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M=[123456789]M=\left[ \begin{matrix} 1&2&3\\ 4&5&6\\ 7&8&9\end{matrix} \right]M=⎣⎡147258369⎦⎤ 用 Mi,jM_{i,j}Mi,j 表示矩"><meta property="og:type" content="article"><meta property="og:title" content="矩阵快速幂运用"><meta property="og:url" content="https://schtonn.github.io/posts/matrix-pow/index.html"><meta property="og:site_name" content="Alex's Blog"><meta property="og:description" content="引入 想要了解矩阵快速幂，就不得不提到矩阵的概念。矩阵就像一个二维数组，存储了一组数据，如： M=[123456789]M=\left[ \begin{matrix} 1&2&3\\ 4&5&6\\ 7&8&9\end{matrix} \right]M=⎣⎡147258369⎦⎤ 用 Mi,jM_{i,j}Mi,j 表示矩"><meta property="og:locale" content="en_US"><meta property="article:published_time" content="2020-03-21T11:20:08.000Z"><meta property="article:modified_time" content="2020-03-29T07:59:04.002Z"><meta property="article:author" content="Alex"><meta property="article:tag" content="math"><meta name="twitter:card" content="summary"><link rel="canonical" href="https://schtonn.github.io/posts/matrix-pow/"><script 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Games</a></li></ul></nav></div></header><div class="back-to-top"><i class="fa fa-arrow-up"></i> <span>0%</span></div><div class="reading-progress-bar"></div><main class="main"><div class="main-inner"><div class="content-wrap"><div class="content post posts-expand"><article itemscope itemtype="http://schema.org/Article" class="post-block" lang="en"><link itemprop="mainEntityOfPage" href="https://schtonn.github.io/posts/matrix-pow/"><span hidden itemprop="author" itemscope itemtype="http://schema.org/Person"><meta itemprop="image" content="/images/avatar.gif"><meta itemprop="name" content="Alex"><meta itemprop="description" content=""></span><script></script><span hidden itemprop="publisher" itemscope itemtype="http://schema.org/Organization"><meta itemprop="name" content="Alex's Blog"></span><header class="post-header"><h1 class="post-title" itemprop="name headline"> 矩阵快速幂运用</h1><div class="post-meta"><span class="post-meta-item"><span class="post-meta-item-icon"><i class="fa fa-calendar-o"></i></span> <span class="post-meta-item-text">Posted on</span> <time title="Created: 2020-Mar-21 19:20:08" itemprop="dateCreated datePublished" datetime="2020-03-21T19:20:08+08:00">2020-Mar-21</time></span><span class="post-meta-item"><span class="post-meta-item-icon"><i class="fa fa-calendar-check-o"></i></span> <span class="post-meta-item-text">Edited on</span> <time title="Modified: 2020-Mar-29 15:59:04" itemprop="dateModified" datetime="2020-03-29T15:59:04+08:00">2020-Mar-29</time></span><span class="post-meta-item"><span class="post-meta-item-icon"><i class="fa fa-comment-o"></i></span> <span class="post-meta-item-text">Valine:</span><a title="valine" href="/posts/matrix-pow/#valine-comments" itemprop="discussionUrl"><span class="post-comments-count valine-comment-count" data-xid="/posts/matrix-pow/" itemprop="commentCount"></span></a></span></div></header><div class="post-body" itemprop="articleBody"><h3 id="引入"><a class="markdownIt-Anchor" href="#引入"></a> 引入</h3><p>想要了解矩阵快速幂，就不得不提到矩阵的概念。矩阵就像一个二维数组，存储了一组数据，如：</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>M</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"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>3</mn></mstyle></mtd></mtr><mtr><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></mtr><mtr><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></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">M=\left[ \begin{matrix} 1&2&3\\ 4&5&6\\ 7&8&9\end{matrix} \right]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathdefault" style="margin-right:.10903em">M</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:3.60004em;vertical-align:-1.55002em"></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:2.05002em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-4.05002em"><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:1.55002em"><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:2.05em"><span style="top:-4.21em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-3.0099999999999993em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-1.8099999999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">7</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em"><span></span></span></span></span></span><span class="arraycolsep" style="width:.5em"></span><span class="arraycolsep" style="width:.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em"><span style="top:-4.21em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.0099999999999993em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">5</span></span></span><span style="top:-1.8099999999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">8</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em"><span></span></span></span></span></span><span class="arraycolsep" style="width:.5em"></span><span class="arraycolsep" style="width:.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em"><span style="top:-4.21em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-3.0099999999999993em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">6</span></span></span><span style="top:-1.8099999999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">9</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em"><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:2.05002em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-4.05002em"><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:1.55002em"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>用<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>M</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">M_{i,j}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.969438em;vertical-align:-.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:.10903em">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.311664em"><span style="top:-2.5500000000000003em;margin-left:-.10903em;margin-right:.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="mpunct mtight">,</span><span class="mord mathdefault mtight" style="margin-right:.05724em">j</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.286108em"><span></span></span></span></span></span></span></span></span></span> 表示矩阵第<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.65952em;vertical-align:0"></span><span class="mord mathdefault">i</span></span></span></span> 行第<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.85396em;vertical-align:-.19444em"></span><span class="mord mathdefault" style="margin-right:.05724em">j</span></span></span></span> 列的数据，如<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>M</mi><mrow><mn>2</mn><mo separator="true">,</mo><mn>3</mn></mrow></msub><mo>=</mo><mn>6</mn></mrow><annotation encoding="application/x-tex">M_{2,3}=6</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.969438em;vertical-align:-.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:.10903em">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.301108em"><span style="top:-2.5500000000000003em;margin-left:-.10903em;margin-right:.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="mpunct mtight">,</span><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.64444em;vertical-align:0"></span><span class="mord">6</span></span></span></span>。</p><a id="more"></a><h3 id="矩阵运算"><a class="markdownIt-Anchor" href="#矩阵运算"></a> 矩阵运算</h3><h4 id="加法"><a class="markdownIt-Anchor" href="#加法"></a> 加法</h4><p>矩阵的加法和实数加法类似，要求运算的两个矩阵大小相等，将对应位置相加即可。</p><h4 id="减法"><a class="markdownIt-Anchor" href="#减法"></a> 减法</h4><p>与加法相同，对应位置相减即可。</p><h4 id="乘法"><a class="markdownIt-Anchor" href="#乘法"></a> 乘法</h4><p>矩阵乘法并非对应位置相乘。</p><p>若矩阵<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>A</mi><mo>×</mo><mi>B</mi><mo>=</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">A\times B=C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.76666em;vertical-align:-.08333em"></span><span class="mord mathdefault">A</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathdefault" style="margin-right:.05017em">B</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathdefault" style="margin-right:.07153em">C</span></span></span></span> 有意义，那么<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathdefault">A</span></span></span></span> 的列数要求与<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathdefault" style="margin-right:.05017em">B</span></span></span></span> 的行数相等。</p><p>若<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathdefault">A</span></span></span></span> 的列数与<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathdefault" style="margin-right:.05017em">B</span></span></span></span> 的行数等于<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.43056em;vertical-align:0"></span><span class="mord mathdefault">n</span></span></span></span>：</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>C</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>j</mi></mrow></msub><mo>=</mo><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>A</mi><mrow><mi>i</mi><mo separator="true">,</mo><mi>k</mi></mrow></msub><mo>×</mo><msub><mi>B</mi><mrow><mi>k</mi><mo separator="true">,</mo><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">C_{i,j}=\sum_{k=1}^{n}A_{i,k}\times B_{k,j}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.969438em;vertical-align:-.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:.07153em">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.311664em"><span style="top:-2.5500000000000003em;margin-left:-.07153em;margin-right:.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="mpunct mtight">,</span><span class="mord mathdefault mtight" style="margin-right:.05724em">j</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.9535100000000005em;vertical-align:-1.302113em"></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:-1.8478869999999998em;margin-left:0"><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:.03148em">k</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.0500049999999996em"><span class="pstrut" style="height:3.05em"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.300005em;margin-left:0"><span class="pstrut" style="height:3.05em"></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:1.302113em"><span></span></span></span></span></span><span class="mspace" style="margin-right:.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:.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:0;margin-right:.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="mpunct mtight">,</span><span class="mord mathdefault mtight" style="margin-right:.03148em">k</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.286108em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.969438em;vertical-align:-.286108em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:.05017em">B</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.3361079999999999em"><span style="top:-2.5500000000000003em;margin-left:-.05017em;margin-right:.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:.03148em">k</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight" style="margin-right:.05724em">j</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.286108em"><span></span></span></span></span></span></span></span></span></span></span></p><p>即结果第<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.65952em;vertical-align:0"></span><span class="mord mathdefault">i</span></span></span></span> 行第<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.85396em;vertical-align:-.19444em"></span><span class="mord mathdefault" style="margin-right:.05724em">j</span></span></span></span> 列的值为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathdefault">A</span></span></span></span> 的第<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.65952em;vertical-align:0"></span><span class="mord mathdefault">i</span></span></span></span> 行与<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathdefault" style="margin-right:.05017em">B</span></span></span></span> 的第<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.85396em;vertical-align:-.19444em"></span><span class="mord mathdefault" style="margin-right:.05724em">j</span></span></span></span> 列的<strong>对应项相乘后求和</strong>。</p><p>一个很直观的理解是，把矩阵<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathdefault">A</span></span></span></span> 放在结果的左侧， 矩阵<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathdefault" style="margin-right:.05017em">B</span></span></span></span> 放在结果的下方，将行与列延长，对应相乘后结果写在交点处。</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn mathvariant="bold">1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn mathvariant="bold">2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn mathvariant="bold">3</mn></mstyle></mtd></mtr><mtr><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></mtr><mtr><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></mtr></mtable><mo fence="true">]</mo></mrow><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mo>→</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>36</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo>↑</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo>↑</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow></mrow></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mspace linebreak="newline"></mspace><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><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"><mn mathvariant="bold">2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>3</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>4</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn mathvariant="bold">5</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>6</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>7</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn mathvariant="bold">8</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>9</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\left[ \begin{matrix} \bold{1}&\bold{2}&\bold{3}\\ 4&5&6\\ 7&8&9\end{matrix} \right]\left[ \begin{matrix} \rightarrow&36&&\\ &\uparrow&&\\ &\uparrow&&\end{matrix} \right]\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \left[ \begin{matrix} 1&\bold{2}&3\\ 4&\bold{5}&6\\ 7&\bold{8}&9\end{matrix} \right]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.60004em;vertical-align:-1.55002em"></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:2.05002em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-4.05002em"><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:1.55002em"><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:2.05em"><span style="top:-4.21em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathbf">1</span></span></span></span><span style="top:-3.0099999999999993em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-1.8099999999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">7</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em"><span></span></span></span></span></span><span class="arraycolsep" style="width:.5em"></span><span class="arraycolsep" style="width:.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em"><span style="top:-4.21em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathbf">2</span></span></span></span><span style="top:-3.0099999999999993em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">5</span></span></span><span style="top:-1.8099999999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">8</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em"><span></span></span></span></span></span><span class="arraycolsep" style="width:.5em"></span><span class="arraycolsep" style="width:.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em"><span style="top:-4.21em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathbf">3</span></span></span></span><span style="top:-3.0099999999999993em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">6</span></span></span><span style="top:-1.8099999999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">9</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em"><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:2.05002em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-4.05002em"><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:1.55002em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:.16666666666666666em"></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:2.05002em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-4.05002em"><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:1.55002em"><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:2.05em"><span style="top:-4.21em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mrel">→</span></span></span><span style="top:-3.0099999999999993em"><span class="pstrut" style="height:3em"></span><span class="mord"></span></span><span style="top:-1.8099999999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em"><span></span></span></span></span></span><span class="arraycolsep" style="width:.5em"></span><span class="arraycolsep" style="width:.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em"><span style="top:-4.21em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">3</span><span class="mord">6</span></span></span><span style="top:-3.0099999999999993em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mrel">↑</span></span></span><span style="top:-1.8099999999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mrel">↑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em"><span></span></span></span></span></span><span class="arraycolsep" style="width:.5em"></span><span class="arraycolsep" style="width:.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em"><span style="top:-4.05em"><span class="pstrut" style="height:2.84em"></span><span class="mord"></span></span><span style="top:-2.849999999999999em"><span class="pstrut" style="height:2.84em"></span><span class="mord"></span></span><span style="top:-1.6499999999999992em"><span class="pstrut" style="height:2.84em"></span><span class="mord"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em"><span></span></span></span></span></span><span class="arraycolsep" style="width:.5em"></span><span class="arraycolsep" style="width:.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em"><span style="top:-4.05em"><span class="pstrut" style="height:2.84em"></span><span class="mord"></span></span><span style="top:-2.849999999999999em"><span class="pstrut" style="height:2.84em"></span><span class="mord"></span></span><span style="top:-1.6499999999999992em"><span class="pstrut" style="height:2.84em"></span><span class="mord"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em"><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:2.05002em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-4.05002em"><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:1.55002em"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:.16666666666666666em"></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:3.60004em;vertical-align:-1.55002em"></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mspace" style="margin-right:.16666666666666666em"></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:2.05002em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-4.05002em"><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:1.55002em"><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:2.05em"><span style="top:-4.21em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-3.0099999999999993em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-1.8099999999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">7</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em"><span></span></span></span></span></span><span class="arraycolsep" style="width:.5em"></span><span class="arraycolsep" style="width:.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em"><span style="top:-4.21em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathbf">2</span></span></span></span><span style="top:-3.0099999999999993em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathbf">5</span></span></span></span><span style="top:-1.8099999999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathbf">8</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em"><span></span></span></span></span></span><span class="arraycolsep" style="width:.5em"></span><span class="arraycolsep" style="width:.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05em"><span style="top:-4.21em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-3.0099999999999993em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">6</span></span></span><span style="top:-1.8099999999999994em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">9</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.5500000000000007em"><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:2.05002em"><span style="top:-2.2500000000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-4.05002em"><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:1.55002em"><span></span></span></span></span></span></span></span></span></span></span></span></p><p>注意矩阵乘法满足结合律，但是<strong>不一定满足交换律</strong>，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>A</mi><mo>×</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A\times B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.76666em;vertical-align:-.08333em"></span><span class="mord mathdefault">A</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathdefault" style="margin-right:.05017em">B</span></span></span></span> 成立不一定代表<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>B</mi><mo>×</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">B\times A</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.76666em;vertical-align:-.08333em"></span><span class="mord mathdefault" style="margin-right:.05017em">B</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathdefault">A</span></span></span></span> 成立。</p><h4 id="除法"><a class="markdownIt-Anchor" href="#除法"></a> 除法</h4><p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.36687em;vertical-align:0"></span><span class="mrel">→</span></span></span></span>To be continued.</p><h3 id="快速幂"><a class="markdownIt-Anchor" href="#快速幂"></a> 快速幂</h3><p>快速幂可以<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>log</mi><mo></mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathdefault" style="margin-right:.02778em">O</span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:.01389em">g</span></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathdefault">n</span><span class="mclose">)</span></span></span></span> 求解<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>a</mi><mi>n</mi></msup></mrow><annotation encoding="application/x-tex">a^n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.664392em;vertical-align:0"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:.664392em"><span style="top:-3.063em;margin-right:.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>，对矩阵同样适用。<br> 代码：</p><figure class="highlight cpp"><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="keyword">int</span> ans=<span class="number">1</span>;</span><br><span class="line"><span class="keyword">while</span>(n){</span><br><span class="line"> <span class="keyword">if</span>(n&<span class="number">1</span>)ans*=a;</span><br><span class="line"> a*=a;</span><br><span class="line"> n>>=<span class="number">1</span>;</span><br><span class="line">}</span><br></pre></td></tr></table></figure><h3 id="矩阵快速幂"><a class="markdownIt-Anchor" href="#矩阵快速幂"></a> 矩阵快速幂</h3><p>可以使用重载运算符的办法直接进行矩阵快速幂，方法和普通快速幂一样，但是要注意乘法顺序。<code>ans</code>的初始值要特别注意。如果想让它充当1，则需要设置成单位矩阵<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathdefault" style="margin-right:.07847em">I</span></span></span></span>。</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><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"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo>⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo>⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><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>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mo>⋯</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></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"><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"><mo>⋱</mo></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><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</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><annotation encoding="application/x-tex">\left[\begin{matrix}1&0&0&\cdots&0\\0&1&0&\cdots&0\\0&0&1&\cdots&0\\\vdots&\vdots&\vdots&\ddots&0\\0&0&0&0&1\end{matrix}\right]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:6.66em;vertical-align:-3.08em"></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:3.554995em"><span style="top:-.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-1.9049750000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.5059750000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.1069750000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.7079750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.308975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.554995em"><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:3.050045em"><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:3.5800000000000005em"><span style="top:-6.427500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-5.2275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-4.0275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.1675000000000004em"><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:0;border-top-width:1.5em;bottom:0"></span></span></span></span><span style="top:-.9675em"><span class="pstrut" style="height:3.6875em"></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:3.08em"><span></span></span></span></span></span><span class="arraycolsep" style="width:.5em"></span><span class="arraycolsep" style="width:.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.427500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-5.2275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-4.0275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.1675000000000004em"><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:0;border-top-width:1.5em;bottom:0"></span></span></span></span><span style="top:-.9675em"><span class="pstrut" style="height:3.6875em"></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:3.08em"><span></span></span></span></span></span><span class="arraycolsep" style="width:.5em"></span><span class="arraycolsep" style="width:.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.427500000000001em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-5.2275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-4.0275em"><span class="pstrut" style="height:3.6875em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-2.1675000000000004em"><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:0;border-top-width:1.5em;bottom:0"></span></span></span></span><span style="top:-.9675em"><span class="pstrut" style="height:3.6875em"></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:3.08em"><span></span></span></span></span></span><span class="arraycolsep" style="width:.5em"></span><span class="arraycolsep" style="width:.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.240000000000001em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋯</span></span></span><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:-3.84em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋯</span></span></span><span style="top:-1.9800000000000006em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="minner">⋱</span></span></span><span style="top:-.78em"><span class="pstrut" style="height:3.5em"></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:3.08em"><span></span></span></span></span></span><span class="arraycolsep" style="width:.5em"></span><span class="arraycolsep" style="width:.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:3.5800000000000005em"><span style="top:-6.240000000000001em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-5.04em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.84em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-1.9800000000000006em"><span class="pstrut" style="height:3.5em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-.78em"><span class="pstrut" style="height:3.5em"></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:3.08em"><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:3.554995em"><span style="top:-.7499750000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-1.9049750000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.5059750000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.1069750000000003em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.7079750000000002em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.308975em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.554995em"><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:3.050045em"><span></span></span></span></span></span></span></span></span></span></span></span></p><h4 id="递推"><a class="markdownIt-Anchor" href="#递推"></a> 递推</h4><p>使用矩阵快速幂，可以用<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>log</mi><mo></mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(\log n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathdefault" style="margin-right:.02778em">O</span><span class="mopen">(</span><span class="mop">lo<span style="margin-right:.01389em">g</span></span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathdefault">n</span><span class="mclose">)</span></span></span></span> 的时间复杂度完成<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.43056em;vertical-align:0"></span><span class="mord mathdefault">n</span></span></span></span> 次递推关系，如斐波那契数列。矩阵可以通过递推关系推算出来。</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><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"><mn>1</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>×</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>F</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>F</mi><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mo>=</mo><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>F</mi><mi>n</mi></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>F</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mspace linebreak="newline"></mspace></mrow><annotation encoding="application/x-tex">\because\left[\begin{matrix}1&1\\0&1\end{matrix}\right] \times \left[\begin{matrix}F_{n-1}\\F_{n-2}\end{matrix}\right] = \left[\begin{matrix}F_n\\F_{n-1}\end{matrix}\right]\\</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.69224em;vertical-align:0"></span><span class="mrel amsrm">∵</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.40003em;vertical-align:-.95003em"></span><span class="minner"><span class="mopen delimcenter" style="top:0"><span class="delimsizing size3">[</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: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:.9500000000000004em"><span></span></span></span></span></span><span class="arraycolsep" style="width:.5em"></span><span class="arraycolsep" style="width:.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">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.9500000000000004em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:2.40003em;vertical-align:-.95003em"></span><span class="minner"><span class="mopen delimcenter" style="top:0"><span class="delimsizing size3">[</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:1.45em"><span style="top:-3.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:.13889em">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.301108em"><span style="top:-2.5500000000000003em;margin-left:-.13889em;margin-right:.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="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:.208331em"><span></span></span></span></span></span></span></span></span><span style="top:-2.4099999999999997em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:.13889em">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.301108em"><span style="top:-2.5500000000000003em;margin-left:-.13889em;margin-right:.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="mbin mtight">−</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:.208331em"><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:.9500000000000004em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0"><span class="delimsizing size3">]</span></span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.40003em;vertical-align:-.95003em"></span><span class="minner"><span class="mopen delimcenter" style="top:0"><span class="delimsizing size3">[</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:1.45em"><span style="top:-3.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:.13889em">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.151392em"><span style="top:-2.5500000000000003em;margin-left:-.13889em;margin-right:.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:.15em"><span></span></span></span></span></span></span></span></span><span style="top:-2.4099999999999997em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:.13889em">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.301108em"><span style="top:-2.5500000000000003em;margin-left:-.13889em;margin-right:.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="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:.208331em"><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:.9500000000000004em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0"><span class="delimsizing size3">]</span></span></span></span><span class="mspace newline"></span></span></span></span></p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>∴</mo><msub><mi>F</mi><mi>n</mi></msub><mo>=</mo><msup><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"><mn>1</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><mi>n</mi></msup><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></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\therefore F_n=\left[\begin{matrix}1&1\\0&1\end{matrix}\right]^n \times \left[\begin{matrix}1\\1\end{matrix}\right]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.69224em;vertical-align:0"></span><span class="mrel amsrm">∴</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.83333em;vertical-align:-.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:.13889em">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.151392em"><span style="top:-2.5500000000000003em;margin-left:-.13889em;margin-right:.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:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.454322em;vertical-align:-.95003em"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0"><span class="delimsizing size3">[</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: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:.9500000000000004em"><span></span></span></span></span></span><span class="arraycolsep" style="width:.5em"></span><span class="arraycolsep" style="width:.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">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.9500000000000004em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0"><span class="delimsizing size3">]</span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.504292em"><span style="top:-3.9029000000000003em;margin-right:.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="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">×</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:2.40003em;vertical-align:-.95003em"></span><span class="minner"><span class="mopen delimcenter" style="top:0"><span class="delimsizing size3">[</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: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">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.9500000000000004em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0"><span class="delimsizing size3">]</span></span></span></span></span></span></span></p><h4 id="推算矩阵"><a class="markdownIt-Anchor" href="#推算矩阵"></a> 推算矩阵</h4><p>矩阵由转移关系得来。以斐波那契数列为例：</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>∵</mo><msub><mi>F</mi><mi>n</mi></msub><mo>=</mo><mn>1</mn><msub><mi>F</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><mn>1</mn><msub><mi>F</mi><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\because F_n=1F_{n-1}+1F_{n-2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.69224em;vertical-align:0"></span><span class="mrel amsrm">∵</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.83333em;vertical-align:-.15em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:.13889em">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.151392em"><span style="top:-2.5500000000000003em;margin-left:-.13889em;margin-right:.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:.15em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.891661em;vertical-align:-.208331em"></span><span class="mord">1</span><span class="mord"><span class="mord mathdefault" style="margin-right:.13889em">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.301108em"><span style="top:-2.5500000000000003em;margin-left:-.13889em;margin-right:.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="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:.208331em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.891661em;vertical-align:-.208331em"></span><span class="mord">1</span><span class="mord"><span class="mord mathdefault" style="margin-right:.13889em">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.301108em"><span style="top:-2.5500000000000003em;margin-left:-.13889em;margin-right:.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="mbin mtight">−</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:.208331em"><span></span></span></span></span></span></span></span></span></span></span></p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>F</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn><msub><mi>F</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><mn>0</mn><msub><mi>F</mi><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub></mrow><annotation encoding="application/x-tex">F_{n-1}=1F_{n-1}+0F_{n-2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.891661em;vertical-align:-.208331em"></span><span class="mord"><span class="mord mathdefault" style="margin-right:.13889em">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.301108em"><span style="top:-2.5500000000000003em;margin-left:-.13889em;margin-right:.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="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:.208331em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.891661em;vertical-align:-.208331em"></span><span class="mord">1</span><span class="mord"><span class="mord mathdefault" style="margin-right:.13889em">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.301108em"><span style="top:-2.5500000000000003em;margin-left:-.13889em;margin-right:.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="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:.208331em"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.891661em;vertical-align:-.208331em"></span><span class="mord">0</span><span class="mord"><span class="mord mathdefault" style="margin-right:.13889em">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:.301108em"><span style="top:-2.5500000000000003em;margin-left:-.13889em;margin-right:.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="mbin mtight">−</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:.208331em"><span></span></span></span></span></span></span></span></span></span></span></p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>∴</mo><mi>M</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"><mn>1</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow></mrow><annotation encoding="application/x-tex">\therefore M=\left[\begin{matrix}1&1\\0&1\end{matrix}\right]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:.69224em;vertical-align:0"></span><span class="mrel amsrm">∴</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.68333em;vertical-align:0"></span><span class="mord mathdefault" style="margin-right:.10903em">M</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:2.40003em;vertical-align:-.95003em"></span><span class="minner"><span class="mopen delimcenter" style="top:0"><span class="delimsizing size3">[</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: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:.9500000000000004em"><span></span></span></span></span></span><span class="arraycolsep" style="width:.5em"></span><span class="arraycolsep" style="width:.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">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:.9500000000000004em"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0"><span class="delimsizing size3">]</span></span></span></span></span></span></span></p><p>其余类似。</p><h3 id="例题"><a class="markdownIt-Anchor" href="#例题"></a> 例题</h3><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>n</mi><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn><mi>F</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>3</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">F(n,1)=F(n-1,2)+2F(n-3,1)+5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathdefault" style="margin-right:.13889em">F</span><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathdefault" style="margin-right:.13889em">F</span><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord">2</span><span class="mclose">)</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord">2</span><span class="mord mathdefault" style="margin-right:.13889em">F</span><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.64444em;vertical-align:0"></span><span class="mord">5</span></span></span></span></span></p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>n</mi><mo separator="true">,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mn>3</mn><mi>F</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>3</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn><mi>F</mi><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>3</mn><mo separator="true">,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>+</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">F(n,2)=F(n-1,1)+3F(n-3,1)+2F(n-3,2)+3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathdefault" style="margin-right:.13889em">F</span><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord">2</span><span class="mclose">)</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathdefault" style="margin-right:.13889em">F</span><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord">3</span><span class="mord mathdefault" style="margin-right:.13889em">F</span><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord">2</span><span class="mord mathdefault" style="margin-right:.13889em">F</span><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">−</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord">2</span><span class="mclose">)</span><span class="mspace" style="margin-right:.2222222222222222em"></span><span class="mbin">+</span><span class="mspace" style="margin-right:.2222222222222222em"></span></span><span class="base"><span class="strut" style="height:.64444em;vertical-align:0"></span><span class="mord">3</span></span></span></span></span></p><p>已知</p><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mo separator="true">,</mo><mi>F</mi><mo stretchy="false">(</mo><mn>1</mn><mo separator="true">,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mn>3</mn><mo separator="true">,</mo><mi>F</mi><mo stretchy="false">(</mo><mn>2</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo separator="true">,</mo><mi>F</mi><mo stretchy="false">(</mo><mn>2</mn><mo separator="true">,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mn>4</mn><mo separator="true">,</mo><mi>F</mi><mo stretchy="false">(</mo><mn>3</mn><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>6</mn><mo separator="true">,</mo><mi>F</mi><mo stretchy="false">(</mo><mn>3</mn><mo separator="true">,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">F(1,1)=2,F(1,2)=3,F(2,1)=1,F(2,2)=4,F(3,1)=6,F(3,2)=5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathdefault" style="margin-right:.13889em">F</span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathdefault" style="margin-right:.13889em">F</span><span class="mopen">(</span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord">2</span><span class="mclose">)</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathdefault" style="margin-right:.13889em">F</span><span class="mopen">(</span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathdefault" style="margin-right:.13889em">F</span><span class="mopen">(</span><span class="mord">2</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord">2</span><span class="mclose">)</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord">4</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathdefault" style="margin-right:.13889em">F</span><span class="mopen">(</span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord">6</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord mathdefault" style="margin-right:.13889em">F</span><span class="mopen">(</span><span class="mord">3</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord">2</span><span class="mclose">)</span><span class="mspace" style="margin-right:.2777777777777778em"></span><span class="mrel">=</span><span class="mspace" style="margin-right:.2777777777777778em"></span></span><span class="base"><span class="strut" style="height:.64444em;vertical-align:0"></span><span class="mord">5</span></span></span></span></span></p><p>求<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>n</mi><mo separator="true">,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(n,1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathdefault" style="margin-right:.13889em">F</span><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span> 和<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>n</mi><mo separator="true">,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(n,2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-.25em"></span><span class="mord mathdefault" style="margin-right:.13889em">F</span><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mpunct">,</span><span class="mspace" style="margin-right:.16666666666666666em"></span><span class="mord">2</span><span class="mclose">)</span></span></span></span>。</p><h4 id="矩阵"><a class="markdownIt-Anchor" href="#矩阵"></a> 矩阵</h4><p class="katex-block"><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo fence="true">[</mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></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>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>5</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></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</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>3</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></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></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</mn></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</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></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</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><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>1</mn></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</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><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</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></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</mn></mstyle></mtd></mtr><mtr><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</mn></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</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr></mtable><mo fence="true">]</mo></mrow><annotation encoding="application/x-tex">\left[ \begin{matrix} 0&1&0&0&2&0&5\\ 1&0&0&0&3&2&3\\ 1&0&0&0&0&0&0\\ 0&1&0&0&0&0&0\\ 0&0&1&0&0&0&0\\ 0&0&0&1&0&0&0\\ 0&0&0&0&0&0&1\\ \end{matrix} \right]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:8.40804em;vertical-align:-3.9500599999999997em"></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:4.45798em"><span style="top:.1500399999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎣</span></span></span><span style="top:-1.0049600000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-1.6059600000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.2069600000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-2.8079600000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-3.4089600000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.00996em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-4.61096em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-5.21196em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎢</span></span></span><span style="top:-6.45798em"><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:3.9500599999999997em"><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:4.450000000000001em"><span style="top:-6.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-5.410000000000001em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-4.210000000000001em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-3.0100000000000002em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-1.8100000000000003em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-.6100000000000001em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:.5900000000000001em"><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:3.95em"><span></span></span></span></span></span><span class="arraycolsep" style="width:.5em"></span><span class="arraycolsep" style="width:.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.450000000000001em"><span style="top:-6.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-5.410000000000001em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-4.210000000000001em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.0100000000000002em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-1.8100000000000003em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-.6100000000000001em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:.5900000000000001em"><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:3.95em"><span></span></span></span></span></span><span class="arraycolsep" style="width:.5em"></span><span class="arraycolsep" style="width:.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.450000000000001em"><span style="top:-6.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-5.410000000000001em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-4.210000000000001em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.0100000000000002em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-1.8100000000000003em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-.6100000000000001em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:.5900000000000001em"><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:3.95em"><span></span></span></span></span></span><span class="arraycolsep" style="width:.5em"></span><span class="arraycolsep" style="width:.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.450000000000001em"><span style="top:-6.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-5.410000000000001em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-4.210000000000001em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.0100000000000002em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-1.8100000000000003em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-.6100000000000001em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:.5900000000000001em"><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:3.95em"><span></span></span></span></span></span><span class="arraycolsep" style="width:.5em"></span><span class="arraycolsep" style="width:.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.450000000000001em"><span style="top:-6.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-5.410000000000001em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-4.210000000000001em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.0100000000000002em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-1.8100000000000003em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-.6100000000000001em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:.5900000000000001em"><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:3.95em"><span></span></span></span></span></span><span class="arraycolsep" style="width:.5em"></span><span class="arraycolsep" style="width:.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.450000000000001em"><span style="top:-6.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-5.410000000000001em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-4.210000000000001em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.0100000000000002em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-1.8100000000000003em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-.6100000000000001em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:.5900000000000001em"><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:3.95em"><span></span></span></span></span></span><span class="arraycolsep" style="width:.5em"></span><span class="arraycolsep" style="width:.5em"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.450000000000001em"><span style="top:-6.61em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">5</span></span></span><span style="top:-5.410000000000001em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">3</span></span></span><span style="top:-4.210000000000001em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.0100000000000002em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-1.8100000000000003em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-.6100000000000001em"><span class="pstrut" style="height:3em"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:.5900000000000001em"><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:3.95em"><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:4.45798em"><span style="top:.1500399999999994em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎦</span></span></span><span style="top:-1.0049600000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-1.6059600000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.2069600000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-2.8079600000000005em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-3.4089600000000004em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.00996em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-4.61096em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-5.21196em"><span class="pstrut" style="height:3.1550000000000002em"></span><span class="delimsizinginner delim-size4"><span>⎥</span></span></span><span style="top:-6.45798em"><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:3.9500599999999997em"><span></span></span></span></span></span></span></span></span></span></span></span></p><h4 id="代码"><a class="markdownIt-Anchor" href="#代码"></a> 代码</h4><figure class="highlight cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br><span class="line">82</span><br><span class="line">83</span><br><span class="line">84</span><br><span class="line">85</span><br><span class="line">86</span><br><span class="line">87</span><br><span class="line">88</span><br><span class="line">89</span><br><span class="line">90</span><br><span class="line">91</span><br><span class="line">92</span><br><span class="line">93</span><br><span class="line">94</span><br><span class="line">95</span><br><span class="line">96</span><br><span class="line">97</span><br><span class="line">98</span><br><span class="line">99</span><br><span class="line">100</span><br><span class="line">101</span><br><span class="line">102</span><br><span class="line">103</span><br><span class="line">104</span><br><span class="line">105</span><br><span class="line">106</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="meta-keyword">include</span> <span class="meta-string">"bits/stdc++.h"</span></span></span><br><span class="line"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="built_in">std</span>;</span><br><span class="line"><span class="keyword">const</span> <span class="keyword">long</span> <span class="keyword">long</span> mod=<span class="number">99999999</span>;</span><br><span class="line"><span class="keyword">long</span> <span class="keyword">long</span> base[<span class="number">7</span>][<span class="number">7</span>]={</span><br><span class="line"> {<span class="number">0</span>,<span class="number">1</span>,<span class="number">0</span>,<span class="number">0</span>,<span class="number">2</span>,<span class="number">0</span>,<span class="number">5</span>},</span><br><span class="line"> {<span class="number">1</span>,<span class="number">0</span>,<span class="number">0</span>,<span class="number">0</span>,<span class="number">3</span>,<span class="number">2</span>,<span class="number">3</span>},</span><br><span class="line"> {<span class="number">1</span>,<span class="number">0</span>,<span class="number">0</span>,<span class="number">0</span>,<span class="number">0</span>,<span class="number">0</span>,<span class="number">0</span>},</span><br><span class="line"> {<span class="number">0</span>,<span class="number">1</span>,<span class="number">0</span>,<span class="number">0</span>,<span class="number">0</span>,<span class="number">0</span>,<span class="number">0</span>},</span><br><span class="line"> {<span class="number">0</span>,<span class="number">0</span>,<span class="number">1</span>,<span class="number">0</span>,<span class="number">0</span>,<span class="number">0</span>,<span class="number">0</span>},</span><br><span class="line"> {<span class="number">0</span>,<span class="number">0</span>,<span class="number">0</span>,<span class="number">1</span>,<span class="number">0</span>,<span class="number">0</span>,<span class="number">0</span>},</span><br><span class="line"> {<span class="number">0</span>,<span class="number">0</span>,<span class="number">0</span>,<span class="number">0</span>,<span class="number">0</span>,<span class="number">0</span>,<span class="number">1</span>},</span><br><span class="line">};</span><br><span class="line"><span class="class"><span class="keyword">struct</span> <span class="title">matrix</span>{</span></span><br><span class="line"> <span class="keyword">long</span> <span class="keyword">long</span> v[<span class="number">30</span>][<span class="number">30</span>];</span><br><span class="line"> <span class="keyword">long</span> <span class="keyword">long</span> x,y;</span><br><span class="line">};</span><br><span class="line">matrix <span class="keyword">operator</span>+(matrix a,matrix b){</span><br><span class="line"> matrix c;</span><br><span class="line"> <span class="keyword">if</span>(a.x!=b.x||a.y!=b.y)<span class="keyword">return</span> c;</span><br><span class="line"> <span class="keyword">long</span> <span class="keyword">long</span> x=a.x,y=a.y;</span><br><span class="line"> c.x=x;c.y=y;</span><br><span class="line"> <span class="keyword">for</span>(<span class="keyword">long</span> <span class="keyword">long</span> i=<span class="number">0</span>;i<x;i++){</span><br><span class="line"> <span class="keyword">for</span>(<span class="keyword">long</span> <span class="keyword">long</span> j=<span class="number">0</span>;j<y;j++){</span><br><span class="line"> c.v[i][j]=a.v[i][j]+b.v[i][j];</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">return</span> c;</span><br><span class="line">}</span><br><span class="line">matrix <span class="keyword">operator</span>-(matrix a,matrix b){</span><br><span class="line"> matrix c;</span><br><span class="line"> <span class="keyword">if</span>(a.x!=b.x||a.y!=b.y)<span class="keyword">return</span> c;</span><br><span class="line"> <span class="keyword">long</span> <span class="keyword">long</span> x=a.x,y=a.y;</span><br><span class="line"> c.x=x;c.y=y;</span><br><span class="line"> <span class="keyword">for</span>(<span class="keyword">long</span> <span class="keyword">long</span> i=<span class="number">0</span>;i<x;i++){</span><br><span class="line"> <span class="keyword">for</span>(<span class="keyword">long</span> <span class="keyword">long</span> j=<span class="number">0</span>;j<y;j++){</span><br><span class="line"> c.v[i][j]=a.v[i][j]-b.v[i][j];</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">return</span> c;</span><br><span class="line">}</span><br><span class="line">matrix <span class="keyword">operator</span>*(matrix a,matrix b){</span><br><span class="line"> matrix c;</span><br><span class="line"> <span class="keyword">if</span>(a.y!=b.x)<span class="keyword">return</span> c;</span><br><span class="line"> <span class="keyword">long</span> <span class="keyword">long</span> x=a.x,y=b.y,z=a.y;</span><br><span class="line"> c.x=x;c.y=y;</span><br><span class="line"> <span class="keyword">for</span>(<span class="keyword">long</span> <span class="keyword">long</span> i=<span class="number">0</span>;i<x;i++){</span><br><span class="line"> <span class="keyword">for</span>(<span class="keyword">long</span> <span class="keyword">long</span> j=<span class="number">0</span>;j<y;j++){</span><br><span class="line"> c.v[i][j]=<span class="number">0</span>;</span><br><span class="line"> <span class="keyword">for</span>(<span class="keyword">long</span> <span class="keyword">long</span> k=<span class="number">0</span>;k<z;k++){</span><br><span class="line"> c.v[i][j]=(c.v[i][j]+a.v[i][k]*b.v[k][j]+mod)%mod;</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">return</span> c;</span><br><span class="line">}</span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">init</span><span class="params">(matrix &a)</span></span>{</span><br><span class="line"> <span class="keyword">for</span>(<span class="keyword">long</span> <span class="keyword">long</span> i=<span class="number">0</span>;i<a.x;i++){</span><br><span class="line"> <span class="keyword">for</span>(<span class="keyword">long</span> <span class="keyword">long</span> j=<span class="number">0</span>;j<a.y;j++){</span><br><span class="line"> a.v[i][j]=base[i][j];</span><br><span class="line"> }</span><br><span class="line"> }</span><br><span class="line">}</span><br><span class="line">ostream& <span class="keyword">operator</span><<(ostream& ous,matrix a){</span><br><span class="line"> <span class="keyword">for</span>(<span class="keyword">long</span> <span class="keyword">long</span> i=<span class="number">0</span>;i<a.x;i++){</span><br><span class="line"> <span class="keyword">for</span>(<span class="keyword">long</span> <span class="keyword">long</span> j=<span class="number">0</span>;j<a.y;j++){</span><br><span class="line"> ous<<a.v[i][j]<<<span class="string">' '</span>;</span><br><span class="line"> }</span><br><span class="line"> ous<<<span class="built_in">endl</span>;</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">return</span> ous;</span><br><span class="line">}</span><br><span class="line"><span class="keyword">long</span> <span class="keyword">long</span> n;</span><br><span class="line">matrix& <span class="keyword">operator</span> *=(matrix &a,matrix n){</span><br><span class="line"> a=a*n;</span><br><span class="line"> <span class="keyword">return</span> a;</span><br><span class="line">}</span><br><span class="line">matrix a,ans;</span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">()</span></span>{</span><br><span class="line"> <span class="built_in">cin</span>>>n;</span><br><span class="line"> a.x=a.y=<span class="number">7</span>;</span><br><span class="line"> init(a);</span><br><span class="line"> ans.x=<span class="number">7</span>;ans.y=<span class="number">1</span>;</span><br><span class="line"> ans.v[<span class="number">0</span>][<span class="number">0</span>]=<span class="number">6</span>;</span><br><span class="line"> ans.v[<span class="number">1</span>][<span class="number">0</span>]=<span class="number">5</span>;</span><br><span class="line"> ans.v[<span class="number">2</span>][<span class="number">0</span>]=<span class="number">1</span>;</span><br><span class="line"> ans.v[<span class="number">3</span>][<span class="number">0</span>]=<span class="number">4</span>;</span><br><span class="line"> ans.v[<span class="number">4</span>][<span class="number">0</span>]=<span class="number">2</span>;</span><br><span class="line"> ans.v[<span class="number">5</span>][<span class="number">0</span>]=<span class="number">3</span>;</span><br><span class="line"> ans.v[<span class="number">6</span>][<span class="number">0</span>]=<span class="number">1</span>;</span><br><span class="line"> <span class="keyword">if</span>(n==<span class="number">1</span>){</span><br><span class="line"> <span class="built_in">cout</span><<<span class="number">2</span><<<span class="built_in">endl</span><<<span class="number">3</span>;</span><br><span class="line"> }<span class="keyword">else</span> <span class="keyword">if</span>(n==<span class="number">2</span>){</span><br><span class="line"> <span class="built_in">cout</span><<<span class="number">1</span><<<span class="built_in">endl</span><<<span class="number">4</span>;</span><br><span class="line"> }<span class="keyword">else</span> <span class="keyword">if</span>(n==<span class="number">3</span>){</span><br><span class="line"> <span class="built_in">cout</span><<<span class="number">6</span><<<span class="built_in">endl</span><<<span class="number">5</span>;</span><br><span class="line"> }</span><br><span class="line"> <span class="keyword">if</span>(n<<span class="number">4</span>)<span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line"> n-=<span class="number">3</span>;</span><br><span class="line"> <span class="keyword">while</span>(n){</span><br><span class="line"> <span class="keyword">if</span>(n&<span class="number">1</span>)ans=a*ans;</span><br><span class="line"> a*=a;</span><br><span class="line"> n>>=<span class="number">1</span>;</span><br><span class="line"> }</span><br><span class="line"> <span class="built_in">cout</span><<ans.v[<span class="number">0</span>][<span class="number">0</span>]<<<span class="built_in">endl</span><<ans.v[<span class="number">1</span>][<span class="number">0</span>];</span><br><span class="line"> <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">}</span><br></pre></td></tr></table></figure></div><div><ul class="post-copyright"><li class="post-copyright-author"> <strong>Post author:</strong> Alex</li><li class="post-copyright-link"> <strong>Post link:</strong> <a href="https://schtonn.github.io/posts/matrix-pow/" title="矩阵快速幂运用">https://schtonn.github.io/posts/matrix-pow/</a></li><li class="post-copyright-license"> <strong>Copyright Notice:</strong> All articles in this blog are licensed under<a href="https://creativecommons.org/licenses/by-nc-sa/4.0/" rel="noopener" target="_blank"><i class="fa fa-fw fa-creative-commons"></i> BY-NC-SA</a> unless stating additionally.</li></ul></div><div class="followme"><p>Welcome to my other publishing channels</p><div class="social-list"><div class="social-item"><a target="_blank" class="social-link" href="/images/wechat_channel.jpg"><span class="icon"><i class="fa fa-wechat"></i></span> <span class="label">WeChat</span></a></div><div class="social-item"><a target="_blank" class="social-link" href="/atom.xml"><span class="icon"><i class="fa fa-rss"></i></span> <span class="label">RSS</span></a></div></div></div><footer class="post-footer"><div 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href="#减法"><span class="nav-number">2.2.</span> <span class="nav-text">减法</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#乘法"><span class="nav-number">2.3.</span> <span class="nav-text">乘法</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#除法"><span class="nav-number">2.4.</span> <span class="nav-text">除法</span></a></li></ol></li><li class="nav-item nav-level-3"><a class="nav-link" href="#快速幂"><span class="nav-number">3.</span> <span class="nav-text">快速幂</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#矩阵快速幂"><span class="nav-number">4.</span> <span class="nav-text">矩阵快速幂</span></a><ol class="nav-child"><li class="nav-item nav-level-4"><a class="nav-link" href="#递推"><span class="nav-number">4.1.</span> <span class="nav-text">递推</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#推算矩阵"><span class="nav-number">4.2.</span> <span class="nav-text">推算矩阵</span></a></li></ol></li><li class="nav-item nav-level-3"><a class="nav-link" href="#例题"><span class="nav-number">5.</span> <span class="nav-text">例题</span></a><ol class="nav-child"><li class="nav-item nav-level-4"><a class="nav-link" href="#矩阵"><span class="nav-number">5.1.</span> <span class="nav-text">矩阵</span></a></li><li class="nav-item nav-level-4"><a class="nav-link" href="#代码"><span class="nav-number">5.2.</span> <span class="nav-text">代码</span></a></li></ol></li></ol></div></div><div class="site-overview-wrap sidebar-panel"><div class="site-author motion-element" itemprop="author" itemscope itemtype="http://schema.org/Person"><p class="site-author-name" itemprop="name">Alex</p><div class="site-description" itemprop="description"></div></div><div class="site-state-wrap motion-element"><nav class="site-state"><div class="site-state-item site-state-posts"> <a href="/archives"><span class="site-state-item-count">10</span> <span class="site-state-item-name">posts</span></a></div><div class="site-state-item 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