">显</span><span class="mord">0</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.088326em;vertical-align:-0.266995em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord cjk_fallback">且</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord cjk_fallback">越</span><span class="mord cjk_fallback">大</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8213309999999999em;"><span style="top:-2.433005em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.266995em;"><span></span></span></span></span></span></span><span class="mord cjk_fallback">越</span><span class="mord cjk_fallback">小</span><span class="mord cjk_fallback">，</span><span class="mord cjk_fallback">即</span><span class="mord cjk_fallback">越</span><span class="mord cjk_fallback">接</span><span class="mord cjk_fallback">近</span><span class="mord cjk_fallback">最</span><span class="mord cjk_fallback">大</span><span class="mord cjk_fallback">值</span><span class="mord cjk_fallback">。</span></span></span></span></span></p><h1 id="topsis法的步骤"><a class="markdownIt-Anchor" href="#topsis法的步骤"></a> TOPSIS法的步骤</h1><h2 id="第一步将原始指标正向化评价标准统一化"><a class="markdownIt-Anchor" href="#第一步将原始指标正向化评价标准统一化"></a> 第一步：将原始指标正向化（评价标准统一化）</h2><p>原始指标分类：</p><p>1.极大型指标（效益型指标）</p><p>2.极小型指标（成本型指标）</p><p>3.中间型指标（越接近某个值越好）</p><p>4.区间型指标（落在区间内为佳）</p><h3 id="转换方式统一为极大型指标"><a class="markdownIt-Anchor" href="#转换方式统一为极大型指标"></a> 转换方式（统一为极大型指标）</h3><h4 id="极大型指标不用转换"><a class="markdownIt-Anchor" href="#极大型指标不用转换"></a> 极大型指标：不用转换</h4><h4 id="极小型指标max-x"><a class="markdownIt-Anchor" href="#极小型指标max-x"></a> 极小型指标：<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>max</mi><mo></mo><mo>−</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\max-x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mop">max</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">−</span><span class="mord mathdefault">x</span></span></span></span></h4><h4 id="中间型指标"><a class="markdownIt-Anchor" href="#中间型指标"></a> 中间型指标：</h4><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">设</mi><mo stretchy="false">{</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">}</mo><mi mathvariant="normal">为</mi><mi mathvariant="normal">一</mi><mi mathvariant="normal">组</mi><mi mathvariant="normal">中</mi><mi mathvariant="normal">间</mi><mi mathvariant="normal">型</mi><mi mathvariant="normal">指</mi><mi mathvariant="normal">标</mi><mi mathvariant="normal">，</mi><mi mathvariant="normal">其</mi><mi mathvariant="normal">中</mi><mi>x</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">最</mi><mi mathvariant="normal">佳</mi><mi mathvariant="normal">取</mi><mi mathvariant="normal">值</mi><mi mathvariant="normal">为</mi><msub><mi>x</mi><mrow><mi>b</mi><mi>e</mi><mi>s</mi><mi>t</mi></mrow></msub><mo separator="true">,</mo><mi mathvariant="normal">则</mi><mi mathvariant="normal">正</mi><mi mathvariant="normal">向</mi><mi mathvariant="normal">化</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">公</mi><mi mathvariant="normal">式</mi><mi mathvariant="normal">如</mi><mi mathvariant="normal">下</mi><mi mathvariant="normal">：</mi><mspace linebreak="newline"></mspace><mi>M</mi><mo>=</mo><mi>max</mi><mo></mo><mo stretchy="false">{</mo><mi mathvariant="normal">∣</mi><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><msub><mi>x</mi><mrow><mi>b</mi><mi>e</mi><mi>s</mi><mi>t</mi></mrow></msub><mi mathvariant="normal">∣</mi><mo stretchy="false">}</mo><mo separator="true">,</mo><msub><mover accent="true"><mi>x</mi><mo>~</mo></mover><mi>i</mi></msub><mo>=</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mi mathvariant="normal">∣</mi><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><msub><mi>x</mi><mrow><mi>b</mi><mi>e</mi><mi>s</mi><mi>t</mi></mrow></msub><mi mathvariant="normal">∣</mi></mrow><mi>M</mi></mfrac></mrow><annotation encoding="application/x-tex">设\{x_i\}为一组中间型指标，其中x的最佳取值为x_{best},则正向化的公式如下：\\M=\max\{|x_i-x_{best}|\},\tilde{x}_{i}=1-\frac{|x_i-x_{best}|}{M}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord cjk_fallback">设</span><span class="mopen">{</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span><span class="mord cjk_fallback">为</span><span class="mord cjk_fallback">一</span><span class="mord cjk_fallback">组</span><span class="mord cjk_fallback">中</span><span class="mord cjk_fallback">间</span><span class="mord cjk_fallback">型</span><span class="mord cjk_fallback">指</span><span class="mord cjk_fallback">标</span><span class="mord cjk_fallback">，</span><span class="mord cjk_fallback">其</span><span class="mord cjk_fallback">中</span><span class="mord mathdefault">x</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">最</span><span class="mord cjk_fallback">佳</span><span class="mord cjk_fallback">取</span><span class="mord cjk_fallback">值</span><span class="mord cjk_fallback">为</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord cjk_fallback">则</span><span class="mord cjk_fallback">正</span><span class="mord cjk_fallback">向</span><span class="mord cjk_fallback">化</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">公</span><span class="mord cjk_fallback">式</span><span class="mord cjk_fallback">如</span><span class="mord cjk_fallback">下</span><span class="mord cjk_fallback">：</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">M</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">max</span><span class="mopen">{</span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mclose">}</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">x</span></span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.22222em;">~</span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10903em;">M</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">b</span><span class="mord mathdefault mtight">e</span><span class="mord mathdefault mtight">s</span><span class="mord mathdefault mtight">t</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><h4 id="区间型指标"><a class="markdownIt-Anchor" href="#区间型指标"></a> 区间型指标：</h4><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi mathvariant="normal">分</mi><mi mathvariant="normal">类</mi><mi mathvariant="normal">讨</mi><mi mathvariant="normal">论</mi><mi mathvariant="normal">：</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mi mathvariant="normal">在</mi><mi mathvariant="normal">区</mi><mi mathvariant="normal">间</mi><mi mathvariant="normal">内</mi><mi mathvariant="normal">：</mi><msub><mover accent="true"><mi>x</mi><mo>~</mo></mover><mi>i</mi></msub><mo>=</mo><mn>1</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mi mathvariant="normal">在</mi><mi mathvariant="normal">区</mi><mi mathvariant="normal">间</mi><mi mathvariant="normal">外</mi><mi mathvariant="normal">：</mi><mi mathvariant="normal">假</mi><mi mathvariant="normal">设</mi><mo stretchy="false">{</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">}</mo><mi mathvariant="normal">为</mi><mi mathvariant="normal">一</mi><mi mathvariant="normal">组</mi><mi mathvariant="normal">区</mi><mi mathvariant="normal">间</mi><mi mathvariant="normal">型</mi><mi mathvariant="normal">指</mi><mi mathvariant="normal">标</mi><mi mathvariant="normal">，</mi><mi mathvariant="normal">且</mi><mi mathvariant="normal">其</mi><mi mathvariant="normal">最</mi><mi mathvariant="normal">佳</mi><mi mathvariant="normal">区</mi><mi mathvariant="normal">间</mi><mi mathvariant="normal">为</mi><mo stretchy="false">[</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">]</mo><mo separator="true">,</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mi mathvariant="normal">令</mi><mi>M</mi><mo>=</mo><mi>max</mi><mo></mo><mo stretchy="false">{</mo><mi>a</mi><mo>−</mo><mi>min</mi><mo></mo><mo stretchy="false">{</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">}</mo><mo separator="true">,</mo><mi>max</mi><mo></mo><mo stretchy="false">{</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">}</mo><mo>−</mo><mi>b</mi><mo stretchy="false">}</mo><mo separator="true">,</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><msub><mi>x</mi><mi>i</mi></msub><mo>=</mo><mrow><mo fence="true">{</mo><mtable rowspacing="0.3599999999999999em" columnalign="left left" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><mo>−</mo><mfrac><mrow><mi>a</mi><mo>−</mo><msub><mi>x</mi><mi>i</mi></msub></mrow><mi>M</mi></mfrac><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>x</mi><mi>i</mi></msub><mo><</mo><mi>a</mi><mo separator="true">;</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>a</mi><mo>≤</mo><msub><mi>x</mi><mi>i</mi></msub><mo>≤</mo><mi>b</mi><mo separator="true">;</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mn>1</mn><mo>−</mo><mfrac><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><mi>b</mi></mrow><mi>M</mi></mfrac><mo separator="true">,</mo></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>></mo><mi>b</mi><mo separator="true">;</mo></mrow></mstyle></mtd></mtr></mtable></mrow></mrow></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{aligned}&分类讨论：\\&(1)在区间内：\tilde{x}_i=1\\&(2)在区间外：假设\{x_i\}为一组区间型指标，且其最佳区间为[a,b],\\&令M=\max\{a-\min\{x_i\},\max\{x_i\}-b\},\\&x_i=\begin{cases}1-\frac{a-x_i}{M},&x_i<a;\\1,&a\leq x_i\leq b;\\1-\frac{x_i-b}{M},&x_i>b;\end{cases}\end{aligned}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:10.620000000000001em;vertical-align:-5.0600000000000005em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.5600000000000005em;"><span style="top:-9.13em;"><span class="pstrut" style="height:4.41em;"></span><span class="mord"></span></span><span style="top:-7.63em;"><span class="pstrut" style="height:4.41em;"></span><span class="mord"></span></span><span style="top:-6.129999999999999em;"><span class="pstrut" style="height:4.41em;"></span><span class="mord"></span></span><span style="top:-4.629999999999999em;"><span class="pstrut" style="height:4.41em;"></span><span class="mord"></span></span><span style="top:-1.5599999999999996em;"><span class="pstrut" style="height:4.41em;"></span><span class="mord"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:5.0600000000000005em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:5.5600000000000005em;"><span style="top:-9.13em;"><span class="pstrut" style="height:4.41em;"></span><span class="mord"><span class="mord"></span><span class="mord cjk_fallback">分</span><span class="mord cjk_fallback">类</span><span class="mord cjk_fallback">讨</span><span class="mord cjk_fallback">论</span><span class="mord cjk_fallback">：</span></span></span><span style="top:-7.63em;"><span class="pstrut" style="height:4.41em;"></span><span class="mord"><span class="mord"></span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span><span class="mord cjk_fallback">在</span><span class="mord cjk_fallback">区</span><span class="mord cjk_fallback">间</span><span class="mord cjk_fallback">内</span><span class="mord cjk_fallback">：</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">x</span></span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.22222em;">~</span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord">1</span></span></span><span style="top:-6.129999999999999em;"><span class="pstrut" style="height:4.41em;"></span><span class="mord"><span class="mord"></span><span class="mopen">(</span><span class="mord">2</span><span class="mclose">)</span><span class="mord cjk_fallback">在</span><span class="mord cjk_fallback">区</span><span class="mord cjk_fallback">间</span><span class="mord cjk_fallback">外</span><span class="mord cjk_fallback">：</span><span class="mord cjk_fallback">假</span><span class="mord cjk_fallback">设</span><span class="mopen">{</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span><span class="mord cjk_fallback">为</span><span class="mord cjk_fallback">一</span><span class="mord cjk_fallback">组</span><span class="mord cjk_fallback">区</span><span class="mord cjk_fallback">间</span><span class="mord cjk_fallback">型</span><span class="mord cjk_fallback">指</span><span class="mord cjk_fallback">标</span><span class="mord cjk_fallback">，</span><span class="mord cjk_fallback">且</span><span class="mord cjk_fallback">其</span><span class="mord cjk_fallback">最</span><span class="mord cjk_fallback">佳</span><span class="mord cjk_fallback">区</span><span class="mord cjk_fallback">间</span><span class="mord cjk_fallback">为</span><span class="mopen">[</span><span class="mord mathdefault">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span><span class="mclose">]</span><span class="mpunct">,</span></span></span><span style="top:-4.629999999999999em;"><span class="pstrut" style="height:4.41em;"></span><span class="mord"><span class="mord"></span><span class="mord cjk_fallback">令</span><span class="mord mathdefault" style="margin-right:0.10903em;">M</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mop">max</span><span class="mopen">{</span><span class="mord mathdefault">a</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mop">min</span><span class="mopen">{</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">max</span><span class="mopen">{</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">}</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault">b</span><span class="mclose">}</span><span class="mpunct">,</span></span></span><span style="top:-1.5599999999999996em;"><span class="pstrut" style="height:4.41em;"></span><span class="mord"><span class="mord"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.35002em;"><span style="top:-2.19999em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.19999em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-3.1500100000000004em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.30001em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎪</span></span></span><span style="top:-4.60002em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.8500199999999998em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em;"><span style="top:-4.41em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8184309999999999em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10903em;">M</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4101em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">a</span><span class="mbin mtight">−</span><span class="mord mtight"><span class="mord mathdefault mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span></span></span><span style="top:-2.97em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">1</span><span class="mpunct">,</span></span></span><span style="top:-1.5300000000000002em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8962079999999999em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10903em;">M</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4101em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mathdefault mtight">b</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.9099999999999997em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.41em;"><span style="top:-4.41em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathdefault">a</span><span class="mpunct">;</span></span></span><span style="top:-2.97em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathdefault">b</span><span class="mpunct">;</span></span></span><span style="top:-1.5300000000000002em;"><span class="pstrut" style="height:3.008em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathdefault">b</span><span class="mpunct">;</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.9099999999999997em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:5.0600000000000005em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><h2 id="第二步正向化矩阵标准化消除量纲不同造成的影响"><a class="markdownIt-Anchor" href="#第二步正向化矩阵标准化消除量纲不同造成的影响"></a> 第二步：正向化矩阵标准化（消除量纲不同造成的影响）</h2><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">假</mi><mi mathvariant="normal">设</mi><mi mathvariant="normal">有</mi><mi>n</mi><mi mathvariant="normal">个</mi><mi mathvariant="normal">要</mi><mi mathvariant="normal">评</mi><mi mathvariant="normal">价</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">对</mi><mi mathvariant="normal">象</mi><mo separator="true">,</mo><mi>m</mi><mi mathvariant="normal">个</mi><mi mathvariant="normal">正</mi><mi mathvariant="normal">向</mi><mi mathvariant="normal">化</mi><mi mathvariant="normal">评</mi><mi mathvariant="normal">价</mi><mi mathvariant="normal">指</mi><mi mathvariant="normal">标</mi><mi mathvariant="normal">构</mi><mi mathvariant="normal">成</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">正</mi><mi mathvariant="normal">向</mi><mi mathvariant="normal">化</mi><mi mathvariant="normal">矩</mi><mi mathvariant="normal">阵</mi><mi mathvariant="normal">如</mi><mi mathvariant="normal">下</mi><mo>:</mo><mspace linebreak="newline"></mspace><mi>A</mi><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mn>11</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mn>12</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo>⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mrow><mn>1</mn><mi>n</mi></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mn>21</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mn>22</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo>⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mrow><mn>2</mn><mi>n</mi></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mrow><mi>n</mi><mn>1</mn></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mrow><mi>n</mi><mn>2</mn></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo>⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mrow><mi>n</mi><mi>n</mi></mrow></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo separator="true">,</mo><mspace linebreak="newline"></mspace><mi mathvariant="normal">那</mi><mi mathvariant="normal">么</mi><mo separator="true">,</mo><mi mathvariant="normal">对</mi><mi mathvariant="normal">其</mi><mi mathvariant="normal">标</mi><mi mathvariant="normal">准</mi><mi mathvariant="normal">化</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">矩</mi><mi mathvariant="normal">阵</mi><mi mathvariant="normal">记</mi><mi mathvariant="normal">为</mi><mi>Z</mi><mo separator="true">,</mo><mi>Z</mi><mi mathvariant="normal">中</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">每</mi><mi mathvariant="normal">一</mi><mi mathvariant="normal">个</mi><mi mathvariant="normal">元</mi><mi mathvariant="normal">素</mi><mi mathvariant="normal">：</mi><msub><mi>z</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mfrac><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msqrt><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>x</mi><mrow><mi>i</mi><mi>j</mi></mrow><mn>2</mn></msubsup></mrow></msqrt></mfrac><mspace linebreak="newline"></mspace><mo stretchy="false">(</mo><mfrac><mrow><mi mathvariant="normal">其</mi><mi mathvariant="normal">每</mi><mi mathvariant="normal">一</mi><mi mathvariant="normal">个</mi><mi mathvariant="normal">元</mi><mi mathvariant="normal">素</mi></mrow><msqrt><mrow><mi mathvariant="normal">列</mi><mi mathvariant="normal">所</mi><mi mathvariant="normal">在</mi><mi mathvariant="normal">元</mi><mi mathvariant="normal">素</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">平</mi><mi mathvariant="normal">方</mi><mi mathvariant="normal">和</mi></mrow></msqrt></mfrac><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">假设有n个要评价的对象,m个正向化评价指标构成的正向化矩阵如下:\\A = \left[ \begin{matrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{matrix} \right],\\ 那么,对其标准化的矩阵记为Z,Z中的每一个元素：z_{ij}=\frac{a_{ij}}{\sqrt{\sum^n_{i=1}x^2_{ij}}}\\(\frac{其每一个元素}{\sqrt{列所在元素的平方和}})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord cjk_fallback">假</span><span class="mord cjk_fallback">设</span><span class="mord cjk_fallback">有</span><span class="mord mathdefault">n</span><span class="mord cjk_fallback">个</span><span class="mord cjk_fallback">要</span><span class="mord cjk_fallback">评</span><span class="mord cjk_fallback">价</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">对</span><span class="mord cjk_fallback">象</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">m</span><span class="mord cjk_fallback">个</span><span class="mord cjk_fallback">正</span><span class="mord cjk_fallback">向</span><span class="mord cjk_fallback">化</span><span class="mord cjk_fallback">评</span><span class="mord cjk_fallback">价</span><span class="mord cjk_fallback">指</span><span class="mord cjk_fallback">标</span><span class="mord cjk_fallback">构</span><span class="mord cjk_fallback">成</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">正</span><span class="mord cjk_fallback">向</span><span class="mord cjk_fallback">化</span><span class="mord cjk_fallback">矩</span><span class="mord cjk_fallback">阵</span><span class="mord cjk_fallback">如</span><span class="mord cjk_fallback">下</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault">A</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:5.459999999999999em;vertical-align:-2.4799999999999995em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.953005em;"><span style="top:-1.3499850000000007em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.5049850000000005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.1059850000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.7069850000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.953005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4500349999999997em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9799999999999995em;"><span style="top:-5.8275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.6275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.7674999999999996em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-1.5675000000000006em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4799999999999995em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9799999999999995em;"><span style="top:-5.8275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.6275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.7674999999999996em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-1.5675000000000006em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4799999999999995em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9799999999999995em;"><span style="top:-5.64em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-4.44em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-2.5799999999999996em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"></span></span><span style="top:-1.3800000000000006em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4799999999999995em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9799999999999995em;"><span style="top:-5.8275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.6275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.7674999999999996em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-1.5675000000000006em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4799999999999995em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.953005em;"><span style="top:-1.3499850000000007em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.5049850000000005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.1059850000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.7069850000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.953005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4500349999999997em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord cjk_fallback">那</span><span class="mord cjk_fallback">么</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord cjk_fallback">对</span><span class="mord cjk_fallback">其</span><span class="mord cjk_fallback">标</span><span class="mord cjk_fallback">准</span><span class="mord cjk_fallback">化</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">矩</span><span class="mord cjk_fallback">阵</span><span class="mord cjk_fallback">记</span><span class="mord cjk_fallback">为</span><span class="mord mathdefault" style="margin-right:0.07153em;">Z</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">Z</span><span class="mord cjk_fallback">中</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">每</span><span class="mord cjk_fallback">一</span><span class="mord cjk_fallback">个</span><span class="mord cjk_fallback">元</span><span class="mord cjk_fallback">素</span><span class="mord cjk_fallback">：</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.04398em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.83756em;vertical-align:-1.7300000000000002em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1075599999999999em;"><span style="top:-2.11em;"><span class="pstrut" style="height:3.18066em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.18066em;"><span class="svg-align" style="top:-3.8em;"><span class="pstrut" style="height:3.8em;"></span><span class="mord" style="padding-left:1em;"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.804292em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.795908em;"><span style="top:-2.4231360000000004em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4129719999999999em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.1406599999999996em;"><span class="pstrut" style="height:3.8em;"></span><span class="hide-tail" style="min-width:1.02em;height:1.8800000000000001em;"><svg width='400em' height='1.8800000000000001em' viewBox='0 0 400000 1944' preserveAspectRatio='xMinYMin slice'><path d='M1001,80H400000v40H1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7s-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744c-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30c26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722c56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5c53.7,-170.3,84.5,-266.8,92.5,-289.5c4,-6.7,10,-10,18,-10zM1001 80H400000v40H1013z'/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.65934em;"><span></span></span></span></span></span></span></span><span style="top:-3.41066em;"><span class="pstrut" style="height:3.18066em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.85766em;"><span class="pstrut" style="height:3.18066em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.7300000000000002em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.68em;vertical-align:-0.9299999999999999em;"></span><span class="mopen">(</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.677em;"><span style="top:-2.3095000000000003em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8005em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord cjk_fallback">列</span><span class="mord cjk_fallback">所</span><span class="mord cjk_fallback">在</span><span class="mord cjk_fallback">元</span><span class="mord cjk_fallback">素</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">平</span><span class="mord cjk_fallback">方</span><span class="mord cjk_fallback">和</span></span></span><span style="top:-2.7605em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80H400000v40H845z'/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2395em;"><span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord cjk_fallback">其</span><span class="mord cjk_fallback">每</span><span class="mord cjk_fallback">一</span><span class="mord cjk_fallback">个</span><span class="mord cjk_fallback">元</span><span class="mord cjk_fallback">素</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9299999999999999em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">)</span></span></span></span></span></p><h2 id="第三步计算得分并归一化缩小样本分布范围"><a class="markdownIt-Anchor" href="#第三步计算得分并归一化缩小样本分布范围"></a> 第三步：计算得分并归一化（缩小样本分布范围）</h2><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">假</mi><mi mathvariant="normal">设</mi><mi mathvariant="normal">有</mi><mi>n</mi><mi mathvariant="normal">个</mi><mi mathvariant="normal">要</mi><mi mathvariant="normal">评</mi><mi mathvariant="normal">价</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">对</mi><mi mathvariant="normal">象</mi><mo separator="true">,</mo><mi>m</mi><mi mathvariant="normal">个</mi><mi mathvariant="normal">正</mi><mi mathvariant="normal">向</mi><mi mathvariant="normal">化</mi><mi mathvariant="normal">评</mi><mi mathvariant="normal">价</mi><mi mathvariant="normal">指</mi><mi mathvariant="normal">标</mi><mi mathvariant="normal">构</mi><mi mathvariant="normal">成</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">正</mi><mi mathvariant="normal">向</mi><mi mathvariant="normal">化</mi><mi mathvariant="normal">矩</mi><mi mathvariant="normal">阵</mi><mi mathvariant="normal">如</mi><mi mathvariant="normal">下</mi><mo>:</mo><mspace linebreak="newline"></mspace><mi>Z</mi><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>z</mi><mn>11</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>z</mi><mn>12</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo>⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>z</mi><mrow><mn>1</mn><mi>n</mi></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>z</mi><mn>21</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>z</mi><mn>22</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo>⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>z</mi><mrow><mn>2</mn><mi>n</mi></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mrow></mrow><mi>z</mi></msub><mrow><mi>n</mi><mn>1</mn></mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>z</mi><mrow><mi>n</mi><mn>2</mn></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo>⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>z</mi><mrow><mi>n</mi><mi>n</mi></mrow></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo separator="true">,</mo></mrow><annotation encoding="application/x-tex">假设有n个要评价的对象,m个正向化评价指标构成的正向化矩阵如下:\\Z = \left[ \begin{matrix} z_{11} & z_{12} & \cdots & z_{1n} \\ z_{21} & z_{22} & \cdots & z_{2n} \\ \vdots & \vdots & & \vdots \\ _z{n1} & z_{n2} & \cdots & z_{nn} \end{matrix} \right],</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord cjk_fallback">假</span><span class="mord cjk_fallback">设</span><span class="mord cjk_fallback">有</span><span class="mord mathdefault">n</span><span class="mord cjk_fallback">个</span><span class="mord cjk_fallback">要</span><span class="mord cjk_fallback">评</span><span class="mord cjk_fallback">价</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">对</span><span class="mord cjk_fallback">象</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">m</span><span class="mord cjk_fallback">个</span><span class="mord cjk_fallback">正</span><span class="mord cjk_fallback">向</span><span class="mord cjk_fallback">化</span><span class="mord cjk_fallback">评</span><span class="mord cjk_fallback">价</span><span class="mord cjk_fallback">指</span><span class="mord cjk_fallback">标</span><span class="mord cjk_fallback">构</span><span class="mord cjk_fallback">成</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">正</span><span class="mord cjk_fallback">向</span><span class="mord cjk_fallback">化</span><span class="mord cjk_fallback">矩</span><span class="mord cjk_fallback">阵</span><span class="mord cjk_fallback">如</span><span class="mord cjk_fallback">下</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">Z</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:5.459999999999999em;vertical-align:-2.4799999999999995em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.953005em;"><span style="top:-1.3499850000000007em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.5049850000000005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.1059850000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.7069850000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.953005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4500349999999997em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9799999999999995em;"><span style="top:-5.8275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.04398em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.6275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.04398em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.7674999999999996em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-1.5675000000000006em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.04398em;">z</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">n</span><span class="mord">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4799999999999995em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9799999999999995em;"><span style="top:-5.8275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.04398em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.6275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.04398em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.7674999999999996em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-1.5675000000000006em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.04398em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4799999999999995em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9799999999999995em;"><span style="top:-5.64em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-4.44em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-2.5799999999999996em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"></span></span><span style="top:-1.3800000000000006em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4799999999999995em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9799999999999995em;"><span style="top:-5.8275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.04398em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.6275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.04398em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.7674999999999996em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-1.5675000000000006em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.04398em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4799999999999995em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.953005em;"><span style="top:-1.3499850000000007em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.5049850000000005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.1059850000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.7069850000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.953005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4500349999999997em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span></span></span></span></span></p> \begin{align}\\ 定义最大值Z^+ &= (Z_1^+,Z_2^+,\cdots,Z_m^+)\\ &=(\max \{z_{11},z_{21},\cdots,z_{n1}\},\max\{z_{12},z_{22},\cdots,z_{n2}\},\cdots, \max\{z_{1m},z_{2m},\cdots,z_{nm}\})\\ 定义最小值Z^- &= (Z_1^-,Z_2^-,\cdots,Z_m^-)\\ &=(\min\{z_{11},z_{21},\cdots,z_{n1}\},\min\{z_{12},z_{22},\cdots,z_{n2}\},\cdots, \min\{z_{1m},z_{2m},\cdots,z_{nm}\})\\ \end{align}<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">定</mi><mi mathvariant="normal">义</mi><mi mathvariant="normal">第</mi><mi>i</mi><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">)</mo><mi mathvariant="normal">个</mi><mi mathvariant="normal">评</mi><mi mathvariant="normal">价</mi><mi mathvariant="normal">对</mi><mi mathvariant="normal">象</mi><mi mathvariant="normal">与</mi><mi mathvariant="normal">对</mi><mi mathvariant="normal">象</mi><mi mathvariant="normal">最</mi><mi mathvariant="normal">大</mi><mi mathvariant="normal">值</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">距</mi><mi mathvariant="normal">离</mi><msubsup><mi>D</mi><mi>i</mi><mo>+</mo></msubsup><mo>=</mo><msqrt><mrow><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><mo stretchy="false">(</mo><msubsup><mi>Z</mi><mi>j</mi><mo>+</mo></msubsup><mo>−</mo><msub><mi>z</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></msqrt><mspace linebreak="newline"></mspace><mi mathvariant="normal">定</mi><mi mathvariant="normal">义</mi><mi mathvariant="normal">第</mi><mi>i</mi><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">)</mo><mi mathvariant="normal">个</mi><mi mathvariant="normal">评</mi><mi mathvariant="normal">价</mi><mi mathvariant="normal">对</mi><mi mathvariant="normal">象</mi><mi mathvariant="normal">与</mi><mi mathvariant="normal">对</mi><mi mathvariant="normal">象</mi><mi mathvariant="normal">最</mi><mi mathvariant="normal">小</mi><mi mathvariant="normal">值</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">距</mi><mi mathvariant="normal">离</mi><msubsup><mi>D</mi><mi>i</mi><mo>−</mo></msubsup><mo>=</mo><msqrt><mrow><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><mo stretchy="false">(</mo><msubsup><mi>Z</mi><mi>j</mi><mo>−</mo></msubsup><mo>−</mo><msub><mi>z</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></msqrt><mspace linebreak="newline"></mspace><mi mathvariant="normal">那</mi><mi mathvariant="normal">么</mi><mi mathvariant="normal">，</mi><mi mathvariant="normal">我</mi><mi mathvariant="normal">们</mi><mi mathvariant="normal">可</mi><mi mathvariant="normal">以</mi><mi mathvariant="normal">计</mi><mi mathvariant="normal">算</mi><mi mathvariant="normal">得</mi><mi mathvariant="normal">出</mi><mi mathvariant="normal">第</mi><mi>i</mi><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">)</mo><mi mathvariant="normal">个</mi><mi mathvariant="normal">评</mi><mi mathvariant="normal">价</mi><mi mathvariant="normal">对</mi><mi mathvariant="normal">象</mi><mi mathvariant="normal">未</mi><mi mathvariant="normal">归</mi><mi mathvariant="normal">一</mi><mi mathvariant="normal">化</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">得</mi><mi mathvariant="normal">分</mi><mi mathvariant="normal">：</mi><msub><mi>S</mi><mi>i</mi></msub><mo>=</mo><mfrac><msubsup><mi>D</mi><mi>i</mi><mo>−</mo></msubsup><mrow><msubsup><mi>D</mi><mi>i</mi><mo>+</mo></msubsup><mo>+</mo><msubsup><mi>D</mi><mi>i</mi><mo>−</mo></msubsup></mrow></mfrac><mspace linebreak="newline"></mspace><mi mathvariant="normal">很</mi><mi mathvariant="normal">明</mi><mi mathvariant="normal">显</mi><mn>0</mn><mo>≤</mo><msub><mi>S</mi><mi>i</mi></msub><mo>≤</mo><mn>1</mn><mo separator="true">,</mo><mi mathvariant="normal">且</mi><msub><mi>S</mi><mi>i</mi></msub><mi mathvariant="normal">越</mi><mi mathvariant="normal">大</mi><msubsup><mi>D</mi><mi>i</mi><mo>+</mo></msubsup><mi mathvariant="normal">越</mi><mi mathvariant="normal">小</mi><mi mathvariant="normal">，</mi><mi mathvariant="normal">即</mi><mi mathvariant="normal">越</mi><mi mathvariant="normal">接</mi><mi mathvariant="normal">近</mi><mi mathvariant="normal">最</mi><mi mathvariant="normal">大</mi><mi mathvariant="normal">值</mi><mi mathvariant="normal">。</mi><mspace linebreak="newline"></mspace><mi mathvariant="normal">我</mi><mi mathvariant="normal">们</mi><mi mathvariant="normal">可</mi><mi mathvariant="normal">以</mi><mi mathvariant="normal">将</mi><mi mathvariant="normal">得</mi><mi mathvariant="normal">分</mi><mi mathvariant="normal">归</mi><mi mathvariant="normal">一</mi><mi mathvariant="normal">化</mi><mo>:</mo><mover accent="true"><msub><mi>S</mi><mi>i</mi></msub><mo>~</mo></mover><mo>=</mo><mfrac><msub><mi>S</mi><mi>i</mi></msub><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>S</mi><mi>i</mi></msub></mrow></mfrac><mo separator="true">,</mo><mi mathvariant="normal">这</mi><mi mathvariant="normal">样</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">话</mi><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mover accent="true"><msub><mi>S</mi><mi>i</mi></msub><mo>~</mo></mover><mo>=</mo><mn>1</mn><mspace linebreak="newline"></mspace><mi mathvariant="normal">如</mi><mi mathvariant="normal">果</mi><mi mathvariant="normal">涉</mi><mi mathvariant="normal">及</mi><mi mathvariant="normal">到</mi><mi mathvariant="normal">各</mi><mi mathvariant="normal">个</mi><mi mathvariant="normal">指</mi><mi mathvariant="normal">标</mi><mi mathvariant="normal">之</mi><mi mathvariant="normal">间</mi><mi mathvariant="normal">有</mi><mi mathvariant="normal">不</mi><mi mathvariant="normal">同</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">权</mi><mi mathvariant="normal">重</mi><mi mathvariant="normal">，</mi><mi mathvariant="normal">则</mi><mi mathvariant="normal">在</mi><mi mathvariant="normal">定</mi><mi mathvariant="normal">义</mi><msubsup><mi>D</mi><mi>i</mi><mrow><mo>+</mo><mo stretchy="false">(</mo><mo>−</mo><mo stretchy="false">)</mo></mrow></msubsup><mi mathvariant="normal">时</mi><mi mathvariant="normal">加</mi><mi mathvariant="normal">入</mi><mi mathvariant="normal">权</mi><mi mathvariant="normal">重</mi><mi mathvariant="normal">指</mi><mi mathvariant="normal">标</mi><msub><mi>ω</mi><mi>i</mi></msub><mo>:</mo><mspace linebreak="newline"></mspace><msubsup><mi>D</mi><mi>i</mi><mrow><mo>+</mo><mo stretchy="false">(</mo><mo>−</mo><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><msqrt><mrow><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><msub><mi>ω</mi><mi>i</mi></msub><mo stretchy="false">(</mo><msubsup><mi>Z</mi><mi>j</mi><mo>+</mo></msubsup><mo>−</mo><msub><mi>z</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></msqrt></mrow><annotation encoding="application/x-tex">定义第i(i=1,2,\cdots,n)个评价对象与对象最大值的距离D_i^+ = \sqrt{\sum^m_{j=1}(Z^+_j-z_{ij})^2}\\定义第i(i=1,2,\cdots,n)个评价对象与对象最小值的距离D_i^- = \sqrt{\sum^m_{j=1}(Z^-_j-z_{ij})^2}\\那么，我们可以计算得出第i(i=1,2,\cdots,n)个评价对象未归一化的得分：S_i=\frac{D_i^-}{D_i^++D_i^-}\\很明显0\leq S_i \leq 1,且S_i越大D_i^+越小，即越接近最大值。\\我们可以将得分归一化:\tilde{S_i}=\frac{S_i}{\sum_{i=1}^n{S_i}}, 这样的话\sum_{i=1}^n\tilde{S_i}=1\\如果涉及到各个指标之间有不同的权重，则在定义D_i^{+(-)}时加入权重指标\omega_i:\\D_i^{+(-)}=\sqrt{\sum^m_{j=1}\omega_i(Z^+_j-z_{ij})^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord cjk_fallback">定</span><span class="mord cjk_fallback">义</span><span class="mord cjk_fallback">第</span><span class="mord mathdefault">i</span><span class="mopen">(</span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.088326em;vertical-align:-0.266995em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">n</span><span class="mclose">)</span><span class="mord cjk_fallback">个</span><span class="mord cjk_fallback">评</span><span class="mord cjk_fallback">价</span><span class="mord cjk_fallback">对</span><span class="mord cjk_fallback">象</span><span class="mord cjk_fallback">与</span><span class="mord cjk_fallback">对</span><span class="mord cjk_fallback">象</span><span class="mord cjk_fallback">最</span><span class="mord cjk_fallback">大</span><span class="mord cjk_fallback">值</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">距</span><span class="mord cjk_fallback">离</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8213309999999999em;"><span style="top:-2.433005em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.266995em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.2929240000000006em;vertical-align:-1.4137769999999998em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8791470000000008em;"><span class="svg-align" style="top:-5.252924em;"><span class="pstrut" style="height:5.252924em;"></span><span class="mord" style="padding-left:1.056em;"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6513970000000007em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.4137769999999998em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.811462em;"><span style="top:-2.4231360000000004em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1031310000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.412972em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.04398em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.8391470000000005em;"><span class="pstrut" style="height:5.252924em;"></span><span class="hide-tail" style="min-width:0.742em;height:3.3329240000000007em;"><svg width='400em' height='3.3329240000000007em' viewBox='0 0 400000 3332' preserveAspectRatio='xMinYMin slice'><path d='M702 80H400000v40H742v3198l-4 4-4 4c-.667.7-2 1.5-4 2.5s-4.167 1.833-6.5 2.5-5.5 1-9.5 1h-12l-28-84c-16.667-52-96.667-294.333-240-727l-212 -643 -85 170c-4-3.333-8.333-7.667-13 -13l-13-13l77-155 77-156c66 199.333 139 419.667 219 661 l218 661zM702 80H400000v40H742z'/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.4137769999999998em;"><span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord cjk_fallback">定</span><span class="mord cjk_fallback">义</span><span class="mord cjk_fallback">第</span><span class="mord mathdefault">i</span><span class="mopen">(</span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.088326em;vertical-align:-0.266995em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">n</span><span class="mclose">)</span><span class="mord cjk_fallback">个</span><span class="mord cjk_fallback">评</span><span class="mord cjk_fallback">价</span><span class="mord cjk_fallback">对</span><span class="mord cjk_fallback">象</span><span class="mord cjk_fallback">与</span><span class="mord cjk_fallback">对</span><span class="mord cjk_fallback">象</span><span class="mord cjk_fallback">最</span><span class="mord cjk_fallback">小</span><span class="mord cjk_fallback">值</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">距</span><span class="mord cjk_fallback">离</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8213309999999999em;"><span style="top:-2.433005em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">−</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.266995em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.2929240000000006em;vertical-align:-1.4137769999999998em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8791470000000008em;"><span class="svg-align" style="top:-5.252924em;"><span class="pstrut" style="height:5.252924em;"></span><span class="mord" style="padding-left:1.056em;"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6513970000000007em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.4137769999999998em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.811462em;"><span style="top:-2.4231360000000004em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1031310000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">−</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.412972em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.04398em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.8391470000000005em;"><span class="pstrut" style="height:5.252924em;"></span><span class="hide-tail" style="min-width:0.742em;height:3.3329240000000007em;"><svg width='400em' height='3.3329240000000007em' viewBox='0 0 400000 3332' preserveAspectRatio='xMinYMin slice'><path d='M702 80H400000v40H742v3198l-4 4-4 4c-.667.7-2 1.5-4 2.5s-4.167 1.833-6.5 2.5-5.5 1-9.5 1h-12l-28-84c-16.667-52-96.667-294.333-240-727l-212 -643 -85 170c-4-3.333-8.333-7.667-13 -13l-13-13l77-155 77-156c66 199.333 139 419.667 219 661 l218 661zM702 80H400000v40H742z'/></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.4137769999999998em;"><span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord cjk_fallback">那</span><span class="mord cjk_fallback">么</span><span class="mord cjk_fallback">，</span><span class="mord cjk_fallback">我</span><span class="mord cjk_fallback">们</span><span class="mord cjk_fallback">可</span><span class="mord cjk_fallback">以</span><span class="mord cjk_fallback">计</span><span class="mord cjk_fallback">算</span><span class="mord cjk_fallback">得</span><span class="mord cjk_fallback">出</span><span class="mord cjk_fallback">第</span><span class="mord mathdefault">i</span><span class="mopen">(</span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">n</span><span class="mclose">)</span><span class="mord cjk_fallback">个</span><span class="mord cjk_fallback">评</span><span class="mord cjk_fallback">价</span><span class="mord cjk_fallback">对</span><span class="mord cjk_fallback">象</span><span class="mord cjk_fallback">未</span><span class="mord cjk_fallback">归</span><span class="mord cjk_fallback">一</span><span class="mord cjk_fallback">化</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">得</span><span class="mord cjk_fallback">分</span><span class="mord cjk_fallback">：</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.466788em;vertical-align:-0.978326em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.488462em;"><span style="top:-2.2985379999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.811462em;"><span style="top:-2.4231360000000004em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1031310000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.811462em;"><span style="top:-2.4231360000000004em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1031310000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">−</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.6769999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.811462em;"><span style="top:-2.4231360000000004em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.1031310000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">−</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.978326em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.78041em;vertical-align:-0.13597em;"></span><span class="mord cjk_fallback">很</span><span class="mord cjk_fallback">明</span><span class="mord cjk_fallback">显</span><span class="mord">0</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.088326em;vertical-align:-0.266995em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord cjk_fallback">且</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord cjk_fallback">越</span><span class="mord cjk_fallback">大</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8213309999999999em;"><span style="top:-2.433005em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.266995em;"><span></span></span></span></span></span></span><span class="mord cjk_fallback">越</span><span class="mord cjk_fallback">小</span><span class="mord cjk_fallback">，</span><span class="mord cjk_fallback">即</span><span class="mord cjk_fallback">越</span><span class="mord cjk_fallback">接</span><span class="mord cjk_fallback">近</span><span class="mord cjk_fallback">最</span><span class="mord cjk_fallback">大</span><span class="mord cjk_fallback">值</span><span class="mord cjk_fallback">。</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord cjk_fallback">我</span><span class="mord cjk_fallback">们</span><span class="mord cjk_fallback">可</span><span class="mord cjk_fallback">以</span><span class="mord cjk_fallback">将</span><span class="mord cjk_fallback">得</span><span class="mord cjk_fallback">分</span><span class="mord cjk_fallback">归</span><span class="mord cjk_fallback">一</span><span class="mord cjk_fallback">化</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0701899999999998em;vertical-align:-0.15em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9201899999999998em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.6023300000000003em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;">~</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.929066em;vertical-align:-1.277669em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em;"><span style="top:-2.305708em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.804292em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.994002em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord cjk_fallback">这</span><span class="mord cjk_fallback">样</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">话</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6513970000000002em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord accent"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9201899999999998em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.6023300000000003em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;">~</span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.276864em;"></span><span class="mord cjk_fallback">如</span><span class="mord cjk_fallback">果</span><span class="mord cjk_fallback">涉</span><span class="mord cjk_fallback">及</span><span class="mord cjk_fallback">到</span><span class="mord cjk_fallback">各</span><span class="mord cjk_fallback">个</span><span class="mord cjk_fallback">指</span><span class="mord cjk_fallback">标</span><span class="mord cjk_fallback">之</span><span class="mord cjk_fallback">间</span><span class="mord cjk_fallback">有</span><span class="mord cjk_fallback">不</span><span class="mord cjk_fallback">同</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">权</span><span class="mord cjk_fallback">重</span><span class="mord cjk_fallback">，</span><span class="mord cjk_fallback">则</span><span class="mord cjk_fallback">在</span><span class="mord cjk_fallback">定</span><span class="mord cjk_fallback">义</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em;"><span style="top:-2.4231360000000004em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mopen mtight">(</span><span class="mord mtight">−</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mord cjk_fallback">时</span><span class="mord cjk_fallback">加</span><span class="mord cjk_fallback">入</span><span class="mord cjk_fallback">权</span><span class="mord cjk_fallback">重</span><span class="mord cjk_fallback">指</span><span class="mord cjk_fallback">标</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.321664em;vertical-align:-0.276864em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em;"><span style="top:-2.4231360000000004em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">+</span><span class="mopen mtight">(</span><span class="mord mtight">−</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.2929240000000006em;vertical-align:-1.4137769999999998em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8791470000000008em;"><span class="svg-align" style="top:-5.252924em;"><span class="pstrut" style="height:5.252924em;"></span><span class="mord" style="padding-left:1.056em;"><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6513970000000007em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.4137769999999998em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.811462em;"><span style="top:-2.4231360000000004em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span><span style="top:-3.1031310000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.412972em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.04398em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.740108em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.8391470000000005em;"><span class="pstrut" style="height:5.252924em;"></span><span class="hide-tail" style="min-width:0.742em;height:3.3329240000000007em;"><svg width='400em' height='3.3329240000000007em' viewBox='0 0 400000 3332' preserveAspectRatio='xMinYMin slice'><path d='M702 80H400000v40H742v3198l-4 4-4 4c-.667.7-2 1.5-4 2.5s-4.167 1.833-6.5 2.5-5.5 1-9.5 1h-12l-28-84c-16.667-52-96.667-294.333-240-727l-212 -643 -85 170c-4-3.333-8.333-7.667-13 -13l-13-13l77-155 77-156c66 199.333 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<categories>
<category> 数学建模 </category>
</categories>
<tags>
<tag> 数学建模 </tag>
<tag> MATLAB </tag>
<tag> 优劣势解距离法 </tag>
</tags>
</entry>
<entry>
<title>层次分析法</title>
<link href="/2022/09/05/%E5%B1%82%E6%AC%A1%E5%88%86%E6%9E%90%E6%B3%95/"/>
<url>/2022/09/05/%E5%B1%82%E6%AC%A1%E5%88%86%E6%9E%90%E6%B3%95/</url>
<content type="html"><![CDATA[<style>#web_bg{ background: url(https://img-baofun.zhhainiao.com/pcwallpaper_ugc/static/0f30dee4161712dce86bcc9affba7f70.jpg?x-oss-process=image%2fresize%2cm_lfit%2cw_1920%2ch_1080)!important; /*重新定义background会导致原有定位属性失效，所以也需要再声明一次加权的定位属性*/ background-position: center !important; background-size: cover !important; background-repeat: no-repeat !important;}</style><h1 id="层次分析法"><a class="markdownIt-Anchor" href="#层次分析法"></a> 层次分析法</h1><h2 id="1判断矩阵"><a class="markdownIt-Anchor" href="#1判断矩阵"></a> 1.判断矩阵</h2><p>判断矩阵就是一个方阵，记为A，对应的元素为a<sub>ij</sub></p><p>(1)a<sub>ij</sub>表示的意义是，与指标j相比，指标i的重要程度</p><p>(2)当i=j时，两个指标相同，因此同等重要，这就解释了为什么主对角线元素为1</p><p>(3)a<sub>ij</sub>>0且满足a<sub>ij</sub> <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>×</mo></mrow><annotation encoding="application/x-tex">\times</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord">×</span></span></span></span>a<sub>ji</sub>=1 (我们称满足这一条件的矩阵为<em>正互反</em>矩阵)</p><p>实际上，这个矩阵即为层次分析法中的判断矩阵。</p><h2 id="2一致矩阵"><a class="markdownIt-Anchor" href="#2一致矩阵"></a> 2.一致矩阵</h2><p>首先我们引入标度，即相比而言，某要素的重要程度。在矩阵中，体现在各个矩阵元素之中。例如：</p><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>i</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">重</mi><mi mathvariant="normal">要</mi><mi mathvariant="normal">程</mi><mi mathvariant="normal">度</mi></mrow><mrow><mi>j</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">重</mi><mi mathvariant="normal">要</mi><mi mathvariant="normal">程</mi><mi mathvariant="normal">度</mi></mrow></mfrac><mi mathvariant="normal">，</mi><msub><mi>a</mi><mrow><mi>j</mi><mi>k</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>j</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">重</mi><mi mathvariant="normal">要</mi><mi mathvariant="normal">程</mi><mi mathvariant="normal">度</mi></mrow><mrow><mi>k</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">重</mi><mi mathvariant="normal">要</mi><mi mathvariant="normal">程</mi><mi mathvariant="normal">度</mi></mrow></mfrac><mi mathvariant="normal">，</mi><msub><mi>a</mi><mrow><mi>i</mi><mi>k</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>i</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">重</mi><mi mathvariant="normal">要</mi><mi mathvariant="normal">程</mi><mi mathvariant="normal">度</mi></mrow><mrow><mi>k</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">重</mi><mi mathvariant="normal">要</mi><mi mathvariant="normal">程</mi><mi mathvariant="normal">度</mi></mrow></mfrac><mo>=</mo><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>×</mo><msub><mi>a</mi><mrow><mi>j</mi><mi>k</mi></mrow></msub></mrow><annotation encoding="application/x-tex">a_{ij}=\frac{i的重要程度}{j的重要程度}，a_{jk}=\frac{j的重要程度}{k的重要程度}，a_{ik}=\frac{i的重要程度}{k的重要程度}=a_{ij}\times a_{jk}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.2169600000000003em;vertical-align:-0.8804400000000001em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3365200000000002em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05724em;">j</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">重</span><span class="mord cjk_fallback">要</span><span class="mord cjk_fallback">程</span><span class="mord cjk_fallback">度</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">i</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">重</span><span class="mord cjk_fallback">要</span><span class="mord cjk_fallback">程</span><span class="mord cjk_fallback">度</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8804400000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord cjk_fallback">，</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.02252em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3365200000000002em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">重</span><span class="mord cjk_fallback">要</span><span class="mord cjk_fallback">程</span><span class="mord cjk_fallback">度</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05724em;">j</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">重</span><span class="mord cjk_fallback">要</span><span class="mord cjk_fallback">程</span><span class="mord cjk_fallback">度</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord cjk_fallback">，</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.02252em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3365200000000002em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">重</span><span class="mord cjk_fallback">要</span><span class="mord cjk_fallback">程</span><span class="mord cjk_fallback">度</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">i</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">重</span><span class="mord cjk_fallback">要</span><span class="mord cjk_fallback">程</span><span class="mord cjk_fallback">度</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8694379999999999em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></p><p>互反的因素之间其标度互为倒数。同时，一致矩阵的各行各列之间成倍数关系。</p><p><strong>定义：若矩阵中的每一个元素a<sub>ij</sub>>0且满足a<sub>ij</sub><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>×</mo></mrow><annotation encoding="application/x-tex">\times</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord">×</span></span></span></span>a<sub>ji</sub>=1，则我们称该矩阵为正互反矩阵。在层次分析法中，我们构造的判断矩阵均是正互反矩阵。</strong></p><p><strong>若正互反矩阵满足a<sub>ij</sub><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>×</mo></mrow><annotation encoding="application/x-tex">\times</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord">×</span></span></span></span>a<sub>jk</sub>=a<sub>ik</sub>，则我们称其为一致矩阵。</strong></p><p><strong>注意：在使用判断矩阵求权重前，必须对其进行一致性检验。</strong></p><p><em><strong>引理：n阶正互反矩阵A为一致矩阵时，当且仅当最大特征值<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>λ</mi><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>=</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">\lambda_{max}=n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">x</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">n</span></span></span></span>.且当正互反矩阵A非一致时，一定满足<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>λ</mi><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>></mo><mi>n</mi></mrow><annotation encoding="application/x-tex">\lambda_{max}>n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">x</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">n</span></span></span></span>.判断矩阵越不一致，最大特征值与n相差就越大。</strong></em></p><h2 id="3一致性检验的步骤"><a class="markdownIt-Anchor" href="#3一致性检验的步骤"></a> 3.一致性检验的步骤</h2><p><strong>第一步：计算一致性指标CI</strong></p><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>C</mi><mi>I</mi><mo>=</mo><mfrac><mrow><msub><mi>λ</mi><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>−</mo><mi>n</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow><annotation encoding="application/x-tex">CI=\frac{\lambda_{max}-n}{n-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="mord mathdefault" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.14077em;vertical-align:-0.7693300000000001em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">1</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">x</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.7693300000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><p><strong>第二步：查找对应的平均随机一致性指标RI</strong></p><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><menclose notation="top bottom left right"><mtable rowspacing="0.15999999999999992em" columnalign="center center center" columnlines="solid solid" columnspacing="1em" rowlines="solid"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>n</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>3</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>4</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>5</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>6</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>7</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>8</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>9</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>10</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>11</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>12</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>13</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>14</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>15</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mi>R</mi><mi>I</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0.52</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0.89</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1.12</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1.26</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1.36</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1.41</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1.46</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1.49</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1.52</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1.54</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1.56</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1.58</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1.59</mn></mstyle></mtd></mtr></mtable></menclose></mrow><annotation encoding="application/x-tex">\begin{array}{|c|c|c|}\hline n&1&2&3&4&5&6&7&8&9&10&11&12&13&14&15\\\hline RI&0&0&0.52&0.89&1.12&1.26&1.36&1.41&1.46&1.49&1.52&1.54&1.56&1.58&1.59\\\hline\end{array}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.45em;vertical-align:-0.9500000000000004em;"></span><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5em;"><span style="top:-3.45em;"><span class="pstrut" style="height:3.45em;"></span><span class="mtable"><span class="vertical-separator" style="height:2.4em;vertical-align:-0.95em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="mord mathdefault" style="margin-right:0.07847em;">I</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9500000000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="vertical-separator" style="height:2.4em;vertical-align:-0.95em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9500000000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="vertical-separator" style="height:2.4em;vertical-align:-0.95em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9500000000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="vertical-separator" style="height:2.4em;vertical-align:-0.95em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span><span class="mord">.</span><span class="mord">5</span><span class="mord">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9500000000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span><span class="mord">.</span><span class="mord">8</span><span class="mord">9</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9500000000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">5</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">.</span><span class="mord">1</span><span class="mord">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9500000000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">6</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">.</span><span class="mord">2</span><span class="mord">6</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9500000000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">7</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">.</span><span class="mord">3</span><span class="mord">6</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9500000000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">8</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">.</span><span class="mord">4</span><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9500000000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">9</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">.</span><span class="mord">4</span><span class="mord">6</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9500000000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">0</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">.</span><span class="mord">4</span><span class="mord">9</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9500000000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">1</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">.</span><span class="mord">5</span><span class="mord">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9500000000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">2</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">.</span><span class="mord">5</span><span class="mord">4</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9500000000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">3</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">.</span><span class="mord">5</span><span class="mord">6</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9500000000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">4</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">.</span><span class="mord">5</span><span class="mord">8</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9500000000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">5</span></span></span><span style="top:-2.4099999999999997em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mord">.</span><span class="mord">5</span><span class="mord">9</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9500000000000004em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span></span></span><span style="top:-2.5em;"><span class="pstrut" style="height:3.45em;"></span><span class="hline" style="border-bottom-width:0.05em;"></span></span><span style="top:-3.7em;"><span class="pstrut" style="height:3.45em;"></span><span class="hline" style="border-bottom-width:0.05em;"></span></span><span style="top:-4.9em;"><span class="pstrut" style="height:3.45em;"></span><span class="hline" style="border-bottom-width:0.05em;"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9500000000000004em;"><span></span></span></span></span></span></span></span></span></span></p><p><strong>第三步：计算一致性比例CR</strong></p><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>C</mi><mi>R</mi><mo>=</mo><mfrac><mrow><mi>C</mi><mi>I</mi></mrow><mrow><mi>R</mi><mi>I</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">CR=\frac{CI}{RI}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.04633em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="mord mathdefault" style="margin-right:0.07847em;">I</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">C</span><span class="mord mathdefault" style="margin-right:0.07847em;">I</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p><p><strong>如果CR<0.1，则可以认为判断矩阵的一致性可以接受；否则需要对判断矩阵进行修正。</strong></p><p><strong>修正方法：</strong></p><p><strong>将一致矩阵上调整，一致矩阵各行呈倍数关系。</strong></p><h2 id="4一致矩阵计算权重的几种方法"><a class="markdownIt-Anchor" href="#4一致矩阵计算权重的几种方法"></a> 4.一致矩阵计算权重的几种方法</h2><h3 id="方法一算术平均法求权重"><a class="markdownIt-Anchor" href="#方法一算术平均法求权重"></a> 方法一：算术平均法求权重</h3><p><strong>第一步：将判断矩阵按照列归一化（每一个元素除以其所在列元素的总和）</strong></p><p><strong>第二步：将归一化的各列相加（按行求和）</strong></p><p><strong>第三步：将相加后得到的向量中，每个元素除以n即可得到权重向量</strong></p><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">假</mi><mi mathvariant="normal">设</mi><mi mathvariant="normal">判</mi><mi mathvariant="normal">断</mi><mi mathvariant="normal">矩</mi><mi mathvariant="normal">阵</mi><mi>A</mi><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mn>11</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mn>12</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo>⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mrow><mn>1</mn><mi>n</mi></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mn>21</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mn>22</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo>⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mrow><mn>2</mn><mi>n</mi></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mrow><mi>n</mi><mn>1</mn></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mrow><mi>n</mi><mn>2</mn></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo>⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mrow><mi>n</mi><mi>n</mi></mrow></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo separator="true">,</mo><mspace linebreak="newline"></mspace><mi mathvariant="normal">那</mi><mi mathvariant="normal">么</mi><mi mathvariant="normal">算</mi><mi mathvariant="normal">术</mi><mi mathvariant="normal">平</mi><mi mathvariant="normal">均</mi><mi mathvariant="normal">法</mi><mi mathvariant="normal">求</mi><mi mathvariant="normal">得</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">权</mi><mi mathvariant="normal">重</mi><mi mathvariant="normal">向</mi><mi mathvariant="normal">量</mi><msub><mi>ω</mi><mi>i</mi></msub><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mfrac><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mrow><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>a</mi><mrow><mi>k</mi><mi>j</mi></mrow></msub></mrow></mfrac><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">假设判断矩阵A=\left[ \begin{matrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{matrix} \right],\\ 那么算术平均法求得的权重向量\omega_i=\frac{1}{n}\sum^n_{j=1}\frac{a_{ij}}{\sum^n_{k=1}a_{kj}} (i=1,2,\cdots,n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord cjk_fallback">假</span><span class="mord cjk_fallback">设</span><span class="mord cjk_fallback">判</span><span class="mord cjk_fallback">断</span><span class="mord cjk_fallback">矩</span><span class="mord cjk_fallback">阵</span><span class="mord mathdefault">A</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:5.459999999999999em;vertical-align:-2.4799999999999995em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.953005em;"><span style="top:-1.3499850000000007em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.5049850000000005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.1059850000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.7069850000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.953005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4500349999999997em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9799999999999995em;"><span style="top:-5.8275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.6275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.7674999999999996em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-1.5675000000000006em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4799999999999995em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9799999999999995em;"><span style="top:-5.8275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.6275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.7674999999999996em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-1.5675000000000006em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4799999999999995em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9799999999999995em;"><span style="top:-5.64em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-4.44em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-2.5799999999999996em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"></span></span><span style="top:-1.3800000000000006em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4799999999999995em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9799999999999995em;"><span style="top:-5.8275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.6275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.7674999999999996em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-1.5675000000000006em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4799999999999995em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.953005em;"><span style="top:-1.3499850000000007em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.5049850000000005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.1059850000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.7069850000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.953005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4500349999999997em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord cjk_fallback">那</span><span class="mord cjk_fallback">么</span><span class="mord cjk_fallback">算</span><span class="mord cjk_fallback">术</span><span class="mord cjk_fallback">平</span><span class="mord cjk_fallback">均</span><span class="mord cjk_fallback">法</span><span class="mord cjk_fallback">求</span><span class="mord cjk_fallback">得</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">权</span><span class="mord cjk_fallback">重</span><span class="mord cjk_fallback">向</span><span class="mord cjk_fallback">量</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.0651740000000007em;vertical-align:-1.4137769999999998em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6513970000000007em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.4137769999999998em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1075599999999999em;"><span style="top:-2.305708em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.804292em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.994002em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">n</span><span class="mclose">)</span></span></span></span></span></p><h3 id="方法二几何平均法求权重"><a class="markdownIt-Anchor" href="#方法二几何平均法求权重"></a> 方法二：几何平均法求权重</h3><p>实质上是将算术平均的方法用几何平均替代。把 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mo>∑</mo><mi>n</mi></mfrac></mrow><annotation encoding="application/x-tex">\frac{\sum}{n}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.3550069999999999em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0100069999999999em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.485007em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mop op-symbol small-op mtight" style="position:relative;top:-0.0000050000000000050004em;">∑</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 替换为 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mo>∏</mo><mfrac><mn>1</mn><mi>n</mi></mfrac></msup></mrow><annotation encoding="application/x-tex">\prod^{\frac{1}{n}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.34393em;vertical-align:-0.25001em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∏</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.09392em;"><span style="top:-3.5029000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8443142857142858em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span style="top:-3.2255000000000003em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span></span></span></span> ,其最终结果在有效层面与算术平均法相同。</p><h3 id="方法三特征值法求权重一般采用"><a class="markdownIt-Anchor" href="#方法三特征值法求权重一般采用"></a> 方法三：特征值法求权重（一般采用）</h3><p><strong>第一步：求出矩阵A的最大特征值以及其对应的特征向量</strong></p><p><strong>第二步：对求出的特征向量进行归一化即可得到权重</strong></p><h2 id="局限性"><a class="markdownIt-Anchor" href="#局限性"></a> 局限性</h2><p><strong>1.评价决策层不能太多，若太多，判断矩阵和一致矩阵的差异可能会很大</strong></p><p><strong>2.数据若已知，用层次分析法会显得有失偏颇</strong></p><h2 id="评价类问题"><a class="markdownIt-Anchor" href="#评价类问题"></a> 评价类问题</h2><p><strong>1.评价的目标是什么？</strong></p><p><strong>2.为了达到目标有哪几种可选择的方案？</strong></p><p><strong>3.评价的准则或者指标是什么？</strong></p>]]></content>
<categories>
<category> 数学建模 </category>
</categories>
<tags>
<tag> 数学建模 </tag>
<tag> MATLAB </tag>
<tag> 层次分析法 </tag>
</tags>
</entry>
<entry>
<title>MATLAB基础操作</title>
<link href="/2022/09/05/matlab%E5%9F%BA%E7%A1%80/"/>
<url>/2022/09/05/matlab%E5%9F%BA%E7%A1%80/</url>
<content type="html"><![CDATA[<style>#web_bg{ background: url(https://img-baofun.zhhainiao.com/pcwallpaper_ugc/static/358108182dc4d59d83a22c63aba0a038.jpg?x-oss-process=image%2fresize%2cm_lfit%2cw_1920%2ch_1080)!important; /*重新定义background会导致原有定位属性失效，所以也需要再声明一次加权的定位属性*/ background-position: center !important; background-size: cover !important; background-repeat: no-repeat !important;}</style><h1 id="matlab基础操作"><a class="markdownIt-Anchor" href="#matlab基础操作"></a> MATLAB基础操作</h1><h2 id="杂项"><a class="markdownIt-Anchor" href="#杂项"></a> 杂项</h2><p>1.每一行的语句后面加上分号，表示不显示运行结果</p><figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><span class="line">a=<span class="number">3</span></span><br><span class="line">a=<span class="number">3</span>;</span><br></pre></td></tr></table></figure><p>2.注释：多行注释：选中若干行，按住Ctrl+R添加注释，按Ctrl+T取消注释</p><p>3.clear可以清楚工作区所有变量</p><figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">clear</span><br></pre></td></tr></table></figure><p>4.clc可以清楚命令行窗口中的所有文本，让屏幕变得更干净</p><figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line">clc</span><br></pre></td></tr></table></figure><h2 id="输入和输出函数"><a class="markdownIt-Anchor" href="#输入和输出函数"></a> 输入和输出函数</h2><h3 id="输出函数-disp函数类似于printf"><a class="markdownIt-Anchor" href="#输出函数-disp函数类似于printf"></a> 输出函数 disp函数（类似于printf）</h3><p>Matlab中字符串用单引号（或者双引号）括起来</p><figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br></pre></td><td class="code"><pre><span class="line"><span class="built_in">disp</span>(<span class="string">'Hello Matlab'</span>)</span><br></pre></td></tr></table></figure><p>向量：</p><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></pre></td><td class="code"><pre><span class="line">a = [<span class="number">1</span> <span class="number">2</span> <span class="number">3</span>]<span class="comment">%同一行中间用逗号分隔，也可以不用逗号，直接用空格</span></span><br><span class="line">a = [<span class="number">1</span>,<span class="number">2</span>,<span class="number">3</span>]</span><br><span class="line">a = [<span class="number">1</span>;<span class="number">2</span>;<span class="number">3</span>]<span class="comment">%分号可以用来分隔每一行的元素</span></span><br></pre></td></tr></table></figure><p>2.字符串合并的函数</p><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></pre></td><td class="code"><pre><span class="line"><span class="comment">%方法一：</span></span><br><span class="line">strcat(<span class="string">'字符串1'</span>,<span class="string">'字符串2'</span>,...)</span><br><span class="line"><span class="comment">%方法二：</span></span><br><span class="line">[str1,str2,str3...]</span><br><span class="line">[str1 str2 str3]</span><br></pre></td></tr></table></figure><p>3.将数字转换为字符串</p><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></pre></td><td class="code"><pre><span class="line">c=<span class="number">100</span></span><br><span class="line">num2str(c)</span><br><span class="line"><span class="built_in">disp</span>([<span class="string">'c的取值为'</span>num2str(c)])</span><br><span class="line"><span class="built_in">disp</span>(strcat(<span class="string">'c的取值为'</span>,num2str(c)))</span><br></pre></td></tr></table></figure><h3 id="输入函数-input函数"><a class="markdownIt-Anchor" href="#输入函数-input函数"></a> 输入函数 input函数</h3><figure class="highlight matlab"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><span class="line">A = input(<span class="string">'请输入A：'</span>)</span><br><span class="line">B = input(<span class="string">'请输入B：'</span>)</span><br></pre></td></tr></table></figure><h3 id="sum函数"><a class="markdownIt-Anchor" href="#sum函数"></a> sum函数</h3><p>1.如果求和对象是向量：各分量之和</p><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></pre></td><td class="code"><pre><span class="line">E = [<span class="number">1</span>,<span class="number">2</span>,<span class="number">3</span>]</span><br><span class="line">sum(E)</span><br><span class="line">E = [<span class="number">1</span>;<span class="number">2</span>;<span class="number">3</span>]</span><br><span class="line">sum(E)</span><br><span class="line"><span class="comment">%最终结果：E=6</span></span><br></pre></td></tr></table></figure><p>2.如果是矩阵，则需要根据行和列的方向作区分</p><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></pre></td><td class="code"><pre><span class="line">E = [<span class="number">1</span>,<span class="number">2</span>;<span class="number">3</span>,<span class="number">4</span>;<span class="number">5</span>,<span class="number">6</span>]</span><br><span class="line">a = sum(E)</span><br><span class="line"><span class="comment">%a = </span></span><br><span class="line"><span class="number">9</span> <span class="number">12</span></span><br><span class="line"><span class="comment">%直接对矩阵求和：对列求和</span></span><br><span class="line">a = sum(E,<span class="number">1</span>) <span class="comment">%参数1表示按列求和</span></span><br><span class="line">a = sum(E,<span class="number">2</span>) <span class="comment">%参数2表示按行求和</span></span><br><span class="line"></span><br><span class="line">sum(E,dim) <span class="comment">%dim表示维数，默认为1</span></span><br><span class="line"></span><br><span class="line">a = sum(sum(E)) <span class="comment">%得到矩阵各个元素之和</span></span><br><span class="line"></span><br><span class="line">E(:) <span class="comment">%将E用列向量拼接</span></span><br><span class="line">sum(E(:))</span><br></pre></td></tr></table></figure><h3 id="matlab提取指定位置的元素"><a class="markdownIt-Anchor" href="#matlab提取指定位置的元素"></a> matlab提取指定位置的元素</h3><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></pre></td><td class="code"><pre><span class="line"><span class="comment">%% 基础：matlab中如何提取矩阵中指定位置的元素？</span></span><br><span class="line"><span class="comment">% （1）取指定行和列的一个元素（输出的是一个值）</span></span><br><span class="line">clc;A=[<span class="number">1</span> <span class="number">1</span> <span class="number">4</span> <span class="number">1</span>/<span class="number">3</span> <span class="number">3</span>;<span class="number">1</span> <span class="number">1</span> <span class="number">4</span> <span class="number">1</span>/<span class="number">3</span> <span class="number">3</span>;<span class="number">1</span>/<span class="number">4</span> <span class="number">1</span>/<span class="number">4</span> <span class="number">1</span> <span class="number">1</span>/<span class="number">3</span> <span class="number">1</span>/<span class="number">2</span>;<span class="number">3</span> <span class="number">3</span> <span class="number">3</span> <span class="number">1</span> <span class="number">3</span>;<span class="number">1</span>/<span class="number">3</span> <span class="number">1</span>/<span class="number">3</span> <span class="number">2</span> <span class="number">1</span>/<span class="number">3</span> <span class="number">1</span>];</span><br><span class="line">A</span><br><span class="line">A(<span class="number">2</span>,<span class="number">1</span>)</span><br><span class="line">A(<span class="number">3</span>,<span class="number">2</span>)</span><br><span class="line"><span class="comment">% （2）取指定的某一行的全部元素（输出的是一个行向量）</span></span><br><span class="line">clc;A</span><br><span class="line">A(<span class="number">2</span>,:)</span><br><span class="line">A(<span class="number">5</span>,:)</span><br><span class="line"><span class="comment">% （3）取指定的某一列的全部元素（输出的是一个列向量）</span></span><br><span class="line">clc;A</span><br><span class="line">A(:,<span class="number">1</span>)</span><br><span class="line">A(:,<span class="number">3</span>)</span><br><span class="line"><span class="comment">% （4）取指定的某些行的全部元素（输出的是一个矩阵）</span></span><br><span class="line">clc;A</span><br><span class="line">A([<span class="number">2</span>,<span class="number">5</span>],:) <span class="comment">% 只取第二行和第五行（一共2行）</span></span><br><span class="line">A(<span class="number">2</span>:<span class="number">5</span>,:) <span class="comment">% 取第二行到第五行（一共4行）</span></span><br><span class="line">A(<span class="number">2</span>:<span class="number">2</span>:<span class="number">5</span>,:) <span class="comment">% 取第二行和第四行 （从2开始，每次递增2个单位，到5结束）</span></span><br><span class="line"><span class="number">1</span>:<span class="number">3</span>:<span class="number">10</span></span><br><span class="line"><span class="number">10</span>:<span class="number">-1</span>:<span class="number">1</span></span><br><span class="line">A(<span class="number">2</span>:<span class="keyword">end</span>,:) <span class="comment">% 取第二行到最后一行</span></span><br><span class="line">A(<span class="number">2</span>:<span class="keyword">end</span><span class="number">-1</span>,:) <span class="comment">% 取第二行到倒数第二行</span></span><br><span class="line"><span class="comment">% （5）取全部元素(按列拼接的，最终输出的是一个列向量)</span></span><br><span class="line">clc;A</span><br><span class="line">A(:)</span><br></pre></td></tr></table></figure><h3 id="size函数"><a class="markdownIt-Anchor" href="#size函数"></a> size函数</h3><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></pre></td><td class="code"><pre><span class="line"><span class="comment">%% size函数</span></span><br><span class="line">clc;</span><br><span class="line">A = [<span class="number">1</span>,<span class="number">2</span>,<span class="number">3</span>;<span class="number">4</span>,<span class="number">5</span>,<span class="number">6</span>]</span><br><span class="line">B = [<span class="number">1</span>,<span class="number">2</span>,<span class="number">3</span>,<span class="number">4</span>,<span class="number">5</span>,<span class="number">6</span>]</span><br><span class="line"><span class="built_in">size</span>(A)</span><br><span class="line"><span class="built_in">size</span>(B)</span><br><span class="line"><span class="comment">% size(A)函数是用来求矩阵A的大小的,它返回一个行向量，第一个元素是矩阵的行数，第二个元素是矩阵的列数</span></span><br><span class="line">[r,c] = <span class="built_in">size</span>(A)</span><br><span class="line"><span class="comment">% 将矩阵A的行数返回到第一个变量r，将矩阵的列数返回到第二个变量c</span></span><br><span class="line">r = <span class="built_in">size</span>(A,<span class="number">1</span>) <span class="comment">%返回行数</span></span><br><span class="line">c = <span class="built_in">size</span>(A,<span class="number">2</span>) <span class="comment">%返回列数</span></span><br></pre></td></tr></table></figure><h3 id="repmat函数"><a class="markdownIt-Anchor" href="#repmat函数"></a> repmat函数</h3><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></pre></td><td class="code"><pre><span class="line"><span class="comment">%% repmat函数</span></span><br><span class="line"><span class="comment">% B = repmat(A,m,n):将矩阵A复制m×n块，即把A作为B的元素，B由m×n个A平铺而成。</span></span><br><span class="line">A = [<span class="number">1</span>,<span class="number">2</span>,<span class="number">3</span>;<span class="number">4</span>,<span class="number">5</span>,<span class="number">6</span>]</span><br><span class="line">B = <span class="built_in">repmat</span>(A,<span class="number">2</span>,<span class="number">1</span>)</span><br><span class="line">B = <span class="built_in">repmat</span>(A,<span class="number">3</span>,<span class="number">2</span>)</span><br></pre></td></tr></table></figure><h3 id="matlab中矩阵的运算"><a class="markdownIt-Anchor" href="#matlab中矩阵的运算"></a> Matlab中矩阵的运算</h3><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></pre></td><td class="code"><pre><span class="line"><span class="comment">%% Matlab中矩阵的运算</span></span><br><span class="line"><span class="comment">% MATLAB在矩阵的运算中，“*”号和“/”号代表矩阵之间的乘法与除法(A/B = A*inv(B))</span></span><br><span class="line">A = [<span class="number">1</span>,<span class="number">2</span>;<span class="number">3</span>,<span class="number">4</span>]</span><br><span class="line">B = [<span class="number">1</span>,<span class="number">0</span>;<span class="number">1</span>,<span class="number">1</span>]</span><br><span class="line">A * B</span><br><span class="line">inv(B) <span class="comment">% 求B的逆矩阵</span></span><br><span class="line">B * inv(B)</span><br><span class="line">A * inv(B)</span><br><span class="line">A / B</span><br><span class="line"></span><br><span class="line"><span class="comment">% 两个形状相同的矩阵对应元素之间的乘除法需要使用“.*”和“./”</span></span><br><span class="line">A = [<span class="number">1</span>,<span class="number">2</span>;<span class="number">3</span>,<span class="number">4</span>]</span><br><span class="line">B = [<span class="number">1</span>,<span class="number">0</span>;<span class="number">1</span>,<span class="number">1</span>]</span><br><span class="line">A .* B</span><br><span class="line">A ./ B</span><br><span class="line"></span><br><span class="line"><span class="comment">% 每个元素同时和常数相乘或相除操作都可以使用</span></span><br><span class="line">A = [<span class="number">1</span>,<span class="number">2</span>;<span class="number">3</span>,<span class="number">4</span>]</span><br><span class="line">A * <span class="number">2</span></span><br><span class="line">A .* <span class="number">2</span></span><br><span class="line">A / <span class="number">2</span> </span><br><span class="line">A ./ <span class="number">2</span></span><br><span class="line"></span><br><span class="line"><span class="comment">% 每个元素同时乘方时只能用 .^</span></span><br><span class="line">A = [<span class="number">1</span>,<span class="number">2</span>;<span class="number">3</span>,<span class="number">4</span>]</span><br><span class="line">A .^ <span class="number">2</span></span><br><span class="line">A ^ <span class="number">2</span> </span><br><span class="line">A * A</span><br></pre></td></tr></table></figure><h3 id="matlab中求特征值和特征向量"><a class="markdownIt-Anchor" href="#matlab中求特征值和特征向量"></a> Matlab中求特征值和特征向量</h3><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></pre></td><td class="code"><pre><span class="line"><span class="comment">%% Matlab中求特征值和特征向量</span></span><br><span class="line"><span class="comment">% 在Matlab中，计算矩阵A的特征值和特征向量的函数是eig(A),其中最常用的两个用法：</span></span><br><span class="line">A = [<span class="number">1</span> <span class="number">2</span> <span class="number">3</span> ;<span class="number">2</span> <span class="number">2</span> <span class="number">1</span>;<span class="number">2</span> <span class="number">0</span> <span class="number">3</span>]</span><br><span class="line"><span class="comment">% （1）E=eig(A)：求矩阵A的全部特征值，构成向量E。</span></span><br><span class="line">E=eig(A)</span><br><span class="line"><span class="comment">% （2）[V,D]=eig(A)：求矩阵A的全部特征值，构成对角阵D，并求A的特征向量构成V的列向量。（V的每一列都是D中与之相同列的特征值的特征向量）</span></span><br><span class="line">[V,D]=eig(A)</span><br></pre></td></tr></table></figure><h3 id="find函数的基本用法"><a class="markdownIt-Anchor" href="#find函数的基本用法"></a> Find函数的基本用法</h3><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></pre></td><td class="code"><pre><span class="line"><span class="comment">%% find函数的基本用法</span></span><br><span class="line"><span class="comment">% 下面例子来自博客：https://www.cnblogs.com/anzhiwu815/p/5907033.html 博客内有更加深入的探究</span></span><br><span class="line"><span class="comment">% find函数，它可以用来返回向量或者矩阵中不为0的元素的位置索引。</span></span><br><span class="line">clc;X = [<span class="number">1</span> <span class="number">0</span> <span class="number">4</span> <span class="number">-3</span> <span class="number">0</span> <span class="number">0</span> <span class="number">0</span> <span class="number">8</span> <span class="number">6</span>]</span><br><span class="line">ind = <span class="built_in">find</span>(X)</span><br><span class="line"><span class="comment">% 其有多种用法，比如返回前2个不为0的元素的位置：</span></span><br><span class="line">ind = <span class="built_in">find</span>(X,<span class="number">2</span>)</span><br><span class="line"></span><br><span class="line"><span class="comment">%上面针对的是向量（一维），若X是一个矩阵（二维，有行和列），索引该如何返回呢？</span></span><br><span class="line">clc;X = [<span class="number">1</span> <span class="number">-3</span> <span class="number">0</span>;<span class="number">0</span> <span class="number">0</span> <span class="number">8</span>;<span class="number">4</span> <span class="number">0</span> <span class="number">6</span>]</span><br><span class="line">ind = <span class="built_in">find</span>(X)</span><br><span class="line"><span class="comment">% 这是因为在Matlab在存储矩阵时，是一列一列存储的，我们可以做一下验证：</span></span><br><span class="line">X(<span class="number">4</span>)</span><br><span class="line"><span class="comment">% 假如你需要按照行列的信息输出该怎么办呢？</span></span><br><span class="line">[r,c] = <span class="built_in">find</span>(X)</span><br><span class="line">[r,c] = <span class="built_in">find</span>(X,<span class="number">1</span>) <span class="comment">%只找第一个非0元素</span></span><br></pre></td></tr></table></figure><h3 id="矩阵的大小与常数的判断"><a class="markdownIt-Anchor" href="#矩阵的大小与常数的判断"></a> 矩阵的大小与常数的判断</h3><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></pre></td><td class="code"><pre><span class="line"><span class="comment">%% 矩阵与常数的大小判断运算</span></span><br><span class="line"><span class="comment">% 共有三种运算符：大于> ;小于< ;等于 == （一个等号表示赋值；两个等号表示判断）</span></span><br><span class="line">clc</span><br><span class="line">X = [<span class="number">1</span> <span class="number">-3</span> <span class="number">0</span>;<span class="number">0</span> <span class="number">0</span> <span class="number">8</span>;<span class="number">4</span> <span class="number">0</span> <span class="number">6</span>]</span><br><span class="line">X > <span class="number">0</span></span><br><span class="line">X == <span class="number">4</span></span><br></pre></td></tr></table></figure><h3 id="判断语句"><a class="markdownIt-Anchor" href="#判断语句"></a> 判断语句</h3><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></pre></td><td class="code"><pre><span class="line"><span class="comment">%% 判断语句</span></span><br><span class="line"><span class="comment">% Matlab的判断语句，if所在的行不需要冒号，语句的最后一定要以end结尾 ；中间的语句要注意缩进。</span></span><br><span class="line">a = input(<span class="string">'请输入考试分数:'</span>)</span><br><span class="line"><span class="keyword">if</span> a >= <span class="number">85</span> </span><br><span class="line"> <span class="built_in">disp</span>(<span class="string">'成绩优秀'</span>)</span><br><span class="line"><span class="keyword">elseif</span> a >= <span class="number">60</span> </span><br><span class="line"> <span class="built_in">disp</span>(<span class="string">'成绩合格'</span>)</span><br><span class="line"><span class="keyword">else</span></span><br><span class="line"> <span class="built_in">disp</span>(<span class="string">'成绩挂科'</span>)</span><br><span class="line"><span class="keyword">end</span></span><br></pre></td></tr></table></figure>]]></content>
<categories>
<category> 数学建模 </category>
</categories>
<tags>
<tag> 数学建模 </tag>
<tag> MATLAB </tag>
</tags>
</entry>
<entry>
<title>层次分析法的Matlab实现</title>
<link href="/2022/09/05/%E5%B1%82%E6%AC%A1%E5%88%86%E6%9E%90%E6%B3%95%E7%9A%84Matlab%E5%AE%9E%E7%8E%B0/"/>
<url>/2022/09/05/%E5%B1%82%E6%AC%A1%E5%88%86%E6%9E%90%E6%B3%95%E7%9A%84Matlab%E5%AE%9E%E7%8E%B0/</url>
<content type="html"><![CDATA[<style>#web_bg{ background: url(https://img-baofun.zhhainiao.com/pcwallpaper_ugc/static/5b729854282ab9b5bc6c578042614ac3.jpg?x-oss-process=image%2fresize%2cm_lfit%2cw_1920%2ch_1080)!important; /*重新定义background会导致原有定位属性失效，所以也需要再声明一次加权的定位属性*/ background-position: center !important; background-size: cover !important; background-repeat: no-repeat !important;}</style><h1 id="层次分析法的matlab实现"><a class="markdownIt-Anchor" href="#层次分析法的matlab实现"></a> 层次分析法的Matlab实现</h1><p><em>注意：在论文写作中，应该先对判断矩阵进行一致性检验，然后再计算权重，因为只有判断矩阵通过了一致性检验，其权重才是有意义的。在下面的代码中，我们先计算了权重，然后再进行了一致性检验，这是为了顺应计算过程，事实上在逻辑上是说不过去的。 因此大家自己写论文中如果用到了层次分析法，一定要先对判断矩阵进行一致性检验。 而且要说明的是，只有非一致矩阵的判断矩阵才需要进行一致性检验。如果你的判断矩阵本身就是一个一致矩阵，那么就没有必要进行一致性检验。</em></p><h2 id="输入判断矩阵"><a class="markdownIt-Anchor" href="#输入判断矩阵"></a> 输入判断矩阵</h2><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></pre></td><td class="code"><pre><span class="line"><span class="comment">%% 输入判断矩阵</span></span><br><span class="line">clear;</span><br><span class="line">clc;</span><br><span class="line"><span class="built_in">disp</span>(<span class="string">'请输入判断矩阵A： '</span>)</span><br><span class="line"><span class="comment">% A = input('判断矩阵A=')</span></span><br><span class="line">A =[<span class="number">1</span> <span class="number">1</span> <span class="number">4</span> <span class="number">1</span>/<span class="number">3</span> <span class="number">3</span>;</span><br><span class="line"> <span class="number">1</span> <span class="number">1</span> <span class="number">4</span> <span class="number">1</span>/<span class="number">3</span> <span class="number">3</span>;</span><br><span class="line"> <span class="number">1</span>/<span class="number">4</span> <span class="number">1</span>/<span class="number">4</span> <span class="number">1</span> <span class="number">1</span>/<span class="number">3</span> <span class="number">1</span>/<span class="number">2</span>;</span><br><span class="line"> <span class="number">3</span> <span class="number">3</span> <span class="number">3</span> <span class="number">1</span> <span class="number">3</span>;</span><br><span class="line"> <span class="number">1</span>/<span class="number">3</span> <span class="number">1</span>/<span class="number">3</span> <span class="number">2</span> <span class="number">1</span>/<span class="number">3</span> <span class="number">1</span>]</span><br><span class="line"><span class="comment">% matlab矩阵有两种写法，可以直接写到一行:</span></span><br><span class="line"><span class="comment">% [1 1 4 1/3 3;1 1 4 1/3 3;1/4 1/4 1 1/3 1/2;3 3 3 1 3;1/3 1/3 2 1/3 1]</span></span><br><span class="line"><span class="comment">% 也可以写成多行:</span></span><br><span class="line">[<span class="number">1</span> <span class="number">1</span> <span class="number">4</span> <span class="number">1</span>/<span class="number">3</span> <span class="number">3</span>;</span><br><span class="line"> <span class="number">1</span> <span class="number">1</span> <span class="number">4</span> <span class="number">1</span>/<span class="number">3</span> <span class="number">3</span>;</span><br><span class="line"> <span class="number">1</span>/<span class="number">4</span> <span class="number">1</span>/<span class="number">4</span> <span class="number">1</span> <span class="number">1</span>/<span class="number">3</span> <span class="number">1</span>/<span class="number">2</span>;</span><br><span class="line"> <span class="number">3</span> <span class="number">3</span> <span class="number">3</span> <span class="number">1</span> <span class="number">3</span>;</span><br><span class="line"> <span class="number">1</span>/<span class="number">3</span> <span class="number">1</span>/<span class="number">3</span> <span class="number">2</span> <span class="number">1</span>/<span class="number">3</span> <span class="number">1</span>]</span><br><span class="line"><span class="comment">% 两行之间以分号结尾（最后一行的分号可加可不加），同行元素之间以空格（或者逗号）分开。</span></span><br></pre></td></tr></table></figure><h2 id="算术平均法求权重"><a class="markdownIt-Anchor" href="#算术平均法求权重"></a> 算术平均法求权重</h2><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></pre></td><td class="code"><pre><span class="line"><span class="comment">%% 方法1：算术平均法求权重</span></span><br><span class="line"><span class="comment">% 第一步：将判断矩阵按照列归一化（每一个元素除以其所在列的和）</span></span><br><span class="line">Sum_A = sum(A)</span><br><span class="line"></span><br><span class="line">[n,n] = <span class="built_in">size</span>(A) <span class="comment">% 也可以写成n = size(A,1)</span></span><br><span class="line"><span class="comment">% 因为我们的判断矩阵A是一个方阵，所以这里的r和c相同，我们可以就用同一个字母n表示</span></span><br><span class="line">SUM_A = <span class="built_in">repmat</span>(Sum_A,n,<span class="number">1</span>) <span class="comment">%repeat matrix的缩写</span></span><br><span class="line"><span class="comment">% 另外一种替代的方法如下：</span></span><br><span class="line"> SUM_A = [];</span><br><span class="line"> <span class="keyword">for</span> <span class="built_in">i</span> = <span class="number">1</span>:n <span class="comment">%循环哦，这一行后面不能加冒号（和Python不同），这里表示循环n次</span></span><br><span class="line"> SUM_A = [SUM_A; Sum_A]</span><br><span class="line"> <span class="keyword">end</span></span><br><span class="line">clc;A</span><br><span class="line">SUM_A</span><br><span class="line">Stand_A = A ./ SUM_A</span><br><span class="line"><span class="comment">% 这里我们直接将两个矩阵对应的元素相除即可</span></span><br><span class="line"></span><br><span class="line"><span class="comment">% 第二步：将归一化的各列相加(按行求和)</span></span><br><span class="line">sum(Stand_A,<span class="number">2</span>)</span><br><span class="line"></span><br><span class="line"><span class="comment">% 第三步：将相加后得到的向量中每个元素除以n即可得到权重向量</span></span><br><span class="line"><span class="built_in">disp</span>(<span class="string">'算术平均法求权重的结果为：'</span>);</span><br><span class="line"><span class="built_in">disp</span>(sum(Stand_A,<span class="number">2</span>) / n)</span><br><span class="line"><span class="comment">% 首先对标准化后的矩阵按照行求和，得到一个列向量</span></span><br><span class="line"><span class="comment">% 然后再将这个列向量的每个元素同时除以n即可（注意这里也可以用./哦）</span></span><br></pre></td></tr></table></figure><h2 id="几何平均法求权重"><a class="markdownIt-Anchor" href="#几何平均法求权重"></a> 几何平均法求权重</h2><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></pre></td><td class="code"><pre><span class="line"><span class="comment">%% 方法2：几何平均法求权重</span></span><br><span class="line"><span class="comment">% 第一步：将A的元素按照行相乘得到一个新的列向量</span></span><br><span class="line">clc;A</span><br><span class="line">Prduct_A = prod(A,<span class="number">2</span>)</span><br><span class="line"><span class="comment">% prod函数和sum函数类似，一个用于乘，一个用于加 dim = 2 维度是行</span></span><br><span class="line"></span><br><span class="line"><span class="comment">% 第二步：将新的向量的每个分量开n次方</span></span><br><span class="line">Prduct_n_A = Prduct_A .^ (<span class="number">1</span>/n)</span><br><span class="line"><span class="comment">% 这里对每个元素进行乘方操作，因此要加.号哦。 ^符号表示乘方哦 这里是开n次方，所以我们等价求1/n次方</span></span><br><span class="line"></span><br><span class="line"><span class="comment">% 第三步：对该列向量进行归一化即可得到权重向量</span></span><br><span class="line"><span class="comment">% 将这个列向量中的每一个元素除以这一个向量的和即可</span></span><br><span class="line"><span class="built_in">disp</span>(<span class="string">'几何平均法求权重的结果为：'</span>);</span><br><span class="line"><span class="built_in">disp</span>(Prduct_n_A ./ sum(Prduct_n_A))</span><br></pre></td></tr></table></figure><h2 id="特征值法求权重"><a class="markdownIt-Anchor" href="#特征值法求权重"></a> 特征值法求权重</h2><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></pre></td><td class="code"><pre><span class="line"><span class="comment">%% 方法3：特征值法求权重</span></span><br><span class="line"><span class="comment">% 第一步：求出矩阵A的最大特征值以及其对应的特征向量</span></span><br><span class="line">clc</span><br><span class="line">[V,D] = eig(A) <span class="comment">%V是特征向量, D是由特征值构成的对角矩阵（除了对角线元素外，其余位置元素全为0）</span></span><br><span class="line">Max_eig = <span class="built_in">max</span>(<span class="built_in">max</span>(D)) <span class="comment">%也可以写成max(D(:))哦~</span></span><br><span class="line"><span class="comment">% 那么怎么找到最大特征值所在的位置了？ 需要用到find函数，它可以用来返回向量或者矩阵中不为0的元素的位置索引。</span></span><br><span class="line"><span class="comment">% 那么问题来了，我们要得到最大特征值的位置，就需要将包含所有特征值的这个对角矩阵D中，不等于最大特征值的位置全变为0</span></span><br><span class="line"><span class="comment">% 这时候可以用到矩阵与常数的大小判断运算</span></span><br><span class="line">D == Max_eig</span><br><span class="line">[r,c] = <span class="built_in">find</span>(D == Max_eig , <span class="number">1</span>)</span><br><span class="line"><span class="comment">% 找到D中第一个与最大特征值相等的元素的位置，记录它的行和列。</span></span><br><span class="line"></span><br><span class="line"><span class="comment">% 第二步：对求出的特征向量进行归一化即可得到我们的权重</span></span><br><span class="line">V(:,c)</span><br><span class="line"><span class="built_in">disp</span>(<span class="string">'特征值法求权重的结果为：'</span>);</span><br><span class="line"><span class="built_in">disp</span>( V(:,c) ./ sum(V(:,c)) )</span><br><span class="line"><span class="comment">% 我们先根据上面找到的最大特征值的列数c找到对应的特征向量，然后再进行标准化。</span></span><br></pre></td></tr></table></figure><h2 id="计算一致性比例"><a class="markdownIt-Anchor" href="#计算一致性比例"></a> 计算一致性比例</h2><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></pre></td><td class="code"><pre><span class="line"><span class="comment">%% 计算一致性比例CR</span></span><br><span class="line">clc</span><br><span class="line">CI = (Max_eig - n) / (n<span class="number">-1</span>);</span><br><span class="line">RI=[<span class="number">0</span> <span class="number">0</span> <span class="number">0.52</span> <span class="number">0.89</span> <span class="number">1.12</span> <span class="number">1.26</span> <span class="number">1.36</span> <span class="number">1.41</span> <span class="number">1.46</span> <span class="number">1.49</span> <span class="number">1.52</span> <span class="number">1.54</span> <span class="number">1.56</span> <span class="number">1.58</span> <span class="number">1.59</span>]; <span class="comment">%注意哦，这里的RI最多支持 n = 15</span></span><br><span class="line">CR=CI/RI(n);</span><br><span class="line"><span class="built_in">disp</span>(<span class="string">'一致性指标CI='</span>);<span class="built_in">disp</span>(CI);</span><br><span class="line"><span class="built_in">disp</span>(<span class="string">'一致性比例CR='</span>);<span class="built_in">disp</span>(CR);</span><br><span class="line"><span class="keyword">if</span> CR<<span class="number">0.10</span></span><br><span class="line"> <span class="built_in">disp</span>(<span class="string">'因为CR < 0.10，所以该判断矩阵A的一致性可以接受!'</span>);</span><br><span class="line"><span class="keyword">else</span></span><br><span class="line"> <span class="built_in">disp</span>(<span class="string">'注意：CR >= 0.10，因此该判断矩阵A需要进行修改!'</span>);</span><br><span class="line"><span class="keyword">end</span></span><br></pre></td></tr></table></figure>]]></content>
<categories>
<category> 数学建模 </category>
</categories>
<tags>
<tag> 数学建模 </tag>
<tag> MATLAB </tag>
<tag> 层次分析法 </tag>
</tags>
</entry>
<entry>
<title>插值算法 - MATLAB - 数学建模</title>
<link href="/2022/09/05/%E6%8F%92%E5%80%BC%E7%AE%97%E6%B3%95/"/>
<url>/2022/09/05/%E6%8F%92%E5%80%BC%E7%AE%97%E6%B3%95/</url>
<content type="html"><![CDATA[<style>#web_bg{ background: url(https://img-baofun.zhhainiao.com/pcwallpaper_ugc/static/6bfa67aef1a28d75d9656ef014c92863.jpeg?x-oss-process=image%2fresize%2cm_lfit%2cw_3840%2ch_2160)!important; /*重新定义background会导致原有定位属性失效，所以也需要再声明一次加权的定位属性*/ background-position: center !important; background-size: cover !important; background-repeat: no-repeat !important;}</style><h1 id="插值算法"><a class="markdownIt-Anchor" href="#插值算法"></a> 插值算法</h1><h2 id="一维插值问题"><a class="markdownIt-Anchor" href="#一维插值问题"></a> 一维插值问题</h2><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">已</mi><mi mathvariant="normal">知</mi><mi mathvariant="normal">有</mi><mi>n</mi><mo>+</mo><mn>1</mn><mi mathvariant="normal">个</mi><mi mathvariant="normal">节</mi><mi mathvariant="normal">点</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>y</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi mathvariant="normal">其</mi><mi mathvariant="normal">中</mi><msub><mi>x</mi><mi>i</mi></msub><mi mathvariant="normal">互</mi><mi mathvariant="normal">不</mi><mi mathvariant="normal">相</mi><mi mathvariant="normal">同</mi><mspace linebreak="newline"></mspace><mi mathvariant="normal">不</mi><mi mathvariant="normal">妨</mi><mi mathvariant="normal">假</mi><mi mathvariant="normal">设</mi><mi>a</mi><mo>=</mo><msub><mi>x</mi><mn>0</mn></msub><mo><</mo><msub><mi>x</mi><mn>1</mn></msub><mo><</mo><mo>⋯</mo><mo><</mo><msub><mi>x</mi><mi>n</mi></msub><mo>=</mo><mi>b</mi><mo separator="true">,</mo><mi mathvariant="normal">求</mi><mi mathvariant="normal">任</mi><mi mathvariant="normal">一</mi><mi mathvariant="normal">插</mi><mi mathvariant="normal">值</mi><mi mathvariant="normal">点</mi><msup><mi>x</mi><mo>∗</mo></msup><mo stretchy="false">(</mo><mi mathvariant="normal">≠</mi><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mi mathvariant="normal">处</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">插</mi><mi mathvariant="normal">值</mi><msup><mi>y</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">已知有n+1个节点(x_i,y_i)(i=0,1,\cdots,n),其中x_i互不相同\\不妨假设a=x_0<x_1<\cdots<x_n=b,求任一插值点x^{*}(\ne x_i)处的插值y^{*}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord cjk_fallback">已</span><span class="mord cjk_fallback">知</span><span class="mord cjk_fallback">有</span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mord cjk_fallback">个</span><span class="mord cjk_fallback">节</span><span class="mord cjk_fallback">点</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mopen">(</span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">n</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord cjk_fallback">其</span><span class="mord cjk_fallback">中</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord cjk_fallback">互</span><span class="mord cjk_fallback">不</span><span class="mord cjk_fallback">相</span><span class="mord cjk_fallback">同</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord cjk_fallback">不</span><span class="mord cjk_fallback">妨</span><span class="mord cjk_fallback">假</span><span class="mord cjk_fallback">设</span><span class="mord mathdefault">a</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">b</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord cjk_fallback">求</span><span class="mord cjk_fallback">任</span><span class="mord cjk_fallback">一</span><span class="mord cjk_fallback">插</span><span class="mord cjk_fallback">值</span><span class="mord cjk_fallback">点</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.738696em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mrel"><span class="mrel"><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.69444em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="rlap"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="inner"><span class="mrel"></span></span><span class="fix"></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"><span></span></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord cjk_fallback">处</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">插</span><span class="mord cjk_fallback">值</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.738696em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span></span></span></span></span></span></span></span></span></p><img src="https://gimg2.baidu.com/image_search/src=http%3A%2F%2Fimg-blog.csdnimg.cn%2F20210211113103599.png%3Fx-oss-process%3Dimage%2Fwatermark%2Ctype_ZmFuZ3poZW5naGVpdGk%2Cshadow_10%2Ctext_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L1NhdGFuX0Rldmls%2Csize_16%2Ccolor_FFFFFF%2Ct_70&refer=http%3A%2F%2Fimg-blog.csdnimg.cn&app=2002&size=f9999,10000&q=a80&n=0&g=0n&fmt=auto?sec=1666412102&t=15965ee3862176602bb372e239ef8f59" alt="插值算法样例图" style="zoom:50%;" /><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">思</mi><mi mathvariant="normal">路</mi><mi mathvariant="normal">：</mi><mi mathvariant="normal">构</mi><mi mathvariant="normal">造</mi><mi mathvariant="normal">函</mi><mi mathvariant="normal">数</mi><mi>y</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi mathvariant="normal">使</mi><mi mathvariant="normal">得</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi mathvariant="normal">过</mi><mi mathvariant="normal">所</mi><mi mathvariant="normal">有</mi><mi mathvariant="normal">节</mi><mi mathvariant="normal">点</mi><mi mathvariant="normal">，</mi><mi mathvariant="normal">求</mi><mi mathvariant="normal">解</mi><mi>f</mi><mo stretchy="false">(</mo><msup><mi>x</mi><mo>∗</mo></msup><mo stretchy="false">)</mo><mi mathvariant="normal">即</mi><mi mathvariant="normal">可</mi><mi mathvariant="normal">得</mi><mi mathvariant="normal">到</mi><msup><mi>y</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">思路：构造函数y=f(x),使得f(x)过所有节点，求解f(x^{*})即可得到y^{*}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord cjk_fallback">思</span><span class="mord cjk_fallback">路</span><span class="mord cjk_fallback">：</span><span class="mord cjk_fallback">构</span><span class="mord cjk_fallback">造</span><span class="mord cjk_fallback">函</span><span class="mord cjk_fallback">数</span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord cjk_fallback">使</span><span class="mord cjk_fallback">得</span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span><span class="mord cjk_fallback">过</span><span class="mord cjk_fallback">所</span><span class="mord cjk_fallback">有</span><span class="mord cjk_fallback">节</span><span class="mord cjk_fallback">点</span><span class="mord cjk_fallback">，</span><span class="mord cjk_fallback">求</span><span class="mord cjk_fallback">解</span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.738696em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord cjk_fallback">即</span><span class="mord cjk_fallback">可</span><span class="mord cjk_fallback">得</span><span class="mord cjk_fallback">到</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.738696em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∗</span></span></span></span></span></span></span></span></span></span></span></span></span></p><h2 id="插值法的定义"><a class="markdownIt-Anchor" href="#插值法的定义"></a> 插值法的定义</h2><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">设</mi><mi mathvariant="normal">函</mi><mi mathvariant="normal">数</mi><mi>y</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi mathvariant="normal">在</mi><mi mathvariant="normal">区</mi><mi mathvariant="normal">间</mi><mo stretchy="false">[</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">]</mo><mi mathvariant="normal">上</mi><mi mathvariant="normal">有</mi><mi mathvariant="normal">定</mi><mi mathvariant="normal">义</mi><mi mathvariant="normal">，</mi><mi mathvariant="normal">且</mi><mi mathvariant="normal">已</mi><mi mathvariant="normal">知</mi><mi mathvariant="normal">在</mi><mi mathvariant="normal">点</mi><mi>a</mi><mo>≤</mo><msub><mi>x</mi><mn>0</mn></msub><mo><</mo><msub><mi>x</mi><mn>1</mn></msub><mo><</mo><mo>⋯</mo><mo><</mo><msub><mi>x</mi><mi>n</mi></msub><mo>≤</mo><mi>b</mi><mi mathvariant="normal">上</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">值</mi><mi mathvariant="normal">分</mi><mi mathvariant="normal">别</mi><mi mathvariant="normal">为</mi><mo>:</mo><msub><mi>y</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><msub><mi>y</mi><mi>n</mi></msub><mo separator="true">,</mo></mrow><annotation encoding="application/x-tex">设函数y=f(x)在区间[a,b]上有定义，且已知在点a\leq x_0<x_1<\cdots<x_n\leq b上的值分别为:y_0,y_1,\cdots,y_n,</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord cjk_fallback">设</span><span class="mord cjk_fallback">函</span><span class="mord cjk_fallback">数</span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span><span class="mord cjk_fallback">在</span><span class="mord cjk_fallback">区</span><span class="mord cjk_fallback">间</span><span class="mopen">[</span><span class="mord mathdefault">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span><span class="mclose">]</span><span class="mord cjk_fallback">上</span><span class="mord cjk_fallback">有</span><span class="mord cjk_fallback">定</span><span class="mord cjk_fallback">义</span><span class="mord cjk_fallback">，</span><span class="mord cjk_fallback">且</span><span class="mord cjk_fallback">已</span><span class="mord cjk_fallback">知</span><span class="mord cjk_fallback">在</span><span class="mord cjk_fallback">点</span><span class="mord mathdefault">a</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.7859700000000001em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">b</span><span class="mord cjk_fallback">上</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">值</span><span class="mord cjk_fallback">分</span><span class="mord cjk_fallback">别</span><span class="mord cjk_fallback">为</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span></span></span></span><br /><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">若</mi><mi mathvariant="normal">存</mi><mi mathvariant="normal">在</mi><mi mathvariant="normal">一</mi><mi mathvariant="normal">简</mi><mi mathvariant="normal">单</mi><mi mathvariant="normal">函</mi><mi mathvariant="normal">数</mi><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi mathvariant="normal">，</mi><mi mathvariant="normal">使</mi><mi mathvariant="normal">得</mi><mi>P</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>y</mi><mi>i</mi></msub><mo separator="true">,</mo><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">若存在一简单函数P(x)，使得P(x_i)=y_i,(i=0,1,2,\cdots,n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord cjk_fallback">若</span><span class="mord cjk_fallback">存</span><span class="mord cjk_fallback">在</span><span class="mord cjk_fallback">一</span><span class="mord cjk_fallback">简</span><span class="mord cjk_fallback">单</span><span class="mord cjk_fallback">函</span><span class="mord cjk_fallback">数</span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span><span class="mord cjk_fallback">，</span><span class="mord cjk_fallback">使</span><span class="mord cjk_fallback">得</span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">n</span><span class="mclose">)</span></span></span></span><br /><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">则</mi><mi mathvariant="normal">称</mi><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi mathvariant="normal">为</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi mathvariant="normal">的</mi><mi mathvariant="normal">插</mi><mi mathvariant="normal">值</mi><mi mathvariant="normal">函</mi><mi mathvariant="normal">数</mi><mo separator="true">,</mo><mi mathvariant="normal">点</mi><msub><mi>x</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><msub><mi>x</mi><mi>n</mi></msub><mi mathvariant="normal">称</mi><mi mathvariant="normal">为</mi><mi mathvariant="normal">插</mi><mi mathvariant="normal">值</mi><mi mathvariant="normal">节</mi><mi mathvariant="normal">点</mi><mi mathvariant="normal">，</mi><mi mathvariant="normal">包</mi><mi mathvariant="normal">含</mi><mi mathvariant="normal">插</mi><mi mathvariant="normal">值</mi><mi mathvariant="normal">节</mi><mi mathvariant="normal">点</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">区</mi><mi mathvariant="normal">间</mi><mo stretchy="false">[</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">]</mo><mi mathvariant="normal">称</mi><mi mathvariant="normal">为</mi><mi mathvariant="normal">插</mi><mi mathvariant="normal">值</mi><mi mathvariant="normal">区</mi><mi mathvariant="normal">间</mi></mrow><annotation encoding="application/x-tex">则称P(x)为f(x)的插值函数,点x_0,x_1,\cdots,x_n 称为插值节点，包含插值节点的区间[a,b]称为插值区间</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord cjk_fallback">则</span><span class="mord cjk_fallback">称</span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span><span class="mord cjk_fallback">为</span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">插</span><span class="mord cjk_fallback">值</span><span class="mord cjk_fallback">函</span><span class="mord cjk_fallback">数</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord cjk_fallback">点</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord cjk_fallback">称</span><span class="mord cjk_fallback">为</span><span class="mord cjk_fallback">插</span><span class="mord cjk_fallback">值</span><span class="mord cjk_fallback">节</span><span class="mord cjk_fallback">点</span><span class="mord cjk_fallback">，</span><span class="mord cjk_fallback">包</span><span class="mord cjk_fallback">含</span><span class="mord cjk_fallback">插</span><span class="mord cjk_fallback">值</span><span class="mord cjk_fallback">节</span><span class="mord cjk_fallback">点</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">区</span><span class="mord cjk_fallback">间</span><span class="mopen">[</span><span class="mord mathdefault">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span><span class="mclose">]</span><span class="mord cjk_fallback">称</span><span class="mord cjk_fallback">为</span><span class="mord cjk_fallback">插</span><span class="mord cjk_fallback">值</span><span class="mord cjk_fallback">区</span><span class="mord cjk_fallback">间</span></span></span></span></p><p><strong>1.若P(x)是次数不超过n的代数多项式,即</strong><br /><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>a</mi><mn>0</mn></msub><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><mi>x</mi><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>a</mi><mi>n</mi></msub><msup><mi>x</mi><mi>n</mi></msup><mo separator="true">,</mo></mrow><annotation encoding="application/x-tex">P(x)=a_0+a_1x+\cdots+a_nx^n,</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.858832em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span></span></span></span><span class="mpunct">,</span></span></span></span><br /><strong>2.若P(x)为分段多项式，就称为分段插值。</strong><br /><strong>3.若P(x)为三角多项式，就称为三角插值。</strong></p><h2 id="一般插值多项式原理"><a class="markdownIt-Anchor" href="#一般插值多项式原理"></a> 一般插值多项式原理</h2><h3 id="定理"><a class="markdownIt-Anchor" href="#定理"></a> 定理：</h3><p>设有<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span>个互不相同的节点<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>y</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo separator="true">,</mo><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x_i,y_i),(i=1,2,\cdots,n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">n</span><span class="mclose">)</span></span></span></span>,则存在唯一的多项式：</p>\begin{align}&L_n(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n \tag{1}\\s.t. \space&L_n(x_j)=y_j &(j=0,1,2,\cdots,n) \tag{2}\\\end{align}<p><strong>证明</strong>：构造方程组</p>\begin{align}\begin{cases}\tag{3}a_0+a_1x_0+a_2x_0^2+\cdots+a_nx_0^n = y_0\\a_0+a_1x_1+a_2x_1^2+\cdots+a_nx_1^n = y_1\\\cdots\cdots\\a_0+a_1x_n+a_2x_n^2+\cdots+a_nx_n^n = y_n\\ \end{cases}\end{align}<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal"></mi><mi mathvariant="normal">令</mi><mi mathvariant="normal">：</mi><mi>A</mi><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>0</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo>⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msubsup><mi>x</mi><mn>0</mn><mi>n</mi></msubsup></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mn>1</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo>⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msubsup><mi>x</mi><mn>1</mn><mi>n</mi></msubsup></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>x</mi><mi>n</mi></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo>⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msubsup><mi>x</mi><mi>n</mi><mi>n</mi></msubsup></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mi>X</mi><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mn>0</mn></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mn>1</mn></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>a</mi><mi>n</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mi>Y</mi><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>y</mi><mn>0</mn></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>y</mi><mn>1</mn></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi mathvariant="normal">⋮</mi><mpadded height="+0em" voffset="0em"><mspace mathbackground="black" width="0em" height="1.5em"></mspace></mpadded></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>y</mi><mi>n</mi></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">令：A=\left[\begin{matrix}1 & x_0 & \cdots & x_0^n\\1 & x_1 & \cdots & x_1^n\\\vdots & \vdots & &\vdots\\1 & x_n & \cdots & x_n^n\end{matrix}\right]X=\left[\begin{matrix}a_0 \\ a_1 \\ \vdots \\ a_n\end{matrix}\right]Y=\left[\begin{matrix}y_0 \\ y_1 \\ \vdots \\ y_n\end{matrix}\right]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord"></span><span class="mord cjk_fallback">令</span><span class="mord cjk_fallback">：</span><span class="mord mathdefault">A</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:5.459999999999999em;vertical-align:-2.4799999999999995em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.953005em;"><span style="top:-1.3499850000000007em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.5049850000000005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.1059850000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.7069850000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.953005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4500349999999997em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9799999999999995em;"><span style="top:-5.8275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-4.6275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-2.7674999999999996em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-1.5675000000000006em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4799999999999995em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9799999999999995em;"><span style="top:-5.8275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.6275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.7674999999999996em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-1.5675000000000006em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4799999999999995em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9799999999999995em;"><span style="top:-5.64em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-4.44em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-2.5799999999999996em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"></span></span><span style="top:-1.3800000000000006em;"><span class="pstrut" style="height:3.5em;"></span><span class="mord"><span class="minner">⋯</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4799999999999995em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9799999999999995em;"><span style="top:-5.8275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-2.4518920000000004em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.24810799999999997em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.6275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-2.4518920000000004em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.24810799999999997em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.7674999999999996em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-1.5675000000000006em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4799999999999995em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.953005em;"><span style="top:-1.3499850000000007em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.5049850000000005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.1059850000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.7069850000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.953005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4500349999999997em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:5.459999999999999em;vertical-align:-2.4799999999999995em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.953005em;"><span style="top:-1.3499850000000007em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.5049850000000005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.1059850000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.7069850000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.953005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4500349999999997em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9799999999999995em;"><span style="top:-5.8275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.6275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.7674999999999996em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-1.5675000000000006em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4799999999999995em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.953005em;"><span style="top:-1.3499850000000007em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.5049850000000005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.1059850000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.7069850000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.953005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4500349999999997em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:5.459999999999999em;vertical-align:-2.4799999999999995em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.953005em;"><span style="top:-1.3499850000000007em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-2.5049850000000005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.1059850000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.7069850000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.953005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎡</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4500349999999997em;"><span></span></span></span></span></span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.9799999999999995em;"><span style="top:-5.8275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-4.6275em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.7674999999999996em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord">⋮</span><span class="mord rule" style="border-right-width:0em;border-top-width:1.5em;bottom:0em;"></span></span></span></span><span style="top:-1.5675000000000006em;"><span class="pstrut" style="height:3.6875em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4799999999999995em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.953005em;"><span style="top:-1.3499850000000007em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-2.5049850000000005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.1059850000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.7069850000000004em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.953005em;"><span class="pstrut" style="height:3.1550000000000002em;"></span><span class="delimsizinginner delim-size4"><span>⎤</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.4500349999999997em;"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>方程组的矩阵形式如下：<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>A</mi><mi>X</mi><mo>=</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">AX=Y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault">A</span><span class="mord mathdefault" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">Y</span></span></span></span></p><p>由于: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">∣</mi><mi>A</mi><mi mathvariant="normal">∣</mi><mo>=</mo><msubsup><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msubsup><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><msub><mi>x</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mi mathvariant="normal">≠</mi><mn>0</mn></mrow><annotation encoding="application/x-tex">|A|=\prod_{i=1}^{n}\prod_{j=0}^{i-1}(x_i-x_j)\neq0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord mathdefault">A</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.400382em;vertical-align:-0.43581800000000004em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∏</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.804292em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∏</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.964564em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.43581800000000004em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.036108em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><span class="mrel"><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.69444em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="rlap"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="inner"><span class="mrel"></span></span><span class="fix"></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"><span></span></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span> (范德蒙行列式),所以方程组有唯一解。</p><p>从而 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>L</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>a</mi><mn>0</mn></msub><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><mi>x</mi><mo>+</mo><msub><mi>a</mi><mn>2</mn></msub><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>a</mi><mi>n</mi></msub><msup><mi>x</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">L_n(x)=a_0 + a_1x + a_2x^2 + \cdots + a_nx^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.964108em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.814392em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span></span></span></span></span></span></span> 唯一存在。</p><h3 id="lagrange插值法"><a class="markdownIt-Anchor" href="#lagrange插值法"></a> Lagrange插值法</h3>\begin{align}\omega_{n+1}(x) &=(x-x_0)(x-x_1)\cdots(x-x_n)\\\omega_{n+1}'(x_k)&=(x_k-x_0)\cdots(x_k-x_{k-1})(x_k-x_{k+1})\cdots(x_k-x_n).\\l_i(x)&=\frac{(x-x_0)\cdots(x-x_{i-1})(x-x_{i+1})\cdots(x-x_n)}{(x_i-x_0)\cdots(x_i-x_{i-1})(x_i-x_{i+1})\cdots(x_i-x_n)}\\&=\frac{(x-x_0)\cdots(x-x_{i-1})(x-x_i)(x-x_{i+1})\cdots(x-x_n)}{(x-x_i)(x_i-x_0)\cdots(x_i-x_{i-1})(x_i-x_{i+1})\cdots(x_i-x_n)}\\&=\frac{\omega_{n+1}(x)}{(x-x_i)\omega_{n+1}'(x_i)}\end{align}<p><strong>Lagrange插值多项式:</strong> <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>L</mi><mi>n</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mi>n</mi></msubsup><msub><mi>y</mi><mi>k</mi></msub><mfrac><mrow><msub><mi>ω</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo><msubsup><mi>ω</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mo mathvariant="normal">′</mo></msubsup><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">L_n(x)=\sum^n_{k=0}y_k\frac{\omega_{n+1}(x)}{(x-x_k)\omega_{n+1}'(x_k)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">L</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.614725em;vertical-align:-0.604725em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.804292em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="mrel mtight">=</span><span class="mord mtight">0</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.01em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">x</span><span class="mbin mtight">−</span><span class="mord mtight"><span class="mord mathdefault mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3487714285714287em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15122857142857138em;"><span></span></span></span></span></span></span><span class="mclose mtight">)</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7416285714285715em;"><span style="top:-2.1884857142857146em;margin-left:-0.03588em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span><span style="top:-2.8448em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.3710357142857143em;"><span></span></span></span></span></span></span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mathdefault mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3487714285714287em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15122857142857138em;"><span></span></span></span></span></span></span><span class="mclose mtight">)</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.485em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173142857142857em;"><span style="top:-2.357em;margin-left:-0.03588em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.20252142857142857em;"><span></span></span></span></span></span></span><span class="mopen mtight">(</span><span class="mord mathdefault mtight">x</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.604725em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></p><p><strong>龙格现象</strong>：高次插值时会产生龙格现象，即在两端出波动极大，产生明显的震荡。在不熟悉曲线运动趋势的前提下，不要轻易使用高次插值。</p><img src="https://gimg2.baidu.com/image_search/src=http%3A%2F%2Fwww.mianfeiwendang.com%2Fpic%2Fac887d106170ec3fa35dc6cd%2F1-810-jpg_6-1080-0-0-1080.jpg&refer=http%3A%2F%2Fwww.mianfeiwendang.com&app=2002&size=f9999,10000&q=a80&n=0&g=0n&fmt=auto?sec=1666412985&t=0034401be770555ef5b6412676d3c318" alt="龙格现象"><p><strong>针对龙格现象的一种解决办法</strong>：分段低次插值.</p><p><strong>分段线性插值</strong>：用分段直线插值.</p><p><strong>分段二次插值</strong>：用分段二次函数插值.</p><h3 id="newton插值法"><a class="markdownIt-Anchor" href="#newton插值法"></a> Newton插值法</h3>\begin{align}f(x) &= f(x_0)+f[x_0,x_1](x-x_0)\\&+f[x_0,x_1,x_2](x-x_0)(x-x_1)\cdots\\&+f[x_0,x_1,\cdots,x_{n-2},x_{n-1}](x-x_0)(x-x_1)\cdots(x-x_{n-3})(x-x_{n-2})\\&+f[x_0,x_1,\cdots,x_{n-1},x_{n}](x-x_0)(x-x_1)\cdots(x-x_{n-2})(x-x_{n-1})\end{align}<p><strong>差商</strong>：</p><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">称</mi><mi>f</mi><mo stretchy="false">[</mo><msub><mi>x</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">]</mo><mo>=</mo><mfrac><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo stretchy="false">)</mo></mrow><mrow><msub><mi>x</mi><mi>k</mi></msub><mo>−</mo><msub><mi>x</mi><mn>0</mn></msub></mrow></mfrac><mi mathvariant="normal">为</mi><mi mathvariant="normal">函</mi><mi mathvariant="normal">数</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi mathvariant="normal">关</mi><mi mathvariant="normal">于</mi><mi mathvariant="normal">点</mi><msub><mi>x</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mi>k</mi></msub><mi mathvariant="normal">的</mi><mi mathvariant="normal">一</mi><mi mathvariant="normal">阶</mi><mi mathvariant="normal">差</mi><mi mathvariant="normal">商</mi><mo stretchy="false">(</mo><mi mathvariant="normal">也</mi><mi mathvariant="normal">称</mi><mi mathvariant="normal">为</mi><mi mathvariant="normal">均</mi><mi mathvariant="normal">差</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">称 f[x_0,x_k]=\frac{f(x_k)-f(x_0)}{x_k-x_0}为函数f(x)关于点x_0,x_k的一阶差商(也称为均差)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord cjk_fallback">称</span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">[</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.4608599999999998em;vertical-align:-0.4508599999999999em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.01em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3487714285714287em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15122857142857138em;"><span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight"><span class="mord mathdefault mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.485em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em;">f</span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mathdefault mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3487714285714287em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15122857142857138em;"><span></span></span></span></span></span></span><span class="mclose mtight">)</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight" style="margin-right:0.10764em;">f</span><span class="mopen mtight">(</span><span class="mord mtight"><span class="mord mathdefault mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4508599999999999em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord cjk_fallback">为</span><span class="mord cjk_fallback">函</span><span class="mord cjk_fallback">数</span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span><span class="mord cjk_fallback">关</span><span class="mord cjk_fallback">于</span><span class="mord cjk_fallback">点</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">一</span><span class="mord cjk_fallback">阶</span><span class="mord cjk_fallback">差</span><span class="mord cjk_fallback">商</span><span class="mopen">(</span><span class="mord cjk_fallback">也</span><span class="mord cjk_fallback">称</span><span class="mord cjk_fallback">为</span><span class="mord cjk_fallback">均</span><span class="mord cjk_fallback">差</span><span class="mclose">)</span></span></span></span></p><p><strong>二阶差商</strong>：</p><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo stretchy="false">[</mo><msub><mi>x</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">]</mo><mo>=</mo><mfrac><mrow><mi>f</mi><mo stretchy="false">[</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo stretchy="false">]</mo><mo>−</mo><mi>f</mi><mo stretchy="false">[</mo><msub><mi>x</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">]</mo></mrow><mrow><msub><mi>x</mi><mn>2</mn></msub><mo>−</mo><msub><mi>x</mi><mn>0</mn></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">f[x_0,x_1,x_2]=\frac{f[x_1,x_2]-f[x_0,x_1]}{x_2-x_0}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">[</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.4550999999999998em;vertical-align:-0.44509999999999994em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.01em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mtight"><span class="mord mathdefault mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.485em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em;">f</span><span class="mopen mtight">[</span><span class="mord mtight"><span class="mord mathdefault mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mpunct mtight">,</span><span class="mord mtight"><span class="mord mathdefault mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mclose mtight">]</span><span class="mbin mtight">−</span><span class="mord mathdefault mtight" style="margin-right:0.10764em;">f</span><span class="mopen mtight">[</span><span class="mord mtight"><span class="mord mathdefault mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mpunct mtight">,</span><span class="mord mtight"><span class="mord mathdefault mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mclose mtight">]</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.44509999999999994em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></p><p><strong>k阶差商</strong>：</p><p>$ f[x_0,x_1,\cdots,x_k]=\frac{f[x_1,x_1,\cdots,x_k]-f[x_0,x_1,\cdots,x_{k-1}]}{x_k-x_0}$</p><h3 id="lagrange插值法与newton插值法的评价"><a class="markdownIt-Anchor" href="#lagrange插值法与newton插值法的评价"></a> Lagrange插值法与Newton插值法的评价</h3><p><strong>评价：Lagrange插值法所依赖的样点是固定的，而Newton插值法的样点可以增加，其插值函数也能相应的将改变融合到新函数中，但是两者均在高次插值时具有龙格现象。</strong></p><p><strong>同时，两种插值法仅仅是在数值上在插值节点处与被插函数有相等的函数值，但是其性质状态与被插函数可能区别很大。在实际问题中，插值函数不仅要求在函数值层面与被插值函数相同，甚至还要求在插值节点处与被插值函数具有相同的一阶或者二阶导数值。两种插值法不能满足这一条件。</strong></p><h2 id="hermite插值"><a class="markdownIt-Anchor" href="#hermite插值"></a> Hermite插值</h2><p><strong>原理：</strong><br /><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">设</mi><mi mathvariant="normal">函</mi><mi mathvariant="normal">数</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi mathvariant="normal">在</mi><mi mathvariant="normal">区</mi><mi mathvariant="normal">间</mi><mo stretchy="false">[</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">]</mo><mi mathvariant="normal">上</mi><mi mathvariant="normal">有</mi><mi>n</mi><mo>+</mo><mn>1</mn><mi mathvariant="normal">个</mi><mi mathvariant="normal">互</mi><mi mathvariant="normal">异</mi><mi mathvariant="normal">节</mi><mi mathvariant="normal">点</mi></mrow><annotation encoding="application/x-tex">设函数 f(x) 在区间 [a,b] 上有 n+1 个互异节点</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord cjk_fallback">设</span><span class="mord cjk_fallback">函</span><span class="mord cjk_fallback">数</span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span><span class="mord cjk_fallback">在</span><span class="mord cjk_fallback">区</span><span class="mord cjk_fallback">间</span><span class="mopen">[</span><span class="mord mathdefault">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span><span class="mclose">]</span><span class="mord cjk_fallback">上</span><span class="mord cjk_fallback">有</span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord cjk_fallback">个</span><span class="mord cjk_fallback">互</span><span class="mord cjk_fallback">异</span><span class="mord cjk_fallback">节</span><span class="mord cjk_fallback">点</span></span></span></span><br /><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>a</mi><mo>=</mo><msub><mi>x</mi><mn>0</mn></msub><mo><</mo><msub><mi>x</mi><mn>1</mn></msub><mo><</mo><msub><mi>x</mi><mn>2</mn></msub><mo><</mo><mo>⋯</mo><mo><</mo><msub><mi>x</mi><mi>n</mi></msub><mo>=</mo><mi>b</mi><mo separator="true">,</mo><mi mathvariant="normal">定</mi><mi mathvariant="normal">义</mi><mi mathvariant="normal">在</mi><mo stretchy="false">[</mo><mi>a</mi><mo separator="true">,</mo><mi>b</mi><mo stretchy="false">]</mo><mi mathvariant="normal">上</mi><mi mathvariant="normal">函</mi><mi mathvariant="normal">数</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi mathvariant="normal">满</mi><mi mathvariant="normal">足</mi><mi mathvariant="normal">：</mi></mrow><annotation encoding="application/x-tex">a=x_0<x_1<x_2<\cdots<x_n=b,定义在 [a,b] 上函数 f(x) 满足：</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">a</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">b</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord cjk_fallback">定</span><span class="mord cjk_fallback">义</span><span class="mord cjk_fallback">在</span><span class="mopen">[</span><span class="mord mathdefault">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span><span class="mclose">]</span><span class="mord cjk_fallback">上</span><span class="mord cjk_fallback">函</span><span class="mord cjk_fallback">数</span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span><span class="mord cjk_fallback">满</span><span class="mord cjk_fallback">足</span><span class="mord cjk_fallback">：</span></span></span></span></p><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>y</mi><mi>i</mi></msub><mo separator="true">,</mo><msup><mi>f</mi><mo mathvariant="normal">′</mo></msup><mo stretchy="false">(</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>y</mi><mi>i</mi><mo mathvariant="normal">′</mo></msubsup><mtext> </mtext><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x_i)=y_i,f'(x_i)=y_i' \space(i=0,1,2,\cdots,n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.051892em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.801892em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.051892em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace"> </span><span class="mopen">(</span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">n</span><span class="mclose">)</span></span></span></span></span></p><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">可</mi><mi mathvariant="normal">以</mi><mi mathvariant="normal">唯</mi><mi mathvariant="normal">一</mi><mi mathvariant="normal">确</mi><mi mathvariant="normal">定</mi><mi mathvariant="normal">一</mi><mi mathvariant="normal">个</mi><mi mathvariant="normal">次</mi><mi mathvariant="normal">数</mi><mi mathvariant="normal">不</mi><mi mathvariant="normal">超</mi><mi mathvariant="normal">过</mi><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mi mathvariant="normal">的</mi><mi mathvariant="normal">多</mi><mi mathvariant="normal">项</mi><mi mathvariant="normal">式</mi><msub><mi>H</mi><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mi>H</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi mathvariant="normal">满</mi><mi mathvariant="normal">足</mi><mi mathvariant="normal">：</mi></mrow><annotation encoding="application/x-tex">可以唯一确定一个次数不超过 2n+1 的多项式 H_{2n+1}=H(x) 满足：</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord cjk_fallback">可</span><span class="mord cjk_fallback">以</span><span class="mord cjk_fallback">唯</span><span class="mord cjk_fallback">一</span><span class="mord cjk_fallback">确</span><span class="mord cjk_fallback">定</span><span class="mord cjk_fallback">一</span><span class="mord cjk_fallback">个</span><span class="mord cjk_fallback">次</span><span class="mord cjk_fallback">数</span><span class="mord cjk_fallback">不</span><span class="mord cjk_fallback">超</span><span class="mord cjk_fallback">过</span><span class="mord">2</span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.891661em;vertical-align:-0.208331em;"></span><span class="mord">1</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">多</span><span class="mord cjk_fallback">项</span><span class="mord cjk_fallback">式</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.08125em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span><span class="mord cjk_fallback">满</span><span class="mord cjk_fallback">足</span><span class="mord cjk_fallback">：</span></span></span></span></p><p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>y</mi><mi>j</mi></msub><mo separator="true">,</mo><mtext> </mtext><msup><mi>H</mi><mo mathvariant="normal">′</mo></msup><mo stretchy="false">(</mo><msub><mi>x</mi><mi>j</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>m</mi><mi>j</mi></msub><mtext> </mtext><mo stretchy="false">(</mo><mi>j</mi><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mn>1</mn><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mi>n</mi><mo stretchy="false">)</mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">H(x_j)=y_j, \space H'(x_j)=m_j\space(j=0,1,\cdots,n).</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.036108em;vertical-align:-0.286108em;"></span><span class="mord mathdefault" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.088em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace"> </span><span class="mord"><span class="mord mathdefault" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.801892em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.036108em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace"> </span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">n</span><span class="mclose">)</span><span class="mord">.</span></span></span></span></span></p>]]></content>
<categories>
<category> 数学建模 </category>
</categories>
<tags>
<tag> 数学建模 </tag>
<tag> MATLAB </tag>
<tag> 插值算法 </tag>
</tags>
</entry>
</search>