{"id":50,"date":"2025-03-06T20:12:38","date_gmt":"2025-03-06T19:12:38","guid":{"rendered":"http:\/\/blog.mathyuan.com\/?p=50"},"modified":"2025-07-06T13:11:16","modified_gmt":"2025-07-06T11:11:16","slug":"%e6%95%b0%e5%80%bc%e5%88%86%e6%9e%90%e7%ac%94%e8%ae%b0","status":"publish","type":"post","link":"https:\/\/blog.mathyuan.com\/?p=50","title":{"rendered":"\u6570\u503c\u5206\u6790\u7b14\u8bb0"},"content":{"rendered":"\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>\u6b64\u7bc7\u5b8c\u5168\u662f\u4e3a\u4e86\u901a\u8fc7\u8003\u8bd5. \u6211\u4e0d\u559c\u6b22\u6570\u503c\u5206\u6790, \u4f46\u662f\u4e0d\u5f97\u4e0d\u5b66. <\/p>\n<\/blockquote>\n\n\n\n<p>\u603b\u7684\u6765\u8bf4\uff0c\u8fd9\u95e8\u8bfe\u4f3c\u4e4e\u5305\u62ec\u4ee5\u4e0b\u51e0\u4e2a\u677f\u5757<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u51fd\u6570\u7684\u903c\u8fd1\uff08\u591a\u9879\u5f0f\u63d2\u503c\uff09<\/li>\n\n\n\n<li>\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u6570\u503c\u89e3\uff08Gauss\u6d88\u5143\u6cd5\uff0c\u590d\u77e9\u9635\u7684QR\u5206\u89e3\uff0cQR\u65b9\u6cd5\u548c\u6700\u5c0f\u4e8c\u4e58\uff09<\/li>\n\n\n\n<li>Fourier\u53d8\u6362\uff08\u79bb\u6563\/\u5feb\u901f\uff0c\u4e09\u89d2\u591a\u9879\u5f0f\uff09<\/li>\n\n\n\n<li>\u6570\u503c\u5fae\u5206<\/li>\n\n\n\n<li>\u6570\u503c\u79ef\u5206\uff08Newton-Cotes\u516c\u5f0f\uff0cGauss\u516c\u5f0f\u548c\u6b63\u4ea4\u57fa\uff09<\/li>\n\n\n\n<li>\u975e\u7ebf\u6027\u65b9\u7a0b\u6570\u503c\u89e3<\/li>\n\n\n\n<li>\u5e38\u5fae\u5206\u65b9\u7a0b\u7684\u6570\u503c\u89e3\uff08\u663e\u5f0fEuler\u6cd5\uff0c\u9690\u5f0fEuler\u6cd5\uff0c\u6539\u8fdb\u663e\u5f0fEuler\u6cd5\u548cRunge-Kutta\u6cd5\uff09<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">\u51fd\u6570\u7684\u903c\u8fd1<\/h2>\n\n\n\n<p>\u51fd\u6570\u7684\u903c\u8fd1\u901a\u8fc7\u63d2\u503c\u591a\u9879\u5f0f\u6765\u5b9e\u73b0\uff0c\u7ed9\u5b9a\u70b9\u96c6\u53ca\u5176\u4e0a\u7684\u51fd\u6570\u503c\uff0c\u8f93\u51fa\u662f\u63d2\u503c\u591a\u9879\u5f0f\uff0c\u6709\u4e24\u79cd\u8868\u793a\u63d2\u503c\u591a\u9879\u5f0f\u7684\u529e\u6cd5<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Newton\u5dee\u5546\u8868\u793a\uff1a\u8ba1\u7b97\u5dee\u5546$[x_0,\\ldots,x_n]f$\uff0c\u7136\u540e\u4ee5\u8fd9\u4e9b\u5dee\u5546\u4f5c\u4e3a\u7cfb\u6570\uff0c\u4e58\u4ee5$(x-x_0)\\cdots(x-x_{n-1})$;<\/li>\n\n\n\n<li>Langrange\u8868\u793a\uff1a\u8003\u8651Lagrange\u591a\u9879\u5f0f$l_i$\uff0c\u7136\u540e\u76f4\u63a5\u7528$f(x_i)l_i(x)$\u7684\u6c42\u548c\u4f5c\u4e3a\u63d2\u503c\u591a\u9879\u5f0f\u3002<\/li>\n<\/ol>\n\n\n\n<p>\u903c\u8fd1\u95ee\u9898\u7684\u597d\u574f\u7531\u6536\u655b\u901f\u5ea6\u6765\u523b\u753b\uff0c\u6536\u655b\u901f\u5ea6\u7684\u5b9a\u4e49\u662f\uff1a\u5f53\u70b9\u96c6\u7684\u6700\u5927\u533a\u95f4\u957f\u5ea6$h$\u8d8b\u8fd1\u4e8e0\u65f6\uff0c\u7edd\u5bf9\u8bef\u5dee\u7684\u6536\u655b\u9636\u6570$h^p$\u3002\u56e0\u6b64\uff0c\u5982\u679c\u5e0c\u671b\u5f97\u5230\u6536\u655b\u9636\u6570\uff0c\u53ef\u4ee5\u7ed8\u5236loglog\u56fe\u770b\u56fe\u50cf\u7684\u659c\u7387\u662f\u591a\u5c11\u3002<\/p>\n\n\n\n<p>\u53e6\u5916\u8fd8\u53ef\u4ee5\u8003\u8651\u5728\u6bcf\u4e2a\u533a\u95f4\u5185\uff0c\u7528\u6837\u6761\u51fd\u6570\u6765\u903c\u8fd1\u3002\u533a\u95f4\u548c\u533a\u95f4\u4e4b\u95f4\u7684\u5149\u6ed1\u6027\u662f\u7ed9\u5b9a\u7684$m$\uff0c\u533a\u95f4\u5185\u90e8\u90fd\u662f\u67d0\u4e2a\u6b21\u6570\u7684\u3001\u7528\u7b49\u8ddd\u70b9\u6765\u63d2\u503c\u7684\u591a\u9879\u5f0f\u3002\u7528\u5f97\u6bd4\u8f83\u591a\u7684\u6709\u4e24\u4e2a\u60c5\u5f62\uff1a\u5e38\u6570\u6837\u6761\u51fd\u6570\u3001\u7ebf\u6027\u6837\u6761\u51fd\u6570\u3002\u7ebf\u6027\u6837\u6761\u51fd\u6570\u7684\u57fa\u672c\u5355\u4f4d\u662f$b_i$\uff0c\u5b83\u662f\u5728\u7b2ci\u4e2a\u533a\u95f4\u5185\u4ece0\u52301\uff0c\u5728\u7b2ci+1\u4e2a\u533a\u95f4\u5185\u4ece1\u52300\uff0c\u5728\u5176\u5b83\u533a\u95f4\u4e0a\u4e3a0\u7684\u5206\u6bb5\u7ebf\u6027\u51fd\u6570\u3002\u5229\u7528\u8fd9\u4e2a\u57fa\u672c\u5355\u4f4d\u51fd\u6570\uff0c\u4e00\u822c\u51fd\u6570$f$\u7684\u7ebf\u6027\u6837\u6761\u51fd\u6570\u662f$$f_\\mathrm{int}=\\sum_{i=0}^N f(x_i)b_i(x).$$<\/p>\n\n\n\n<p>\u53e6\u5916\uff0c\u7528\u8fde\u7eed\u7ebf\u6027\u6837\u6761\u51fd\u6570\u6765\u63d2\u503c\uff0c\u5176\u8bef\u5dee\u8fd8\u548c\u8282\u70b9\u7684\u9009\u53d6\u6709\u5173\uff1a\u5982\u679c\u8282\u70b9\u9009\u53d6\u5f97\u6bd4\u8f83\u597d\uff0c\u90a3\u4e48\u8bef\u5dee\u53ef\u80fd\u6bd4\u8f83\u5c0f\u3002\u90a3\u4e48\u6211\u4eec\u53ef\u4ee5\u5f52\u7eb3\u5730\u9010\u6e10\u589e\u52a0\u8282\u70b9\uff0c\u4f7f\u5f97\u6bcf\u4e00\u6b65\u589e\u52a0\u8282\u70b9\u65f6\uff0c\u8bef\u5dee\u90fd\u53d8\u5316\u6bd4\u8f83\u5c0f\uff0c\u6700\u7ec8\u5728\u7b2cN\u6b65\u65f6\u505c\u6b62\uff0c\u8fd9\u79cd\u529e\u6cd5\u53eb\u505a<strong>\u81ea\u9002\u5e94\u7ec6\u5316\uff08adaptive refinement\uff09\u3002<\/strong>\u8003\u8651\u67d0\u4e2a\u5c0f\u533a\u95f4$\\tau$\uff0c\u5b9a\u4e49\u5176\u4e2d\u7684\u53d8\u5316\u4e3a$d_\\tau=\\max |u_n(M_\\tau)-u_{n-1}(M_\\tau)|$\uff0c\u5176\u4e2d$M_\\tau$\u662f\u533a\u95f4$\\tau$\u7684\u4e2d\u95f4\u70b9\u3002\u5047\u8bbe\u8fd9\u4e9b\u53d8\u5316\u7684\u6700\u5927\u503c\u662f$D$\uff0c\u90a3\u4e48\u5bf9\u4e8e\u7ed9\u5b9a$\\alpha&gt;0$\uff0c\u53ef\u4ee5\u628a\u6240\u6709\u6ee1\u8db3$d_\\tau\\geq\\alpha D$\u7684\u533a\u95f4\u4e2d\u70b9\u90fd\u6536\u96c6\u8d77\u6765\uff0c\u548c\u539f\u6765\u5c31\u6709\u7684\u533a\u95f4\u7aef\u70b9\u4e00\u8d77\uff0c\u5171\u540c\u6784\u6210\u65b0\u7684\u652f\u6491\u70b9\u96c6\u5408\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u6570\u503c\u89e3<\/h2>\n\n\n\n<p>\u8003\u8651\u590d\u6570\u57df\u5185\u7684\u7ebf\u6027\u65b9\u7a0b\u7ec4$Ax=b$\uff0c\u5e0c\u671b\u5c06$A$\u4f5cQR\u5206\u89e3\uff0c\u7136\u540e\u5c31\u53ef\u4ee5\u901a\u8fc7\u5de6\u4e58$Q*$\u6765\u62b5\u6d88Q\uff0c\u5f97\u5230\u65b9\u7a0b$Rx=y$\uff0c\u6700\u540e\u4ece\u540e\u5f80\u524d\u9010\u6e10\u6c42\u89e3\u3002\u8fd9\u91cc$Q$\u662f\u4e00\u4e2a\u9149\u77e9\u9635\uff0c\u800c\u4e00\u822c\u7684\u6d88\u5143\u5f97\u5230\u7684\u77e9\u9635\u662f\u4e0b\u4e09\u89d2\u77e9\u9635\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u8003\u8651Householder\u53d8\u6362\uff1a\u9010\u6b65\u8ba1\u7b97$P_j=I-2ww^*$\uff0c\u5176\u4e2d$P_j A$\u7684\u7b2c\u4e00\u5217\u53ea\u6709\u7b2c\u4e00\u4e2a\u5143\u7d20\u4e0d\u4e3a0. \u8ba1\u7b97\u6b21\u6570\u662f$4\/3n^3+O(n^2)$<\/p>\n\n\n\n<p>\u5f53\u7136\uff0c\u624b\u7b97QR\u5206\u89e3\u4f9d\u65e7\u53ef\u4ee5\u7528Gram-Schmidt\u6b63\u4ea4\u5316\u7684\u601d\u8def\uff0c\u9010\u4e2a\u8ba1\u7b97\u3002\u8fd9\u91cc\u6ce8\u610f\u590d\u6570\u77e9\u9635\u548c\uff08\u5217\uff09\u5411\u91cf\u7684\u5185\u79ef\u662f\u81ea\u5df1\u548c\u5171\u8f6d\u8f6c\u7f6e\u7684\u4e58\u79ef\u3002<\/p>\n\n\n\n<h1 class=\"wp-block-heading\">\u5feb\u901fFourier\u53d8\u6362<\/h1>\n\n\n\n<h3 class=\"wp-block-heading\">1. \u79bb\u6563\u5085\u91cc\u53f6\u53d8\u6362<\/h3>\n\n\n\n<p>\u9996\u5148, \u6211\u4eec\u8003\u8651\u4e00\u822c\u7684\u79bb\u6563\u5085\u91cc\u53f6\u53d8\u6362: \u4e00\u822c\u5730, \u590d\u53d8\u51fd\u6570$f:\\C\\rightarrow \\C$\u7684\u5085\u91cc\u53f6\u53d8\u6362\u4e3a<br>$$<br>F(t):=\\int_\\R f(s)\\exp(-is\\cdot t)\\mathrm{d}s<br>$$<br>\u5176\u53cd\u53d8\u6362\u4e3a<br>$$<br>f(t)=\\frac{1}{2\\pi} \\int_\\R F(s)\\exp(is\\cdot t)\\mathrm{d}t<br>$$<br>\u7ed9\u5b9a\u95f4\u9694$T_s$, \u5982\u679c\u6211\u4eec\u63a5\u53d7Dirac\u51fd\u6570$\\delta:\\R\\rightarrow \\R$\u7684\u5b58\u5728\u6027, \u90a3\u4e48$f$\u7684\u79bb\u6563\u5316 (\u91c7\u6837\u51fd\u6570) \u53ef\u4ee5\u7528Dirac\u51fd\u6570\u8868\u793a\u4e3a<br>$$<br>f_s(t)=\\sum_{n=-\\infty}^\\infty f(t)\\delta(t-nT_s)<br>$$<br>\u4e8e\u662f$f_s$\u7684Fourier\u53d8\u6362 (\u4ee3\u5165\u5b9a\u4e49\u516c\u5f0f) \u5c31\u662f<br>$$<br>F_s(t)= \\sum_{n=-\\infty}^\\infty f(nT_s)\\exp(-inT_s\\cdot t).<br>$$<br>\u7136\u800c, \u5728\u73b0\u5b9e\u751f\u6d3b\u4e2d, \u5982\u679c\u6211\u4eec\u5e76\u4e0d\u77e5\u9053$f$\u672c\u8eab\u7684\u8868\u8fbe\u5f0f, \u6211\u4eec\u6ca1\u6709\u529e\u6cd5\u7cbe\u786e\u8ba1\u7b97\u51fa\u65e0\u9650\u591a\u4e2a$f(nT_s)$. \u6240\u4ee5\u6211\u4eec\u7684\u601d\u8def\u662f: \u5728\u6709\u9650\u591a\u4e2a\u7b49\u8ddd\u7684\u70b9${0,T_s,2T_s,\\ldots,(N-1)T_s}$\u4e0a\u8fdb\u884c\u91c7\u6837, \u4ece\u800c\u5f97\u5230\u79bb\u6563\u7684Fourier\u53d8\u6362<br>$$<br>F[k]=\\sum_{n=0}^{N-1} f(nT_s)\\exp(-\\frac{2k\\pi i}{N}\\cdot n),\\quad 0\\leq k\\leq N-1.<br>$$<br>\u8fd9\u4e2a\u7b97\u6cd5\u7684\u590d\u6742\u5ea6\u662f$O(N^2)$.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">2. \u5feb\u901fFourier\u53d8\u6362<\/h2>\n\n\n\n<p>\u4ece\u79bb\u6563Fourier\u53d8\u6362\u4e2d, \u6211\u4eec\u4e8b\u5b9e\u4e0a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u6ca1\u6709\u7528\u5230\u6c42\u548c\u4e2d\u5355\u4f4d\u6839\u7684\u6027\u8d28;<\/li>\n\n\n\n<li>\u6307\u6570\u51fd\u6570\u7684\u5468\u671f\u6027;<\/li>\n<\/ul>\n\n\n\n<p>\u4e8e\u662f, \u4e3a\u4e86\u51cf\u5c11\u7b97\u6cd5\u7684\u590d\u6742\u6027, \u6211\u4eec\u5e0c\u671b\u5c06\u8fd9\u4e9b\u6027\u8d28\u7528\u4e0a, \u6700\u7ec8\u5c06\u7b97\u6cd5\u590d\u6742\u5ea6\u51cf\u5c11\u5230$O(N\\log N)$. \u6ce8\u610f\u5230\u79bb\u6563Fourier\u53d8\u6362\u4e2d, \u5bf9\u4e8e\u6bcf\u4e00\u4e2a$k$, Fourier\u7cfb\u6570$F[k]$\u90fd\u662f\u4e00\u4e2a$N$\u9879\u7684\u6307\u6570\u6c42\u548c. \u9996\u5148, \u6211\u4eec\u56fa\u5b9a$k$, \u8bb0<br>$$<br>f[n]:=f(nT_s),\\quad 0\\leq n\\leq N-1.<br>$$<br>\u5047\u8bbe\u6211\u4eec\u5df2\u7ecf\u5f52\u7eb3\u5730\u6709\u4e86\u8ba1\u7b97: \u7ed9\u5b9a\u95f4\u9694$T_s$\u548c\u91c7\u6837\u503c$(F[0],F[1],\\ldots,F[N-1])$ \u7684&#8221;\u9ed1\u7bb1&#8221;<br>$$<br>\\mathrm{FFT}_k(T_s; (F[0],F[1],\\ldots,F[N-1]) ) \\in \\C, \\quad 0\\leq k\\leq N-1.<br>$$<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u5982\u679c$N$\u200b\u662f\u5076\u6570, \u90a3\u4e48\u53ef\u4ee5\u5c06\u6c42\u548c\u5206\u62c6\u6210\u5947\u6570\u9879\u6c42\u548c\u4e0e\u5076\u6570\u9879\u6c42\u548c<\/li>\n<\/ol>\n\n\n\n<p>$$\\begin{aligned}<br>F[k] &amp;=\\sum^{N\/2-1}_{m=0} f[2m]\\exp(-\\frac{2k\\pi i}{N}\\cdot 2m) +\\sum^{N\/2-1}_{m=0} f[2m+1]\\exp(-\\frac{2k\\pi i}{N}\\cdot (2m+1))\\\\<br>&amp;=\\mathrm{FFT}_k(2T_s; x[0],x[2],\\ldots,x[N-2])+\\xi_N^{-k}\\mathrm{FFT}_k(2T_s; x[1],x[3],\\ldots, x[N-1])\\\\<br>&amp;=:O[k]+\\xi_N^{-k} E[k].<br>\\end{aligned}$$<\/p>\n\n\n\n<p>\u73b0\u5728\u6539\u53d8$k$: \u6ce8\u610f\u5230$O[k]=O[k+\\frac{N}{2}]$, $E[k]=E[k+\\frac{N}{2}]$, \u4e8e\u662f\u7531$N$\u6b21\u5355\u4f4d\u6839\u7684\u6027\u8d28<br>$$F[k+\\frac{N}{2}]=O(k)-\\xi_N^k E[k].$$<\/p>\n\n\n\n<ol start=\"2\" class=\"wp-block-list\">\n<li>\u5982\u679c$N$\u662f\u5947\u6570, \u90a3\u4e48\u53ef\u4ee5\u5c06\u9996\u9879\u5ffd\u7565, \u5176\u4f59\u9879\u6570\u90fd\u662f$\\xi^{-k}_N$\u200b\u7684\u500d\u6570. \u91cd\u65b0\u63d0\u53d6\u51fa\u6765\u5373\u53ef, \u548c\u5076\u6570\u9879\u53ea\u5dee\u4e00\u6b21\u52a0\u6cd5\u548c\u4e00\u6b21\u4e58\u6cd5.<\/li>\n<\/ol>\n\n\n\n<p><strong>\u4f8b.<\/strong> \u5047\u8bbe\u6211\u4eec\u7ed9\u5b9a\u4e868\u4e2a\u91c7\u6837, \u9700\u8981\u8ba1\u7b97\u5176Fourier\u7cfb\u6570$F[0],\\ldots, F[7]$\u200b\u200b, \u90a3\u4e48\u6211\u4eec\u53ea\u9700\u8981\u77e5\u9053<br>$$<br>E(0),\\ldots,E(3), O(0),\\ldots,O(3)<br>$$<br>\u5171\u8ba18\u4e2a&#8221;\u56db\u91c7\u6837\u70b9FFT&#8221; \u7684\u8ba1\u7b97, \u518d\u6709$2N=16$\u6b21\u52a0\u6cd5\u548c\u4e58\u6cd5\u7ec4\u5408. \u56db\u91c7\u6837\u70b9FFT\u53c8\u53ef\u4ee5\u53d8\u62104\u4e2a\u4e8c\u91c7\u6837\u70b9+8\u6b21\u52a0\u6cd5\u548c\u4e58\u6cd5\u7ec4\u5408. \u800c\u6bcf\u4e2a\u4e8c\u91c7\u6837\u70b9\u90fd\u9700\u8981\u8ba1\u7b972\u6b21, \u4e8e\u662f\u6bcf\u4e2a\u56db\u91c7\u6837\u70b9\u8ba1\u7b9716\u6b21,<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p>\u73b0\u5728\u6765\u8ba1\u7b97\u4e0a\u9762\u8fd9\u4e2a\u9012\u5f52\u7b97\u6cd5\u7684\u590d\u6742\u5ea6: \u5047\u8bbe$N$\u4e2a\u70b9\u7684\u590d\u6742\u5ea6\u662f$T(N)$, \u90a3\u4e48<br>$$<br>T(N)\\sim2T(\\frac{N}{2})+N\\leq \\cdots\\sim N\\sum\\frac<br>1 n\\sim N\\log N.<br>$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u6570\u503c\u5fae\u5206<\/h2>\n\n\n\n<p>\u8ba1\u7b97\u51fd\u6570\u7684\u6570\u503c\u5fae\u5206\u53ef\u4ee5\u7528\u63d2\u503c\u591a\u9879\u5f0f\u7684\u5fae\u5206\u6765\u5b9e\u73b0\u3002\u7279\u522b\u5730\uff0c\u5982\u679c\u6211\u4eec\u8981\u8ba1\u7b97\u5fae\u5206\u5728$x_0$\u5904\u7684\u53d6\u503c\uff0c\u6211\u4eec\u53ef\u4ee5\u76f4\u63a5\u53d6$x_0$\u4e3a\u7b2c\u4e00\u4e2a\u63d2\u503c\u70b9\uff0c\u7136\u540e\u540e\u9762\u7684\u5fae\u5206\u8ba1\u7b97\u5c31\u4f1a\u66f4\u7b80\u4fbf\uff1a\u56e0\u4e3aNewton\u63d2\u503c\u591a\u9879\u5f0f\u4e2d\u6bcf\u4e00\u9879\u90fd\u5305\u542b$x-x_0$\uff0c\u6c42\u5bfc\u5728$x_0$\u5904\u8ba1\u7b97\u66f4\u4e3a\u7b80\u4fbf\u3002\u8fd9\u65f6\uff0c\u8bef\u5dee\u662f$\\Vert f^{(n+1)}\\Vert_\\infty\\cdot |I|^n$.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u6570\u503c\u79ef\u5206<\/h2>\n\n\n\n<p>\u7ed9\u5b9a\u51fd\u6570$f$\u548c\u6743\u91cd\u51fd\u6570$w$\uff0c\u5e0c\u671b\u901a\u8fc7\u51fd\u6570$f$\u5728\u67d0\u4e9b\u70b9\u4e0a\u7684\u53d6\u503c\uff0c\u903c\u8fd1\u79ef\u5206$\\int wf$\u3002\u9996\u5148\u5b9a\u4e49\u67d0\u4e2a\u6c42\u79ef\u516c\u5f0f\u7684\u8bef\u5dee\u9636\u6570\u4e3a\uff1a\u6700\u5927\u7684\u4fdd\u8bc1\u591a\u9879\u5f0f\u79ef\u5206\u4e0d\u53d8\u7684\u591a\u9879\u5f0f\u9636\u6570\u3002\u6bd4\u5982\u67d0\u4e2a\u6c42\u79ef\u516c\u5f0f\u5bf9\u6b21\u6570$\\leq 3$\u7684\u591a\u9879\u5f0f\u90fd\u7cbe\u786e\uff0c\u4f46\u5bf9\u591a\u9879\u5f0f\u6b21\u6570=4\u65f6\u4e0d\u6210\u7acb\uff0c\u90a3\u4e48\u8fd9\u4e2a\u6c42\u79ef\u516c\u5f0f\u5c31\u662f3\u6b21\u7684\u3002\u4e00\u822c\u800c\u8a00\uff0c\u6c42\u79ef\u516c\u5f0f\u5177\u6709\u4e0b\u9762\u7684\u7ed3\u6784<\/p>\n\n\n\n<p>$$I(f)=\\sum_{j=1}^n \\lambda_j f(x_j).$$<\/p>\n\n\n\n<p>\u6709\u4e00\u4e9b\u5e38\u89c1\u7684\u6c42\u79ef\u516c\u5f0f\u5217\u5728\u4e0b\u9762<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\uff08Riemann\u516c\u5f0f\uff0c0\u9636\uff09$$f(x_j)(x_{j+1}-x_j)$$<\/li>\n\n\n\n<li>\uff08Midpoint\u516c\u5f0f\uff0c1\u9636\uff09$$f(\\frac{x_j+x_{j+1}}{2})(x_{j+1}-x_j)$$<\/li>\n\n\n\n<li>\uff08Trapezoidal\u516c\u5f0f\uff0c1\u9636\uff09$$\\frac{f(x_j)+f(x_{j+1})}{2}\\cdot (x_{j+1}-x_j)$$<\/li>\n\n\n\n<li>\uff08Simpson\u516c\u5f0f\uff0c3\u9636\uff09$$\\frac 1 6 (f(x_j)+4f(\\frac{x_j+x_{j+1}}{2}) +f(x_{j+1}))(x_{j+1}-x_j)$$<\/li>\n<\/ol>\n\n\n\n<p>\u4e8b\u5b9e\u4e0a\uff0c\u5982\u679c\u8282\u70b9\u53d6\u5b9a\uff0c\u90a3\u4e48\u6709\u4e00\u79cd\u552f\u4e00\u7684\u65b9\u5f0f\u9009\u53d6\u53c2\u6570$\\lambda_j$\uff0c\u4f7f\u5f97\u9636\u6570\u6700\u5927\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Newton-Cotes\u516c\u5f0f\uff1a\u7b49\u8ddd\u8282\u70b9\uff0c$w(x)=1$<\/li>\n\n\n\n<li>Gauss\u516c\u5f0f\uff1a\u5e0c\u671b\u80fd\u5bfb\u627e\u5408\u9002\u7684\u8282\u70b9\uff0c\u4f7f\u5f97\u5176\u6784\u9020\u7684\u79ef\u5206\u516c\u5f0f\u8bef\u5dee\u6700\u5c0f\u3002\u7ed9\u5b9a\u63d2\u503c\u70b9\u6570\u91cfn\uff0c\u8fd9n\u4e2a\u70b9\u5e94\u8be5\u9009\u53d6\u6210\u201cn\u6b21\u6b63\u4ea4\u591a\u9879\u5f0f\u201d\u7684\u57fa\u5e95\uff0c\u5176\u4e2d\u8fd9\u91cc\u6b63\u4ea4\u7684\u542b\u4e49\u662f\u548c$w$\u6709\u5173\u7684\uff0c\u56e0\u4e3a\u5185\u79ef\u662f$\\int (fg)(x) w(x)\\mathrm{d}x$.\n<ul class=\"wp-block-list\">\n<li>\u5982\u679c$w=1$\uff0c\u6b63\u4ea4\u591a\u9879\u5f0f\u5c31\u662fLegendre\u591a\u9879\u5f0f$$\\pi_n(x)=\\sqrt{n+\\frac 1 2+\\frac{1}{2^n n!} \\frac{\\mathrm{d}^n}{\\mathrm{d}x^n} ((x^2-1)^n).$$<\/li>\n\n\n\n<li>\u5982\u679c$w=\\frac{1}{\\sqrt{1-x^2}}$\uff0c\u6b63\u4ea4\u591a\u9879\u5f0f\u5c31\u662fChebyshev\u591a\u9879\u5f0f$$Q_n(x)=\\cos(n\\arccos x).$$<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">\u6570\u503c\u6c42\u89e3\u975e\u7ebf\u6027\u65b9\u7a0b<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u4e8c\u5206\u6cd5\uff1a\u65e0\u9700\u8d58\u8ff0\uff0c\u6ce8\u610f\u8bef\u5dee\u4e3a$\\epsilon$\u7684\u8ba1\u7b97\u6b21\u6570\uff1b<\/li>\n\n\n\n<li>Newton\u6cd5\uff1a\u7ebf\u6027\u5316+\u8fed\u4ee3\uff0c\u5148\u9009\u62e9\u521d\u503c\uff0c\u7136\u540e\u7528\u7ebf\u6027\u5316\u903c\u8fd1$f$\uff0c$x_{n+1}$\u5c31\u662f\u7ebf\u6027\u5316f\u540e\u7684\u7ebf\u6027\u65b9\u7a0b\u7684\u89e3\u3002\u8be5\u65b9\u6cd5\u4f9d\u8d56\u4e8e\u521d\u503c\u7684\u9009\u53d6\uff1a\u5982\u679c\u521d\u503c\u7684\u9009\u62e9\u4e0d\u597d\uff0c\u53ef\u80fd\u65b9\u6cd5\u4e0d\u6536\u655b\u3002\u4e3a\u6b64\uff0c\u6211\u4eec\u4e3a\u4e86\u4fdd\u8bc1\u6536\u655b\u6027\uff0c\u9700\u8981\u9996\u5148<br>1. \u901a\u8fc7\u4e8c\u5206\u6cd5\u627e\u5230\u4e00\u4e2a\u89e3\u5b58\u5728\u7684\u533a\u95f4\uff1b<br>2. \u9009\u62e9\u8fd9\u4e2a\u533a\u95f4\uff0c\u4f7f\u5f97\u533a\u95f4\u957f\u5ea6\u5145\u5206\u5c0f\uff0c\u5e76\u4e14\u5728\u8fd9\u4e2a\u533a\u95f4\u4e0a\uff0c$f&#8217;$\u7684\u4e0b\u754c\u548c$f&#8221;$\u7684\u4e0a\u754c\u4e5f\u6709\u63a7\u5236\uff1b<\/li>\n<\/ul>\n\n\n\n<p>\u5177\u4f53\u800c\u8a00\uff0c\u4e0b\u9762\u7684\u5b9a\u7406\u548c\u63a8\u8bba\u4fdd\u8bc1\u7684\uff1a<\/p>\n\n\n\n<div class='latex_thm'><span class='latex_thm_h'>\u5b9a\u7406 1<\/span><span class='latex_thm_h'>.<\/span> \u5982\u679cx\u662f\u65b9\u7a0b\u7684\u89e3\uff0c\u5b9a\u4e49<\/p>\n\n\n\n<p>$$M_\\varepsilon=\\max_{s,t\\in B_\\varepsilon(x^*)}|\\frac{f&#8221;(s)}{2f'(t)}|$$ <\/p>\n\n\n\n<p>\u5982\u679c$2\\varepsilon M_{\\varepsilon}&lt;1$\uff0c\u90a3\u4e48\u5bf9\u4e8e\u4efb\u610f$B_\\varepsilon(x^*)$\u4e2d\u7684\u521d\u503c\uff0cNewton\u6cd5\u90fd\u6536\u655b\u3002<\/div>\n\n\n\n<p>\u7136\u800c\uff0c\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u6211\u4eec\u4e0d\u53ef\u80fd\u4e00\u5f00\u59cb\u5c31\u77e5\u9053\u96f6\u70b9\u5728\u54ea\uff0c\u4ece\u800c\u4e5f\u6ca1\u529e\u6cd5\u786e\u5b9a\u4ee5\u96f6\u70b9\u4e3a\u4e2d\u5fc3\u7684\u90bb\u57df\u3002\u4e3a\u6b64\u6211\u4eec\u53ea\u80fd\u6269\u5927\u533a\u95f4\u957f\u5ea6\uff1a<\/p>\n\n\n\n<div class='latex_cor'><span class='latex_cor_h'>\u63a8\u8bba 2<\/span><span class='latex_cor_h'>.<\/span> \u5982\u679c\u96f6\u70b9\u5728$a$\u7684$3\\delta$\u90bb\u57df\u5185\uff0c\u4e14<\/p>\n\n\n\n<p>$$4\\delta \\max_{s,t\\in B_{3\\delta} (a) } |\\frac{f&#8221;(s)}{2f'(t)}|&lt;1$$<\/p>\n\n\n\n<p>\u90a3\u4e48\u5bf9\u4e8e$a$\u7684$\\delta$\u90bb\u57df\u5185\u7684\u4efb\u610f\u70b9\uff0c\u8be5\u70b9\u4e3a\u521d\u503c\u7684Newton\u6cd5\u6536\u655b\u3002<\/div>\n\n\n\n<p>\u5229\u7528\u8fd9\u4e2a\u63a8\u8bba\uff0c\u5728\u5b9e\u9645\u64cd\u4f5c\u4e2d\u5c31\u53ef\u4ee5\u7528\u725b\u987f\u6cd5\u6765\u786e\u5b9a\u96f6\u70b9\uff1a<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u627e\u5230\u4e00\u4e2a\u533a\u95f4\uff0c\u4f7f\u5f97\u96f6\u70b9\u5728\u8fd9\u4e2a\u533a\u95f4\u5185\uff1b<\/li>\n\n\n\n<li>\u7b97\u51fa\u8fd9\u4e2a\u5927\u533a\u95f4\u7684M\uff1b<\/li>\n\n\n\n<li>\u901a\u8fc7\u4e8c\u5206\u6cd5\u7f29\u5c0f\u5927\u533a\u95f4\uff0c\u4f7f\u5f97$4\\delta M&lt;1$, \u5176\u4e2d$3\\delta$\u662f\u5c0f\u533a\u95f4\u7684\u533a\u95f4\u534a\u5f84\uff1b<\/li>\n\n\n\n<li>\u4efb\u9009\u5c0f\u533a\u95f4\u5185\u7684\u70b9\u4f5c\u4e3a\u521d\u503c\u70b9\uff0c\u8fdb\u884c\u725b\u987f\u6cd5\u3002<\/li>\n<\/ol>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u5b9a\u70b9\u6cd5\uff1a\u5c06$f(x)=0$\u8f6c\u6362\u4e3a\u9002\u5f53\u7684\u4e0d\u52a8\u70b9\u65b9\u7a0b$x=\\varphi(x)$\uff0c\u7136\u540e\u7528$\\varphi$\u6765\u8fed\u4ee3\u3002\u4e8b\u5b9e\u4e0a\uff0c\u725b\u987f\u6cd5\u4e5f\u662f\u4e00\u79cd\u5b9a\u70b9\u6cd5\uff0c\u5b83\u5c06\u65b9\u7a0b\u8f6c\u5316\u4e3a\u4e86$x=x-\\frac{f(x)}{f'(x)}$.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">\u6570\u503c\u6c42\u89e3ODE<\/h2>\n\n\n\n<p>\u4e00\u822c\u6c42\u89e3\u5b9e\u6570\u57df\u4e0a\u7684\u4e00\u9636ODE\u3002\u5982\u679c\u53f3\u4fa7\u7684\u51fd\u6570$f$\u662fLipschitz\u8fde\u7eed\u7684\uff0c\u90a3\u4e48\u89e3\u5b58\u5728\u4e14\u552f\u4e00\uff0c\u4e14\u8fde\u7eed\u4f9d\u8d56\u4e8e\u521d\u503c$x_0,y_0$\u3002\u6709\u4e86\u8fd9\u4e2a\u7406\u8bba\u4fdd\u8bc1\uff0c\u53ef\u4ee5\u8003\u8651\u5355\u6b65\u6cd5\u7684\u6570\u503c\u6c42\u89e3\u3002\u4e00\u4e2a\u5355\u6b65\u6cd5\u7531\u51fd\u6570\\[\\Phi:[a,b]\\times\\mathbb{R}\\times\\mathbb{R}^+\\rightarrow \\mathbb{R}\\]\u51b3\u5b9a\uff1a\u8be5\u51fd\u6570\u7ed9\u51fa\u7684\u8fed\u4ee3\u662f\\[y_{\\mathrm{next}}=y+h\\Phi(x,y,h).\\]\u5047\u8bbeODE\u7684\u7cbe\u786e\u89e3\u662f$\\kappa$\uff0c\u90a3\u4e48\u5355\u6b65\u6cd5\u7684\u5c40\u90e8\u79bb\u6563\u8bef\u5dee\u5b9a\u4e49\u4e3a\\[\\begin{aligned}T(x,y,h)&amp;:=\\frac 1 h (y_{\\mathrm{next}}-\\kappa(x+h))\\\\&amp;=\\Phi(x,y,h)-\\frac{1}{h}\\left(\\kappa(x+h)-\\kappa(x)\\right). \\end{aligned}\\] \u5982\u679c\u5c40\u90e8\u79bb\u6563\u8bef\u5dee\u5728$h$\u8d8b\u8fd1\u4e8e0\u65f6\uff0c\u80fd\u591f\u4e00\u81f4\u6536\u655b\u52300\uff0c\u90a3\u4e48\u5c31\u79f0\u8fd9\u4e2a\u5355\u6b65\u6cd5\u662f\u4e00\u81f4\u7684\u3002\u8fd9\u4e2a\u5c40\u90e8\u79bb\u6563\u8bef\u5dee\u4e00\u81f4\u6536\u655b\u52300\u7684\u9636\u6570\u5c31\u662f\u5355\u6b65\u6cd5\u7684\u9636\u6570\u3002<\/p>\n\n\n\n<p>\u6839\u636e\u9009\u62e9\u51fd\u6570$\\Phi$\u7684\u4e0d\u540c\uff0c\u6709\u5982\u4e0b\u51e0\u79cd\u4e0d\u540c\u7684\u5355\u6b65\u6cd5<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u663e\u5f0f\u6b27\u62c9\u6cd5\uff1a$\\Phi(x,y,h)=f(x,y)$;<\/li>\n\n\n\n<li>\u9690\u5f0f\u6b27\u62c9\u6cd5\uff1a\\[\\Phi(x,y,h)=f(x+h,y(x+h))=f(x+h,y_{\\mathrm{next}})\\]<\/li>\n\n\n\n<li>Crank-Nicolson\u65b9\u6cd5\uff1a\\[\\Phi(x,y,h)=\\frac 1 2 (f(x,y)+f(x+h,y(x+h)) \\] \u6216\u8005\\[\\Phi(x,y,h)=\\frac 1 2 (f(x,y)+f(x+h, x+hf(x,y) ) \\] \u5176\u4e2d\uff0c\u7b2c\u4e00\u4e2a\u662f\u9690\u5f0f\u7684\uff0c\u7b2c\u4e8c\u4e2a\u662f\u663e\u5f0f\u7684\uff1b<\/li>\n\n\n\n<li>\u6539\u8fdb\u6b27\u62c9\u6cd5\uff082\u9636\uff09\uff1a\u4ee4$k_1=f(x,y)$\uff0c$k_2=f(x+h\/2, y+1\/2 hk_1))$\uff0c\u7136\u540e\u5b9a\u4e49\\[\\Phi(x,y,h)=k_2\\] \u5c31\u662f\u5148\u7528\u663e\u5f0f\u6b27\u62c9\u6cd5\u8d70\u534a\u6b65\uff0c\u518d\u7528\u8d70\u7684\u534a\u6b65\u8d70\u5230\u7684\u70b9\u5bf9\u5e94\u7684$f$\u4f5c\u4e3a\u201c\u901f\u5ea6\u201d\uff0c\u4ece\u539f\u5730\u8d70\u4e00\u6b65\uff1b<\/li>\n\n\n\n<li>Runge-Kutta\u6cd5\uff1a\u4e8b\u5b9e\u4e0a\u662f\u6539\u8fdb\u6b27\u62c9\u6cd5\u7684\u63a8\u5e7f\u2014\u2014\u8d70\u4e24\u6b21\u63a8\u5e7f\u4e3a\u8d70s\u6b21\uff0c\u5b9a\u4e49\\[k_1=f(x,y),\\]\\[k_s=f(x+\\mu_s h,y+h\\sum_{j=1}^{s-1} \\mu_{s,j} k_j)\\]\u7136\u540e\u5b9a\u4e49\\[y_{\\mathrm{next}} = y+h\\sum_{s=1}^r \\alpha_s k_s\\]\u5e76\u4e14\u53d6\u53c2\u6570$\\sum \\alpha_s=1$\u6765\u4fdd\u8bc1\u4e00\u81f4\u6027.<\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<!--nextpage-->\n\n\n\n<p><span id=\"contents\"  style=\"text-align:center; font-size:18px; font-variant:small-caps;display:block;\">\u76ee\u5f55<\/span><br \/>\n          <span id=\"sec:content\"><a href=\"#contents\">\u76ee\u5f55<\/a><\/span><br \/><span>&#x00A0;1.&#x00A0;&#x00A0;<a href=\"#sec:1\">Lagrangian\u63d2\u503c\u591a\u9879\u5f0f<\/a><\/span><br \/><span>&#x00A0;2.&#x00A0;&#x00A0;<a href=\"#sec:2\">Lagrangian\u63d2\u503c\u591a\u9879\u5f0f\u7684\u8ba1\u7b97<\/a><\/span><br \/><span>&#x00A0;3.&#x00A0;&#x00A0;<a href=\"#sec:3\">\u8bef\u5dee\u4f30\u8ba1<\/a><\/span><br \/><span>&#x00A0;4.&#x00A0;&#x00A0;<a href=\"#sec:4\">\u6837\u6761\u51fd\u6570<\/a><\/span><br \/><\/p>\n\n\n\n<p><span class=\"latex_section\">1.&#x00A0;Lagrangian\u63d2\u503c\u591a\u9879\u5f0f<a id=\"sec:1\"><\/a><\/span>\n\u7ed9\u5b9a\u95ed\u533a\u95f4$I=[a,b]$, \u5b9a\u4e49\u5728$I$\u4e0a\u7684\u51fd\u6570$f$, \u4ee5\u53ca$I$\u7684\u4e00\u4e2a\u5206\u5272\\[\\{x_0, x_1,\\ldots, x_n\\},\\]\u5176\u4e2d$x_0,\\ldots,x_n$\u662f\u4e92\u5f02\u7684. Lagrangian\u591a\u9879\u5f0f\u6240\u505a\u5230\u7684, \u5c31\u662f\u7ed9\u51fa\u4e00\u4e2a$n$\u6b21\u7684\u591a\u9879\u5f0f, \u4f7f\u5f97\u8be5\u591a\u9879\u5f0f\u5728\u8fd9\u4e9b\u7ed9\u5b9a\u7684\u70b9$\\{x_i\\}$\u5904\u548c$f$\u76f8\u7b49. \u5173\u4e8e\u8fd9\u6837\u7684\u591a\u9879\u5f0f, \u6211\u4eec\u6709\u5982\u4e0b\u552f\u4e00\u6027\u5b9a\u7406:<br><div class='latex_thm'><span class='latex_thm_h'>\u5b9a\u7406 3<\/span><span class='latex_thm_h'>.<\/span> \u5b58\u5728\u552f\u4e00\u7684\u591a\u9879\u5f0f\u51fd\u6570$p_n\\in\\mathbb{P}_n$, \u4f7f\u5f97\u5bf9\u4efb\u610f$i$, \u90fd\u6709\\[p_n(x_i)=f(x_i).\\]More precisely, \u8fd9\u4e00\u591a\u9879\u5f0f\u51fd\u6570\u53ef\u4ee5\u8868\u793a\u4e3a<br>\\[p_n(x)=\\sum_{i=1}^n f(x_i)L_i(x),\\]\u5176\u4e2d\\[L_i(x)=\\prod_{j\\neq i}\\left(\\frac{x-x_j}{x_i-x_j}\\right).\\]<br><\/div><br>\u5982\u679c\u8bb0\\(\\pi(x)=\\prod_i(x-x_i)\\), \u90a3\u4e48\\(L_i(x)=\\frac{\\pi(x)}{\\pi'(x_i)(x-x_i)}\\).<span class=\"latex_section\">2.&#x00A0;Lagrangian\u63d2\u503c\u591a\u9879\u5f0f\u7684\u8ba1\u7b97<a id=\"sec:2\"><\/a><\/span>\n\u4efb\u4f55\u9886\u57df\u7684experts\u90fd\u4f1a\u6ce8\u610f\u5230, \u5f88\u96be\u76f4\u63a5\u7528\u5b9a\u4e49\u7684\u65b9\u5f0f\u6765\u8ba1\u7b97\u4e00\u4e2a\u5bf9\u8c61. \u5728\u5b9e\u9645\u5e94\u7528\u4e2d, \u4e00\u822c\u7528\u5f52\u7eb3\u6cd5\u8ba1\u7b97Lagrangian\u63d2\u503c\u591a\u9879\u5f0f:<br><ol><br><li>$p_0(x)=f(x_0)$\u662f\u5b8c\u5b8c\u5168\u5168\u7684\u5e38\u6570\u51fd\u6570;<br><\/li><li>\u6ce8\u610f\u5230$p_k-p_{k-1}$\u5728$x_0,\\ldots,x_{k-1}$\u5904vanishes, \u90a3\u4e48\u53ef\u4ee5\u4ee4\\[(p_k-p_{k-1})(x)=f[x_0,\\ldots,x_k](x-x_0)\\cdots(x-x_{k-1}),\\]\u5176\u4e2d, $f[x_0,\\ldots,x_k]\\in\\mathbb{R}$\u662f$p_k(x)$\u7684$x^k$-\u7cfb\u6570.<br><\/li><\/ol><br>\u8fd9\u6837\u4e00\u6765,<br>\\[\\begin{aligned}<br>p_n(x)&amp;=p_{n-1}(x)+f[x_0,\\ldots,x_n](x-x_0)\\cdots(x-x_{n-1})\\\\<br>&amp;=\\cdots=p_0(x)+\\sum_{i=1}^n f[x_0,\\ldots,x_i](x-x_0)\\cdots(x-x_{i-1}).<br>\\end{aligned}\\]<br>\u800c\u5bf9\u4e8e\u7cfb\u6570$f[x_0,\\ldots,x_k]$, \u6709\u5982\u4e0bNewton\u516c\u5f0f\u5f52\u7eb3<br>\\[\\boxed{f[x_0,\\ldots,x_k]=\\frac{f[x_1,\\ldots,x_k]-f[x_0,\\ldots,x_{k-1}]}{x_k-x_0}}\\]<br>\u4eceNewton\u516c\u5f0f\u77e5, $f[x_0,\\ldots,x_k]$\u4e0d\u4f9d\u8d56\u4e8e$x_0,\\ldots,x_k$\u7684\u987a\u5e8f.<\/p>\n\n\n\n<p><span class=\"latex_section\">3.&#x00A0;\u8bef\u5dee\u4f30\u8ba1<a id=\"sec:3\"><\/a><\/span>\n\u5728\u5e94\u7528\u4e2d, \u6211\u4eec\u4f1a\u91c7\u53d6\u65e0\u7a77\u8303\u6570\u6765\u8fdb\u884c\u51fd\u6570\u7684\u8bef\u5dee\u4f30\u8ba1. \u7ed9\u5b9a\u63d2\u503c\u70b9$\\Theta_n=\\{x_0,\\ldots,x_n\\}$, \u8fd9\u4e9b\u63d2\u503c\u70b9\u786e\u5b9a\u51fd\u6570$\\omega_n(x)=(x-x_0)\\cdots(x-x_n)$, \u548c$n$\u6b21\u7684\u725b\u987f\u591a\u9879\u5f0f$p_n$. \u90a3\u4e48\u51fd\u6570$f$\u548c$p_n$\u7684\u8bef\u5dee\u6ee1\u8db3<br>\\[f(x)-p_n(x)=\\frac{f^{(n)}(\\xi)}{(n+1)!}\\omega_n(\\xi).\\]<br>\u7279\u522b\u5730, \u6570\u503c\u4e0a\u8bf4,<br>\\[\\boxed{\\Vert f-p_n\\Vert\\leq \\frac{C_n}{(n+1)!}\\Vert\\omega_n\\Vert_\\infty},\\]<br>\u5176\u4e2d$C_n:=\\Vert f^{(n+1)}\\Vert_\\infty$.<\/p>\n\n\n\n<p><span class=\"latex_section\">4.&#x00A0;\u6837\u6761\u51fd\u6570<a id=\"sec:4\"><\/a><\/span>\n\u5982\u679c\u51fd\u6570\u7684\u5149\u6ed1\u6027\u4e0d\u597d, \u90a3\u4e48\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u7f29\u77ed\u533a\u95f4\u957f\u5ea6\u6765\u4f30\u8ba1, \u8fd9\u6837\u5f97\u5230\u7684\u5206\u6bb5\u51fd\u6570\u79f0\u4e3a\u6837\u6761\u51fd\u6570 (spline function). \u7ed9\u5b9a\u533a\u95f4$I=[a,b]$\u7684\u652f\u6491\u70b9\u96c6<br>\\[\\Theta_n= \\{a=x_0&lt;x_1&lt;\\cdots&lt;x_n=b\\}, \\]<\/p>\n\n\n\n<p>\u5b9a\u4e49\\(h_i=x_i-x_{i-1}, \\quad h=\\max_{1\\leq i\\leq n}h_i\\), \u4ee5\u53ca\u6240\u6709\u652f\u6491\u70b9\u6784\u6210\u7684\u96c6\u5408\u4e3a<br>\\[\\mathscr{G}:=\\{ [x_{i-1},x_i] : i=1,2,\\ldots, n\\}. \\]<br>\u7ed9\u5b9a\u5149\u6ed1\u5ea6$k$, \u6211\u4eec\u5b9a\u4e49\u6240\u6709\u6837\u6761\u51fd\u6570\u6784\u6210\u7684\u7a7a\u95f4\u4e3a<br>\\[\\mathcal{S}_{\\mathscr{G}}^{k,m} := \\{u\\in C^k(I) | \\forall \\tau\\in\\mathscr{G}, u|_\\tau \\in \\mathbb{P}_m \\}. \\]<\/p>\n\n\n\n<p>\u7279\u522b\u5730, \u5982\u679c$k&gt;m$, \u5373\u5168\u5c40\u5149\u6ed1\u5ea6\u5927\u4e8e\u591a\u9879\u5f0f\u6b21\u6570, \u5219\u6837\u6761\u51fd\u6570\u7a7a\u95f4\u5c31\u662f\u5168\u5c40\u7684\u591a\u9879\u5f0f\u51fd\u6570\u7a7a\u95f4, \u5373$\\mathcal{S}_{\\mathscr{G}}^{k,m}=\\mathbb{P}_m$. \u56e0\u6b64, \u5728\u4e0b\u9762\u7684\u8ba8\u8bba\u4e2d, \u6211\u4eec\u603b\u662f\u5047\u8bbe$k\\leq m$.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u4e3a\u4e86\u901a\u8fc7\u8003\u8bd5, \u6982\u62ec\u6027\u8bb0\u5f55\u4e00\u4e0b.  <a class=\"more-link\" href=\"https:\/\/blog.mathyuan.com\/?p=50\">\u7ee7\u7eed\u9605\u8bfb <span class=\"screen-reader-text\">  \u6570\u503c\u5206\u6790\u7b14\u8bb0<\/span><span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9],"tags":[8],"class_list":["post-50","post","type-post","status-publish","format-standard","hentry","category-notes","tag-8"],"_links":{"self":[{"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=\/wp\/v2\/posts\/50","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=50"}],"version-history":[{"count":54,"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=\/wp\/v2\/posts\/50\/revisions"}],"predecessor-version":[{"id":800,"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=\/wp\/v2\/posts\/50\/revisions\/800"}],"wp:attachment":[{"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=50"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=50"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.mathyuan.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=50"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}