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            <h1 id="seo-header">FFT的有趣应用：计算整数乘法</h1>
            
            
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                <p>今天刷知乎，看到了一个有趣的回答，来验证一下</p>
<h2 id="回答摘录"><a href="#回答摘录" class="headerlink" title="回答摘录"></a>回答摘录</h2><p><a target="_blank" rel="noopener" href="https://www.zhihu.com/question/543229591/answer/2760302705">怎么判断一个孩子有没有数学天赋？</a></p>
<p><a target="_blank" rel="noopener" href="https://www.zhihu.com/people/jian-gong-yu-xiao-sama">间宫羽咲sama</a>的回答以及一些有趣的评论</p>
<p>抖个机灵</p>
<p>考他34×77&#x3D;?</p>
<p>如果他不说解是什么，只告诉你解一定是一个有界的常数，那么他适合搞纯数学。</p>
<p>如果他偷偷拿出计算器算出结果，然后不给过程只说结果是2618，那么他适合搞工科。</p>
<p>如果他说对{4,3,0,0}和{7,7,0,0}做FFT得到{7,4-3i,1,4+3i}和{14,7-7i,0,7+7i}，相乘后逆FFT得到{28,49,21,0}，加权相加得到2618，并给你证明了用FFT算法计算大数乘法复杂度为O(n logn loglogn)优于平方复杂度的竖式乘法，那么恭喜你，你的孩子起码有八核。</p>
<p>评论1：34x77&#x3D;2618 如果结果是1010 0011 1010，那么恭喜你，你家的孩子一定是x16位以上计算机系统</p>
<p>评论2：等于8234，我确定数量级是对的。恭喜你的孩子可以当天体物理学家。</p>
<p>评论3：如果回答你2000，恭喜你这是个实验物理奇才</p>
<p>评论4：有人……啊，问了我这么一个问题，嗯，问我……34*77是多少啊，我说，你们都知道是多少，啊，但是，为什么我要说这个（喝水，呸），我们说，工作，不是简单的加法，而是什么？啊？我问问你们，是什么？对，乘法！如果说，一位同志奉献了34，啊，那么，那么，注意啊，我们77位同志，该是多少啊？这是非~常大的能量，这个就是乘法精神！我们……不好意思我接个电话</p>
<p>评论5：不知道34×77结果是多少，但是发现在（34-δφ）（77+77&#x2F;34δφ）对称性下结果不变，那就适合去当理论物理学家</p>
<p>评论6：如果他说2618小于2000，可以当个经济学家</p>
<p>评论7：如果他会从这条算式中萌生找出任意一个数的所有4n+1型素因子的想法的话，那他可能适合搞数论</p>
<p>评论8：闭上眼大概七八秒之后说2618，适合考试</p>
<p>评论9：如果说你家孩子发现了34和77之间的这种运算能满足交换律结合律，在R上封闭，甚至不仅仅34和77有这结论还能推广到任意其他的两个R上的元素上，还发现R上这种运算存在单位元，甚至这俩任意元素在另一种被记为“+”的算子下也满足交换律结合律居然有单位元有零元有逆元，甚至能证明“+”和你的“*”居然满足分配率，那恭喜你家的孩子居然独立证明了R是一个线性空间，日后一定是个代数天才</p>
<p>评论10：孩子找了几个老师家长同学都问了一下，然后算了个均值，那么她适合搞统计</p>
<p>评论11：如果是他用77÷3，算出约等于26，再添2个0，说明他适合当公务员</p>
<h2 id="配置环境"><a href="#配置环境" class="headerlink" title="配置环境"></a>配置环境</h2><ul>
<li>conda create -n scipy -c conda-forge scipy -y</li>
<li>conda activate scipy</li>
<li>conda install -c conda-forge ipykernel -y</li>
<li>python -m ipykernel install –user –name scipy</li>
</ul>
<h2 id="回答验证"><a href="#回答验证" class="headerlink" title="回答验证"></a>回答验证</h2><figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br></pre></td><td class="code"><pre><code class="hljs python"><span class="hljs-keyword">import</span> numpy <span class="hljs-keyword">as</span> np<br><span class="hljs-keyword">from</span> scipy.fftpack <span class="hljs-keyword">import</span> fft, ifft<br><span class="hljs-keyword">def</span> <span class="hljs-title function_">nextPower2</span>(<span class="hljs-params">L</span>):<br>    <span class="hljs-keyword">return</span> np.power(<span class="hljs-number">2</span>,np.ceil(np.log2(L))).astype(<span class="hljs-built_in">int</span>)<br><span class="hljs-keyword">def</span> <span class="hljs-title function_">int2Array</span>(<span class="hljs-params">n, size</span>):<br>    res = np.zeros(nextPower2(size), dtype = np.int8)<br>    i = <span class="hljs-number">0</span><br>    <span class="hljs-keyword">while</span> n &gt; <span class="hljs-number">0</span>:<br>        n, res[i] = <span class="hljs-built_in">divmod</span>(n, <span class="hljs-number">10</span>)<br>        i += <span class="hljs-number">1</span><br>    <span class="hljs-keyword">return</span> res<br><span class="hljs-keyword">def</span> <span class="hljs-title function_">array2Int</span>(<span class="hljs-params">arr</span>):<br>    <span class="hljs-keyword">return</span> np.dot(np.around(arr,<span class="hljs-number">0</span>),<span class="hljs-number">10</span>**np.arange(<span class="hljs-built_in">len</span>(arr)))<br></code></pre></td></tr></table></figure>

<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br></pre></td><td class="code"><pre><code class="hljs python">a = fft(int2Array(<span class="hljs-number">34</span>, <span class="hljs-number">4</span>))<br>b = fft(int2Array(<span class="hljs-number">77</span>, <span class="hljs-number">4</span>))<br>c = a * b<br>d = ifft(c)<br>e = array2Int(d)<br>a,b,c,d,e<br><span class="hljs-comment"># (array([7.-0.j, 4.-3.j, 1.-0.j, 4.+3.j]),</span><br><span class="hljs-comment">#  array([14.-0.j,  7.-7.j,  0.-0.j,  7.+7.j]),</span><br><span class="hljs-comment">#  array([98. -0.j,  7.-49.j,  0. -0.j,  7.+49.j]),</span><br><span class="hljs-comment">#  array([28.+0.j, 49.+0.j, 21.-0.j,  0.+0.j]),</span><br><span class="hljs-comment">#  (2618+0j))</span><br> <br>array2Int(ifft(fft(int2Array(<span class="hljs-number">457</span>, <span class="hljs-number">6</span>))*fft(int2Array(<span class="hljs-number">756</span>, <span class="hljs-number">6</span>))))<br><span class="hljs-comment"># (345492+0j)</span><br></code></pre></td></tr></table></figure>
                
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