{"id":109285,"date":"2026-09-23T20:57:09","date_gmt":"2026-09-23T12:57:09","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/109285.html"},"modified":"2026-09-23T20:57:09","modified_gmt":"2026-09-23T12:57:09","slug":"%e6%8a%8a%e7%9f%a9%e9%98%b5%e6%8b%86%e6%88%90%e7%a7%a9%e4%b8%80%e5%8f%a0%e5%8a%a0%ef%bc%9a%e5%a5%87%e5%bc%82%e5%80%bc%e5%88%86%e8%a7%a3%e3%80%81%e6%9d%a1%e4%bb%b6%e6%95%b0%e4%b8%8e%e4%bd%8e%e7%a7%a9","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/109285.html","title":{"rendered":"\u628a\u77e9\u9635\u62c6\u6210\u79e9\u4e00\u53e0\u52a0\uff1a\u5947\u5f02\u503c\u5206\u89e3\u3001\u6761\u4ef6\u6570\u4e0e\u4f4e\u79e9\u903c\u8fd1\u5728\u7b97\u4ec0\u4e48"},"content":{"rendered":"<h2>\u628a\u77e9\u9635\u62c6\u6210\u79e9\u4e00\u53e0\u52a0&#xff1a;\u5947\u5f02\u503c\u5206\u89e3\u3001\u6761\u4ef6\u6570\u4e0e\u4f4e\u79e9\u903c\u8fd1\u5728\u7b97\u4ec0\u4e48<\/h2>\n<h3>\u4e00\u3001\u4e00\u4e2a\u5177\u4f53\u7684\u77e9\u9635&#xff0c;\u628a\u5355\u4f4d\u7403\u62c9\u6210\u4e86\u4ec0\u4e48\u5f62\u72b6<\/h3>\n<p>\u5148\u6446\u4e00\u4e2a 3\u00d73 \u7684\u6574\u6570\u77e9\u9635&#xff0c;\u5b83\u53ea\u6709\u4e5d\u4e2a\u6570\u5b57&#xff1a;<\/p>\n<p>A &#061; [[ 1,  2,  3],<br \/>\n     [ 4,  5,  6],<br \/>\n     [ 7,  8, 10]]<\/p>\n<p>\u77e9\u9635\u53ef\u4ee5\u770b\u6210\u4e00\u4e2a\u52a8\u4f5c&#xff1a;\u7ed9\u5b83\u4e00\u4e2a\u5411\u91cf&#xff0c;\u5b83\u5410\u51fa\u53e6\u4e00\u4e2a\u5411\u91cf\u3002\u628a\u5355\u4f4d\u7403\u9762\u4e0a\u6240\u6709\u7684\u5411\u91cf\u90fd\u5582\u8fdb\u53bb&#xff0c;\u51fa\u6765\u7684\u50cf\u662f\u4e00\u4e2a\u692d\u7403\u2014\u2014\u77e9\u9635\u505a\u7684\u4e8b\u60c5\u65e0\u975e\u662f\u628a\u7403\u634f\u6210\u692d\u7403&#xff0c;\u692d\u7403\u7684\u4e09\u4e2a\u534a\u8f74\u957f\u5ea6\u5c31\u662f\u4e09\u4e2a\u5947\u5f02\u503c&#xff1a;<\/p>\n<p>17.412505166809    0.875161350110    0.196866521117<\/p>\n<p>\u4e09\u4e2a\u6570\u653e\u5728\u4e00\u8d77\u5c31\u5df2\u7ecf\u8bf4\u6e05\u4e86\u8fd9\u53f0\u300c\u673a\u5668\u300d\u7684\u6027\u683c\u3002\u6700\u957f\u7684\u90a3\u6839\u534a\u8f74\u662f 17.412505166809&#xff0c;\u6700\u77ed\u7684\u90a3\u6839\u53ea\u6709 0.196866521117&#xff0c;\u76f8\u5dee 88 \u500d\u591a\u3002\u4efb\u4f55\u843d\u5728\u6700\u77ed\u534a\u8f74\u65b9\u5411\u4e0a\u7684\u8f93\u5165&#xff0c;\u90fd\u4f1a\u88ab\u538b\u5230\u539f\u6765\u7684\u4e94\u5206\u4e4b\u4e00\u5de6\u53f3&#xff1b;\u4efb\u4f55\u843d\u5728\u6700\u957f\u534a\u8f74\u65b9\u5411\u4e0a\u7684\u8f93\u5165&#xff0c;\u4f1a\u88ab\u653e\u5927\u5341\u4e03\u500d\u591a\u3002\u8fd9\u4e24\u4e2a\u65b9\u5411\u7684\u6bd4\u503c\u6709\u4e00\u4e2a\u4e13\u95e8\u7684\u540d\u5b57\u2014\u2014\u6761\u4ef6\u6570&#xff1a;<\/p>\n<p>\u6761\u4ef6\u6570 &#061; \u6700\u957f\u534a\u8f74 \/ \u6700\u77ed\u534a\u8f74 &#061; 17.412505166809 \/ 0.196866521117 &#061; 88.448279920699<\/p>\n<p>\u8fd8\u6709\u4e00\u6761\u968f\u624b\u53ef\u9a8c\u7684\u5173\u7cfb&#xff1a;\u4e09\u4e2a\u5947\u5f02\u503c\u4e58\u8d77\u6765\u7b49\u4e8e 3.000000000000&#xff0c;\u800c\u8fd9\u4e2a\u77e9\u9635\u7684\u884c\u5217\u5f0f\u662f \u22123\u3002\u8fd9\u4e0d\u662f\u5de7\u5408&#xff0c;\u4e0b\u4e00\u8282\u4f1a\u628a\u5b83\u8bf4\u6e05\u695a\u3002<\/p>\n<p>\u4e09\u7ef4\u7684\u692d\u7403\u753b\u5728\u7eb8\u4e0a\u4e0d\u597d\u770b&#xff0c;\u6240\u4ee5\u5148\u7528\u4e00\u4e2a\u4e8c\u7ef4\u7684\u540c\u7c7b\u4f8b\u5b50\u628a\u56fe\u666f\u7acb\u8d77\u6765\u3002\u53d6<\/p>\n<p>B &#061; [[1, 1],<br \/>\n     [0, 1]]<\/p>\n<p>\u5b83\u628a\u5e73\u9762\u4e0a\u7684\u5355\u4f4d\u5706&#xff08;\u5706\u4e0a\u6bcf\u4e00\u70b9\u5230\u539f\u70b9\u8ddd\u79bb\u90fd\u662f 1&#xff09;\u53d8\u6210\u4e00\u4e2a\u692d\u5706\u3002\u8fd9\u4e2a\u692d\u5706\u7684\u534a\u8f74\u957f\u662f 1.618033988750 \u4e0e 0.618033988750\u2014\u2014\u6b63\u597d\u662f\u9ec4\u91d1\u6bd4 \u03c6 \u4e0e\u5b83\u7684\u5012\u6570 1\/\u03c6&#xff0c;\u6761\u4ef6\u6570\u662f 2.618033988750&#xff0c;\u4e5f\u5c31\u662f \u03c6\u00b2\u3002<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/09\/20260923125703-6ab3cc9f1e444.png\" alt=\"\u5355\u4f4d\u5706\u3001\u5b83\u5728\u526a\u5207\u77e9\u9635\u4e0b\u7684\u50cf\uff0c\u4ee5\u53ca\u4e24\u6761\u5947\u5f02\u65b9\u5411\u4e0a\u7684\u534a\u8f74\" \/><\/p>\n<p>\u56fe 1&#xff1a;\u5355\u4f4d\u5706\u3001\u5b83\u5728\u526a\u5207\u77e9\u9635\u4e0b\u7684\u50cf&#xff0c;\u4ee5\u53ca\u4e24\u6761\u5947\u5f02\u65b9\u5411\u4e0a\u7684\u534a\u8f74<\/p>\n<p>\u56fe 1 \u91cc\u6709\u4e24\u4e2a\u5706\u548c\u4e24\u79cd\u7ebf\u3002\u865a\u7ebf\u662f\u539f\u6765\u7684\u5355\u4f4d\u5706&#xff1b;\u5b9e\u7ebf\u662f\u5b83\u88ab B \u4f5c\u7528\u4e4b\u540e\u7684\u50cf\u2014\u2014\u4e00\u4e2a\u659c\u7740\u8eba\u7684\u692d\u5706\u3002\u4e24\u6761\u77ed\u7ebf\u6bb5\u662f\u692d\u5706\u7684\u4e24\u6761\u534a\u8f74&#xff0c;\u5b83\u4eec\u7684\u65b9\u5411\u5c31\u662f\u300c\u8f93\u5165\u7a7a\u95f4\u91cc\u6700\u503c\u5f97\u5173\u5fc3\u300d\u7684\u4e24\u4e2a\u65b9\u5411&#xff1a;\u6cbf\u5176\u4e2d\u4e00\u6761\u8f93\u5165&#xff0c;\u8f93\u51fa\u88ab\u62c9\u5f97\u6700\u957f&#xff1b;\u6cbf\u53e6\u4e00\u6761\u8f93\u5165&#xff0c;\u8f93\u51fa\u7f29\u5f97\u6700\u77ed\u3002\u503c\u5f97\u7559\u610f\u7684\u662f\u8fd9\u4e24\u6761\u534a\u8f74\u5f7c\u6b64\u5782\u76f4\u2014\u2014\u8fd9\u4e00\u70b9\u4e0d\u662f\u8fd9\u4e2a\u4f8b\u5b50\u7684\u5de7\u5408&#xff0c;\u800c\u662f\u6240\u6709\u77e9\u9635\u5171\u540c\u7684\u6027\u8d28&#xff1a;\u692d\u5706\u7684\u4e24\u6761\u534a\u8f74\u6c38\u8fdc\u662f\u5782\u76f4\u7684&#xff0c;\u800c\u5b83\u4eec\u5bf9\u5e94\u7684\u8f93\u5165\u65b9\u5411\u4e5f\u6c38\u8fdc\u662f\u5782\u76f4\u7684\u3002\u8fd9\u4e24\u7ec4\u4e92\u76f8\u5782\u76f4\u7684\u65b9\u5411&#xff0c;\u5c31\u662f\u4e0b\u4e00\u8282\u7684\u4e3b\u89d2\u3002<\/p>\n<h3>\u4e8c\u3001\u4e09\u4e2a\u52a8\u4f5c&#xff1a;\u8f6c\u4e00\u4e0b\u3001\u6cbf\u4e92\u76f8\u5782\u76f4\u7684\u65b9\u5411\u5404\u81ea\u62c9\u4f38\u3001\u518d\u8f6c\u4e00\u4e0b<\/h3>\n<p>\u5947\u5f02\u503c\u5206\u89e3\u8bf4\u7684\u5c31\u662f\u8fd9\u53e5\u8bdd&#xff1a;<\/p>\n<p>A &#061; U \u03a3 V\u1d40<\/p>\n<p>\u5176\u4e2d V \u7684\u5217\u662f\u4e00\u7ec4\u4e92\u76f8\u5782\u76f4\u7684\u5355\u4f4d\u5411\u91cf&#xff08;\u8f93\u5165\u65b9\u5411&#xff09;&#xff0c;U \u7684\u5217\u662f\u53e6\u4e00\u7ec4\u4e92\u76f8\u5782\u76f4\u7684\u5355\u4f4d\u5411\u91cf&#xff08;\u8f93\u51fa\u65b9\u5411&#xff09;&#xff0c;\u03a3 \u662f\u4e00\u4e2a\u5bf9\u89d2\u77e9\u9635&#xff0c;\u5bf9\u89d2\u5143\u5c31\u662f\u90a3\u4e32\u6309\u4ece\u5927\u5230\u5c0f\u6392\u597d\u7684\u5947\u5f02\u503c\u3002\u628a\u8fd9\u4e2a\u5f0f\u5b50\u4ece\u53f3\u5f80\u5de6\u8bfb&#xff0c;\u5c31\u662f\u4e09\u4e2a\u52a8\u4f5c&#xff1a;<\/p>\n<ul>\n<li>\u5148\u628a\u8f93\u5165\u5411\u91cf\u6309 V \u7684\u8fd9\u7ec4\u65b9\u5411\u91cd\u65b0\u63cf\u8ff0\u4e00\u904d&#xff08;\u76f8\u5f53\u4e8e\u8f6c\u4e00\u4e0b\u5750\u6807\u67b6&#xff09;&#xff1b;<\/li>\n<li>\u7136\u540e\u6bcf\u4e2a\u65b9\u5411\u4e0a\u7684\u5206\u91cf\u5404\u81ea\u4e58\u4e00\u4e2a\u7cfb\u6570 \u03c3\u1d62\u2014\u2014\u6709\u7684\u88ab\u653e\u5927&#xff0c;\u6709\u7684\u88ab\u7f29\u5c0f&#xff0c;\u4e92\u4e0d\u5e72\u6270&#xff1b;<\/li>\n<li>\u6700\u540e\u6309 U \u7684\u8fd9\u7ec4\u65b9\u5411\u62fc\u56de\u8f93\u51fa\u5411\u91cf&#xff08;\u518d\u8f6c\u4e00\u4e0b&#xff09;\u3002<\/li>\n<\/ul>\n<p>\u4e24\u4e2a\u300c\u8f6c\u300d\u90fd\u662f\u6b63\u4ea4\u53d8\u6362&#xff0c;\u4e0d\u6539\u53d8\u957f\u5ea6&#xff1b;\u6240\u6709\u6539\u53d8\u957f\u5ea6\u7684\u6d3b\u90fd\u4ea4\u7ed9\u4e2d\u95f4\u90a3\u4e00\u6b65\u7684\u5bf9\u89d2\u77e9\u9635 [1]\u3002\u8fd9\u4e5f\u89e3\u91ca\u4e86\u4e3a\u4ec0\u4e48\u5947\u5f02\u503c\u4e00\u5b9a\u662f\u975e\u8d1f\u7684\u3001\u800c\u4e14\u603b\u53ef\u4ee5\u6309\u964d\u5e8f\u6392\u5217&#xff1a;\u5b83\u4eec\u662f\u4e00\u7ec4\u957f\u5ea6&#xff0c;\u987a\u5e8f\u53ea\u662f\u4e66\u5199\u4e60\u60ef\u3002<\/p>\n<p>\u4e09\u4e2a\u6982\u5ff5\u5728\u8fd9\u91cc\u8981\u5206\u6e05\u695a&#xff0c;\u5b83\u4eec\u7684\u5730\u4f4d\u5e76\u4e0d\u76f8\u540c&#xff1a;<\/p>\n<ul>\n<li>\u4e24\u7ec4\u65b9\u5411&#xff08;U \u4e0e V&#xff09;\u56de\u7b54\u300c\u54ea\u4e9b\u65b9\u5411\u662f\u7279\u6b8a\u7684\u300d&#xff1b;<\/li>\n<li>\u5947\u5f02\u503c&#xff08;\u03a3 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m\u00d7n \u77e9\u9635&#xff0c;\u4e2d\u95f4\u662f\u5b83\u7684\u5206\u89e3&#xff0c;\u53f3\u8fb9\u4e09\u4e2a\u65b9\u6846\u662f\u4ece\u540c\u4e00\u4e32\u5947\u5f02\u503c\u4e0a\u8bfb\u51fa\u6765\u7684\u4e09\u4e2a\u91cf&#xff0c;\u6700\u53f3\u8fb9\u662f\u6309\u9608\u503c\u622a\u65ad\u4e4b\u540e\u5f97\u5230\u7684\u89e3\u3002\u6574\u7bc7\u6587\u7ae0\u5c31\u662f\u6cbf\u7740\u8fd9\u6761\u7ebf\u8d70&#xff1a;\u5148\u5b66\u4f1a\u8bfb\u8fd9\u4e32\u6570&#xff0c;\u518d\u7528\u5b83\u56de\u7b54\u4e24\u4e2a\u4e0d\u540c\u7684\u95ee\u9898\u2014\u2014\u300c\u54ea\u4e2a\u4f4e\u79e9\u77e9\u9635\u79bb\u5b83\u6700\u8fd1\u300d\u548c\u300c\u89e3\u65b9\u7a0b\u65f6\u8f93\u5165\u8bef\u5dee\u4f1a\u88ab\u653e\u5927\u591a\u5c11\u300d\u3002<\/p>\n<p>\u987a\u4fbf\u8bf4\u4e00\u53e5\u8fd9\u4ef6\u5de5\u5177\u7684\u6765\u8def\u3002\u628a\u77e9\u9635\u5148\u5316\u6210\u4e24\u6761\u5e26\u3001\u518d\u628a\u5e26\u6d88\u6210\u5bf9\u89d2&#xff0c;\u8fd9\u6761\u8ba1\u7b97\u8def\u7ebf\u662f Golub \u4e0e Kahan \u5728 1965 \u5e74\u7ed9\u51fa\u7684 [1]&#xff1b;\u90a3\u7bc7\u6587\u7ae0\u76ee\u524d\u5728\u4f5c\u8005\u4e4b\u4e00\u7684\u4e3b\u9875\u4e0a\u53ef\u4ee5\u516c\u5f00\u8bfb\u5230&#xff0c;\u5b83\u5f00\u7bc7\u5c31\u628a\u4e24\u4e2a\u7528\u9014\u5199\u5728\u4e00\u8d77&#xff1a;\u7b97\u51fa\u5947\u5f02\u503c&#xff0c;\u4ee5\u53ca\u7528\u4f2a\u9006\u89e3\u6700\u5c0f\u4e8c\u4e58 [4]\u3002\u800c\u300c\u7528\u4e00\u4e2a\u4f4e\u79e9\u77e9\u9635\u53bb\u903c\u8fd1\u7ed9\u5b9a\u77e9\u9635\u300d\u8fd9\u4e2a\u95ee\u9898\u672c\u8eab\u66f4\u65e9&#xff0c;1936 \u5e74\u5c31\u6709\u4eba\u628a\u5b83\u5199\u6210\u4e86\u6700\u5c0f\u4e8c\u4e58\u95ee\u9898 [2]\u3002<\/p>\n<p>\u8981\u628a\u4e24\u7ec4\u65b9\u5411\u771f\u7684\u7b97\u51fa\u6765&#xff0c;\u7528\u7684\u662f\u4e00\u79cd\u53eb Jacobi \u8fed\u4ee3\u7684\u65b9\u6cd5\u3002\u5b83\u7684\u60f3\u6cd5\u5f88\u6734\u7d20&#xff1a;\u4e00\u6b21\u53ea\u76ef\u4f4f\u4e24\u4e2a\u65b9\u5411&#xff0c;\u5982\u679c\u8fd9\u4e24\u4e2a\u65b9\u5411\u5728\u8f93\u51fa\u91cc\u8fd8\u4e0d\u591f\u5782\u76f4&#xff0c;\u5c31\u628a\u5b83\u4fe9\u4e00\u8d77\u8f6c\u4e00\u4e2a\u5c0f\u89d2\u5ea6&#xff0c;\u76f4\u5230\u6240\u6709\u65b9\u5411\u4e24\u4e24\u5782\u76f4\u4e3a\u6b62\u3002<\/p>\n<p>\u6982\u5ff5\u6027\u4f2a\u4ee3\u7801&#xff1a;\u5355\u8fb9 Jacobi \u8fed\u4ee3&#xff08;\u4e0d\u662f\u8bba\u6587\u7b97\u6cd5\u6846\u7684\u590d\u73b0&#xff09;<br \/>\n\u8f93\u5165&#xff1a;m\u00d7n \u5b9e\u77e9\u9635 A&#xff08;m \u2265 n&#xff09;&#xff0c;\u6536\u655b\u9608\u503c tol<br \/>\n\u8f93\u51fa&#xff1a;\u4e09\u4e2a\u91cf U\u3001S\u3001V&#xff0c;\u6ee1\u8db3 A \u2248 U \u00b7 diag(S) \u00b7 V\u1d40<\/p>\n<p>\u521d\u59cb\u5316&#xff1a;\u5de5\u4f5c\u77e9\u9635 B \u2190 A&#xff1b;\u65b9\u5411\u77e9\u9635 V \u2190 n \u9636\u5355\u4f4d\u9635<br \/>\n\u91cd\u590d\u4ee5\u4e0b\u4e00\u8f6e&#xff08;\u6bcf\u8f6e\u628a\u6240\u6709\u65b9\u5411\u5bf9\u626b\u4e00\u904d&#xff09;&#xff1a;<br \/>\n    \u5bf9\u6bcf\u4e00\u5bf9\u4e0b\u6807 (p, q)&#xff0c;p &lt; q&#xff1a;<br \/>\n        \u7b97\u51fa\u7b2c p \u5217\u4e0e\u7b2c q \u5217\u7684\u957f\u5ea6\u5e73\u65b9 \u03b1\u3001\u03b2&#xff0c;\u4ee5\u53ca\u4e24\u5217\u7684\u5185\u79ef \u03b3<br \/>\n        \u82e5 |\u03b3| \u4e0d\u8d85\u8fc7 tol \u00b7 \u221a(\u03b1\u00b7\u03b2)&#xff1a;\u8fd9\u4e00\u5bf9\u5df2\u7ecf\u8db3\u591f\u5782\u76f4&#xff0c;\u8df3\u8fc7<br \/>\n        \u5426\u5219&#xff1a;<br \/>\n            \u7531 \u03b1\u3001\u03b2\u3001\u03b3 \u7b97\u51fa\u4e00\u4e2a\u65cb\u8f6c\u89d2\u7684\u6b63\u5207 t<br \/>\n            \u7528\u8fd9\u4e2a\u89d2\u540c\u65f6\u65cb\u8f6c B \u7684\u7b2c p\u3001q \u4e24\u5217<br \/>\n            \u5bf9 V \u7684\u7b2c p\u3001q \u4e24\u5217\u505a\u540c\u4e00\u4e2a\u65cb\u8f6c&#xff08;\u628a\u8fd9\u6b21\u65cb\u8f6c\u8bb0\u4e0b\u6765&#xff09;<br \/>\n    \u82e5\u672c\u8f6e\u6240\u6709\u65b9\u5411\u5bf9\u90fd\u5df2\u88ab\u8df3\u8fc7&#xff1a;\u7ed3\u675f\u5faa\u73af<br \/>\n\u6536\u5c3e&#xff1a;<br \/>\n    S \u2190 B \u5404\u5217\u7684\u957f\u5ea6&#xff0c;\u6309\u4ece\u5927\u5230\u5c0f\u6392\u5e8f<br \/>\n    U \u7684\u5404\u5217 \u2190 B \u7684\u5404\u5217\u9664\u4ee5\u5bf9\u5e94\u7684\u957f\u5ea6<br \/>\n    V \u7684\u5217\u6309\u540c\u4e00\u987a\u5e8f\u91cd\u6392<\/p>\n<p>\u8fd9\u6bb5\u4f2a\u4ee3\u7801\u91cc\u6709\u4e24\u4e2a\u7ec6\u8282\u503c\u5f97\u505c\u4e00\u4e0b\u3002\u7b2c\u4e00&#xff0c;\u65cb\u8f6c\u53ea\u4f5c\u7528\u5728\u4e24\u5217\u4e0a&#xff0c;\u6240\u4ee5\u6bcf\u4e00\u5c0f\u6b65\u90fd\u662f\u53ef\u9006\u7684\u3001\u4e0d\u4f1a\u653e\u5927\u5df2\u6709\u7684\u8bef\u5dee&#xff1b;\u8fd9\u4e5f\u662f\u8fd9\u7c7b\u65b9\u6cd5\u5728\u6570\u503c\u4e0a\u7a33\u7684\u539f\u56e0\u3002\u7b2c\u4e8c&#xff0c;\u4ee3\u7801\u4ece\u6765\u6ca1\u6709\u663e\u5f0f\u7b97\u8fc7 A\u1d40A\u2014\u2014\u5b83\u53ea\u6bd4\u8f83\u5217\u7684\u957f\u5ea6\u4e0e\u5185\u79ef&#xff0c;\u800c \u03b1\u3001\u03b2\u3001\u03b3 \u6070\u597d\u5c31\u662f A\u1d40A \u5728\u8fd9\u4e00\u5bf9\u65b9\u5411\u4e0a\u5bf9\u5e94\u7684\u4e09\u4e2a\u5143\u7d20\u3002\u7ed5\u5f00 A\u1d40A \u76f4\u63a5\u7b97&#xff0c;\u6b63\u662f\u8fd9\u5957\u529e\u6cd5\u6bd4\u300c\u5148\u628a A\u1d40A \u4e58\u51fa\u6765\u518d\u6c42\u7279\u5f81\u503c\u300d\u66f4\u53ef\u9760\u7684\u5730\u65b9&#xff1b;\u7b2c\u4e03\u8282\u4f1a\u770b\u5230\u663e\u5f0f\u4e58\u51fa A\u1d40A \u7684\u4ee3\u4ef7\u3002<\/p>\n<h3>\u4e09\u3001\u79e9\u4e00\u5c55\u5f00&#xff1a;\u628a\u77e9\u9635\u62c6\u6210\u4e00\u5c42\u5c42\u7684\u53e0\u52a0<\/h3>\n<p>\u628a A &#061; U\u03a3V\u1d40 \u6309\u5217\u644a\u5f00&#xff0c;\u5c31\u5f97\u5230\u53e6\u4e00\u79cd\u5199\u6cd5\u3002\u8bbe \u03c3\u2081 \u2265 \u03c3\u2082 \u2265 \u2026 \u2265 \u03c3_min&#xff0c;u\u1d62 \u662f U \u7684\u7b2c i \u5217&#xff0c;v\u1d62 \u662f V \u7684\u7b2c i \u5217&#xff0c;\u90a3\u4e48<\/p>\n<p>A &#061; \u03c3\u2081u\u2081v\u2081\u1d40 &#043; \u03c3\u2082u\u2082v\u2082\u1d40 &#043; \u2026 &#043; \u03c3_min u_min v_min\u1d40<\/p>\n<p>\u6bcf\u4e00\u9879 \u03c3\u1d62u\u1d62v\u1d62\u1d40 \u90fd\u662f\u4e00\u4e2a\u79e9\u4e00\u77e9\u9635&#xff1a;\u5b83\u628a\u4efb\u4f55\u8f93\u5165\u5148\u6295\u5f71\u5230 v\u1d62 \u8fd9\u4e2a\u65b9\u5411\u4e0a&#xff08;\u7b97\u4e00\u4e2a\u7cfb\u6570&#xff09;&#xff0c;\u518d\u628a\u8fd9\u4e2a\u7cfb\u6570\u4e58\u4e0a \u03c3\u1d62 \u653e\u5230 u\u1d62 \u8fd9\u4e2a\u65b9\u5411\u4e0a\u3002\u6362\u53e5\u8bdd\u8bf4&#xff0c;\u6bcf\u4e00\u5c42\u53ea\u8ba4\u4e00\u4e2a\u8f93\u5165\u65b9\u5411\u3001\u53ea\u4ea7\u51fa\u4e00\u4e2a\u8f93\u51fa\u65b9\u5411\u3002\u6574\u4e2a\u77e9\u9635\u5c31\u662f\u8fd9\u4e9b\u5c42\u7684\u53e0\u52a0\u3002<\/p>\n<p>\u5bf9\u7b2c\u4e00\u8282\u90a3\u4e2a 3\u00d73 \u77e9\u9635&#xff0c;\u4e09\u5c42\u5404\u81ea\u7684\u539a\u5ea6\u662f&#xff1a;<\/p>\n<table>\n<tr>\u5c42\u5947\u5f02\u503c\u8fd9\u4e00\u5c42\u5360 \u2016A\u2016\u00b2&#xff08;Frobenius \u8303\u6570\u5e73\u65b9&#xff09;\u7684\u4efd\u989d<\/tr>\n<tbody>\n<tr>\n<td>\u03c3\u2081u\u2081v\u2081\u1d40<\/td>\n<td>17.412505166809<\/td>\n<td>0.99735308<\/td>\n<\/tr>\n<tr>\n<td>\u03c3\u2082u\u2082v\u2082\u1d40<\/td>\n<td>0.875161350110<\/td>\n<td>0.00251943<\/td>\n<\/tr>\n<tr>\n<td>\u03c3\u2083u\u2083v\u2083\u1d40<\/td>\n<td>0.196866521117<\/td>\n<td>0.00012749<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u7b2c\u4e00\u5c42\u4e00\u5c42\u5c31\u5360\u4e86\u5168\u90e8\u300c\u80fd\u91cf\u300d\u7684 99.74%&#xff0c;\u4e2d\u95f4\u90a3\u5c42\u5360 0.25%&#xff0c;\u6700\u540e\u4e00\u5c42\u53ea\u5269 0.013%\u3002\u8fd9\u4e2a\u4efd\u989d\u662f\u53ef\u4ee5\u624b\u7b97\u7684&#xff1a;\u56e0\u4e3a\u4e24\u7ec4\u65b9\u5411\u90fd\u662f\u5355\u4f4d\u5411\u91cf\u4e14\u5f7c\u6b64\u5782\u76f4&#xff0c;\u628a A \u7684\u5e73\u65b9\u548c\u5c55\u5f00\u65f6\u4ea4\u53c9\u9879\u5168\u90e8\u6d88\u5931&#xff0c;\u53ea\u5269\u4e0b \u03c3\u1d62\u00b2 \u76f8\u52a0&#xff0c;\u6240\u4ee5\u7b2c i \u5c42\u7684\u4efd\u989d\u5c31\u662f \u03c3\u1d62\u00b2 \u9664\u4ee5\u5168\u90e8 \u03c3\u1d62\u00b2 \u7684\u548c\u3002\u8fd9\u4efd\u300c\u80fd\u91cf\u8d26\u300d\u662f\u6574\u4e2a\u4f4e\u79e9\u903c\u8fd1\u7684\u4f9d\u636e\u2014\u2014\u5982\u679c\u53ea\u7559\u4e00\u5c42&#xff0c;\u4e22\u6389\u7684\u6b63\u597d\u662f\u540e\u9762\u4e24\u5c42\u3002<\/p>\n<p>\u987a\u5e26\u628a\u7b2c\u4e00\u8282\u7559\u4e0b\u7684\u90a3\u53e5\u8bf4\u6e05\u695a&#xff1a;\u628a A &#061; U\u03a3V\u1d40 \u4ee3\u8fdb\u884c\u5217\u5f0f&#xff0c;\u4e24\u7aef\u7684\u6b63\u4ea4\u77e9\u9635\u884c\u5217\u5f0f\u7edd\u5bf9\u503c\u90fd\u662f 1&#xff0c;\u5269\u4e0b\u7684\u53ea\u6709\u5bf9\u89d2\u9635&#xff0c;\u4e8e\u662f |det A| \u5c31\u7b49\u4e8e\u4e09\u4e2a\u5947\u5f02\u503c\u7684\u4e58\u79ef\u30023.000000000000 \u4e0e |\u22123| \u76f8\u7b49&#xff0c;\u6b63\u662f\u8fd9\u6761\u5173\u7cfb&#xff1b;\u6362\u4e2a\u89d2\u5ea6\u8bf4&#xff0c;\u884c\u5217\u5f0f\u300c\u4f53\u79ef\u7f29\u653e\u500d\u6570\u300d\u672c\u8eab\u5c31\u662f\u5404\u65b9\u5411\u62c9\u4f38\u500d\u6570\u7684\u4e58\u79ef\u3002<\/p>\n<h3>\u56db\u3001\u79e9\u5230\u5e95\u662f\u591a\u5c11&#xff1a;\u4ece\u8c31\u4e0a\u770b&#xff0c;\u4ee5\u53ca\u6d6e\u70b9\u4e0b\u4ec0\u4e48\u53eb\u96f6<\/h3>\n<p>\u77e9\u9635\u7684\u79e9&#xff0c;\u6700\u6734\u7d20\u7684\u5b9a\u4e49\u662f\u300c\u7ebf\u6027\u65e0\u5173\u7684\u5217\u6570\u300d\u3002\u653e\u5230\u5947\u5f02\u503c\u5206\u89e3\u4e0a\u770b&#xff0c;\u5b83\u6709\u4e00\u4e2a\u7b49\u4ef7\u7684\u8bf4\u6cd5&#xff1a;\u79e9\u7b49\u4e8e\u975e\u96f6\u5947\u5f02\u503c\u7684\u4e2a\u6570\u3002\u9053\u7406\u5f88\u76f4\u63a5&#xff1a;\u6bcf\u4e00\u5c42 \u03c3\u1d62u\u1d62v\u1d62\u1d40 \u7684\u79e9\u662f 1&#xff08;\u53ea\u8981 \u03c3\u1d62 \u4e0d\u4e3a\u96f6&#xff09;&#xff0c;\u800c\u4e92\u76f8\u5782\u76f4\u7684\u65b9\u5411\u4e4b\u95f4\u65e0\u6cd5\u4e92\u76f8\u8868\u793a&#xff0c;\u6240\u4ee5\u8fd9\u4e9b\u5c42\u52a0\u8d77\u6765\u7684\u79e9\u6070\u597d\u7b49\u4e8e\u975e\u96f6\u5c42\u7684\u6570\u91cf\u3002<\/p>\n<p>\u4e8e\u662f\u79e9\u4ece\u4e00\u4e2a\u6d88\u5143\u8fc7\u7a0b\u7684\u526f\u4ea7\u7269&#xff0c;\u53d8\u6210\u4e86\u8c31\u4e0a\u7684\u4e00\u6b21\u8bfb\u6570\u3002\u53ef\u300c\u975e\u96f6\u300d\u8fd9\u4e24\u4e2a\u5b57\u5728\u6d6e\u70b9\u8ba1\u7b97\u91cc\u4f1a\u7acb\u523b\u53d8\u6210\u9ebb\u70e6\u3002\u770b\u4e00\u4e2a\u5177\u4f53\u7684 6\u00d74 \u77e9\u9635&#xff1a;<\/p>\n<p>C &#061; [[1, 2, 3, 4],<br \/>\n     [2, 3, 4, 5],<br \/>\n     [3, 4, 5, 6],<br \/>\n     [4, 5, 6, 7],<br \/>\n     [5, 6, 7, 8],<br \/>\n     [6, 7, 8, 9]]<\/p>\n<p>\u5b83\u7684\u6bcf\u4e2a\u5143\u7d20\u90fd\u7b49\u4e8e\u884c\u53f7\u52a0\u5217\u53f7\u518d\u52a0\u4e00&#xff08;\u884c\u53f7\u3001\u5217\u53f7\u90fd\u4ece 0 \u5f00\u59cb&#xff09;&#xff0c;\u6240\u4ee5\u6bcf\u4e00\u884c\u90fd\u662f\u4e0a\u4e00\u884c\u9010\u5143\u7d20\u52a0\u4e00\u3001\u6bcf\u4e00\u5217\u90fd\u662f\u5de6\u8fb9\u4e00\u5217\u9010\u5143\u7d20\u52a0\u4e00\u3002\u8fd9\u79cd\u7ed3\u6784\u610f\u5473\u7740\u5b83\u53ef\u4ee5\u5199\u6210\u4e24\u90e8\u5206\u4e4b\u548c&#xff1a;\u4e00\u90e8\u5206\u53ea\u8ddf\u884c\u53f7\u6709\u5173&#xff0c;\u4e00\u90e8\u5206\u53ea\u8ddf\u5217\u53f7\u6709\u5173\u3002\u5199\u51fa\u6765\u662f<\/p>\n<p>C &#061; u \u00b7 1\u1d40 &#043; 1 \u00b7 w\u1d40,    u &#061; (1,2,3,4,5,6)\u1d40,  w &#061; (0,1,2,3)<\/p>\n<p>\u524d\u4e00\u9879\u7684\u6bcf\u4e00\u884c\u90fd\u662f\u540c\u4e00\u4e2a\u884c\u5411\u91cf\u7684\u500d\u6570&#xff0c;\u79e9\u662f 1&#xff1b;\u540e\u4e00\u9879\u7684\u6bcf\u4e00\u5217\u90fd\u662f\u540c\u4e00\u4e2a\u5217\u5411\u91cf\u7684\u500d\u6570&#xff0c;\u79e9\u4e5f\u662f 1\u3002\u4e24\u9879\u4e4b\u548c\u7684\u79e9\u6700\u591a\u662f 2&#xff0c;\u800c\u5b83\u663e\u7136\u4e0d\u662f\u79e9 1&#xff08;\u884c\u4e0e\u884c\u4e0d\u6210\u6bd4\u4f8b&#xff09;&#xff0c;\u6240\u4ee5\u5b83\u7684\u7cbe\u786e\u79e9\u5c31\u662f 2\u3002\u6309\u8fd9\u4e2a\u5f0f\u5b50\u590d\u7b97&#xff0c;\u6700\u5927\u7edd\u5bf9\u8bef\u5dee\u662f 0\u2014\u2014\u4e0d\u662f\u300c\u5f88\u5c0f\u300d&#xff0c;\u662f\u9010\u4f4d\u76f8\u7b49\u3002<\/p>\n<p>\u6309\u5b9a\u4e49\u7b97\u51fa\u5b83\u7684\u56db\u4e2a\u5947\u5f02\u503c&#xff0c;\u4f1a\u5f97\u5230&#xff1a;<\/p>\n<p>26.400511954467    1.735790466056    8.887071185757784\u00d710\u207b\u00b9\u2076    5.948736790578536\u00d710\u207b\u00b9\u2076<\/p>\n<p>\u540e\u4e24\u4e2a\u6570\u4e0d\u662f\u96f6&#xff0c;\u4f46\u76f8\u5bf9\u4e8e\u6700\u5927\u7684\u90a3\u4e2a\u53ea\u6709 3.366\u00d710\u207b\u00b9\u2077 \u4e0e 2.253\u00d710\u207b\u00b9\u2077\u2014\u2014\u8fd9\u662f\u53cc\u7cbe\u5ea6\u6d6e\u70b9\u5728\u6b64\u89c4\u6a21\u4e0b\u7684\u820d\u5165\u566a\u58f0\u91cf\u7ea7\u3002\u5b83\u4eec\u672c\u8be5\u7cbe\u786e\u4e3a\u96f6&#xff0c;\u673a\u5668\u7ed9\u51fa\u7684\u5374\u662f\u4e00\u70b9\u70b9\u7070\u5c18\u3002\u5982\u679c\u76f4\u63a5\u62ff\u300c\u5947\u5f02\u503c\u662f\u5426\u7b49\u4e8e 0\u300d\u5f53\u5224\u636e&#xff0c;\u8fd9\u4e2a\u79e9 2 \u7684\u77e9\u9635\u4f1a\u88ab\u5224\u6210\u79e9 4\u3002<\/p>\n<p>\u5de5\u4e1a\u4e0a\u7684\u505a\u6cd5\u662f\u6362\u4e00\u4e2a\u5224\u636e&#xff1a;\u7ed9\u5b9a\u4e00\u4e2a\u9608\u503c \u03c4&#xff0c;\u628a\u6ee1\u8db3 \u03c3\u1d62 &gt; \u03c4\u00b7\u03c3_max \u7684\u4e2a\u6570\u5f53\u4f5c\u79e9\u3002\u8fd9\u6837\u5b9a\u4e49\u7684\u91cf\u53eb\u6570\u503c\u79e9&#xff0c;\u5b83\u4e0d\u662f\u4e00\u4e2a\u6570&#xff0c;\u800c\u662f\u4e00\u4e2a\u300c\u6570 &#043; \u53e3\u5f84\u300d\u7684\u7ec4\u5408\u3002\u03c4 \u7684\u5e38\u89c1\u53d6\u6cd5\u8ddf\u77e9\u9635\u7ef4\u6570\u4e0e\u673a\u5668\u7cbe\u5ea6\u540c\u91cf\u7ea7&#xff08;\u4f8b\u5982\u7ef4\u6570\u4e58\u4e0a\u673a\u5668 epsilon \u518d\u4e58\u6700\u5927\u5947\u5f02\u503c&#xff09;&#xff0c;\u4f46\u5177\u4f53\u53d6\u591a\u5c11\u662f\u4f7f\u7528\u8005\u7684\u9009\u62e9&#xff0c;\u4e0d\u662f\u6570\u5b66\u5b9a\u7406\u89c4\u5b9a\u7684\u3002\u7b2c\u56db\u8282\u8fd9\u4e2a\u77e9\u9635\u5728 10\u207b\u00b9\u2076 \u91cf\u7ea7\u7684\u9608\u503c\u4e0b\u6570\u503c\u79e9\u662f 2&#xff0c;\u4e0e\u7cbe\u786e\u79e9\u4e00\u81f4&#xff1b;\u7b2c\u516b\u8282\u7684\u62df\u5408\u7b97\u4f8b\u91cc&#xff0c;\u90a3\u4e2a 11\u00d710 \u7684\u8bbe\u8ba1\u77e9\u9635\u5728\u4e0d\u540c\u9608\u503c\u4e0b\u4f1a\u7ed9\u51fa 10 \u4e0e 9 \u4e24\u4e2a\u4e0d\u540c\u7684\u6570\u503c\u79e9\u2014\u2014\u90a3\u662f\u540c\u4e00\u4ef6\u4e8b\u7684\u53e6\u4e00\u9762\u3002<\/p>\n<p>\u8fd9\u6761\u300c\u79e9\u5f88\u96be\u5224\u300d\u7684\u56f0\u96be\u4e0d\u662f\u6211\u7f16\u51fa\u6765\u7684\u30021965 \u5e74\u90a3\u7bc7\u6587\u7ae0\u7528\u6574\u6574\u4e24\u9875\u8ba8\u8bba\u5b83&#xff1a;\u4f5c\u8005\u5148\u6307\u51fa\u7528 A\u1d40A \u7684\u9006\u53bb\u7b97\u4f2a\u9006\u662f\u300c\u6211\u4eec\u5e38\u505a\u7684\u4e8b\u300d&#xff0c;\u4f46\u5f53 A\u1d40A \u8fd1\u4f3c\u5947\u5f02\u65f6\u4f1a\u6709\u65e0\u7a77\u591a\u4e2a\u5411\u91cf\u8fd1\u4f3c\u5730\u6700\u5c0f\u5316 \u2016b \u2212 Ax\u2016&#xff0c;\u5fc5\u987b\u7528\u67d0\u79cd\u65b9\u5f0f\u628a\u79e9\u8003\u8651\u8fdb\u53bb&#xff1b;\u63a5\u7740\u6307\u51fa\u4f20\u7edf\u505a\u6cd5\u662f\u770b\u884c\u9636\u68af\u5f62\u5bf9\u89d2\u7ebf\u4e0a\u7684\u975e\u96f6\u5143&#xff0c;\u53ef\u6d6e\u70b9\u8ba1\u7b97\u4e0b\u300c\u67d0\u4e2a\u6570\u662f\u5426\u53ef\u89c6\u4e3a\u96f6\u300d\u672c\u8eab\u5c31\u5f88\u96be\u5224\u65ad&#xff0c;\u4e8e\u662f\u6539\u7528\u300c\u7b2c r \u884c\u4e4b\u540e\u662f\u5426\u76f8\u5bf9\u524d r \u884c\u53ef\u5ffd\u7565\u300d\u8fd9\u4e2a\u5224\u636e&#xff0c;\u5e76\u627f\u8ba4\u8fd9\u4e2a\u5224\u636e\u540c\u6837\u96be\u7528&#xff1b;\u6700\u540e\u7ed9\u4e86\u4e00\u4e2a\u53cd\u4f8b\u2014\u2014\u4e00\u4e2a\u884c\u9636\u68af\u5f62\u5bf9\u89d2\u5143\u5168\u4e3a \u00b11 \u7684\u77e9\u9635&#xff0c;\u628a\u5176\u4e2d\u6240\u6709 \u22121 \u6362\u6210 &#043;1 \u4e4b\u540e\u77e9\u9635\u5c31\u53d8\u5f97\u6e29\u987a&#xff0c;\u8bf4\u660e\u53ea\u770b\u6d88\u5143\u7ed3\u679c\u6839\u672c\u65e0\u6cd5\u5224\u65ad\u539f\u77e9\u9635\u662f\u5426\u542b\u8db3\u591f\u5c0f\u3001\u53ef\u4ee5\u88ab\u5220\u6389\u7684\u5947\u5f02\u503c\u3002\u4ed6\u4eec\u7684\u7ed3\u8bba\u662f&#xff1a;\u4e0d\u663e\u5f0f\u770b\u5947\u5f02\u503c&#xff0c;\u4f3c\u4e4e\u6ca1\u6709\u6ee1\u610f\u7684\u5b9a\u79e9\u65b9\u6cd5 [1]\u3002<\/p>\n<h3>\u4e94\u3001\u4f4e\u79e9\u903c\u8fd1&#xff1a;\u4e22\u6389\u7684\u90a3\u90e8\u5206\u8bef\u5dee\u7b49\u4e8e\u591a\u5c11<\/h3>\n<p>\u73b0\u5728\u53ef\u4ee5\u56de\u7b54\u7b2c\u4e00\u4e2a\u6838\u5fc3\u95ee\u9898\u4e86&#xff1a;\u4e00\u4e2a\u77e9\u9635\u5728\u4ec0\u4e48\u610f\u4e49\u4e0b\u6700\u63a5\u8fd1\u67d0\u4e2a\u4f4e\u79e9\u77e9\u9635\u3002<\/p>\n<p>\u628a\u79e9\u4e00\u5c55\u5f00\u6309\u5c42\u4ece\u539a\u5230\u8584\u6392\u597d&#xff0c;\u7136\u540e\u53ea\u4fdd\u7559\u524d r \u5c42&#xff1a;<\/p>\n<p>A_r &#061; \u03c3\u2081u\u2081v\u2081\u1d40 &#043; \u03c3\u2082u\u2082v\u2082\u1d40 &#043; \u2026 &#043; \u03c3_r u_r v_r\u1d40<\/p>\n<p>\u4e22\u6389\u7684\u662f\u7b2c r&#043;1 \u5c42\u5f80\u540e\u7684\u5168\u90e8\u5185\u5bb9\u30021936 \u5e74\u90a3\u7bc7\u6587\u7ae0\u628a\u8fd9\u4e2a\u95ee\u9898\u5199\u6210\u4e86\u6700\u5c0f\u4e8c\u4e58\u95ee\u9898&#xff0c;\u5e76\u6307\u51fa\u5b83\u603b\u6709\u89e3\u3001\u800c\u4e14\u901a\u5e38\u552f\u4e00 [2]&#xff1b;\u4e8c\u5341\u591a\u5e74\u540e\u53c8\u6709\u4eba\u4e13\u95e8\u4e3a\u5b83\u8865\u4e86\u4e00\u4e2a\u8bc1\u660e&#xff0c;\u90a3\u7bc7\u6587\u7ae0\u7684\u6458\u8981\u5199\u5f97\u5f88\u76f4\u767d&#xff1a;\u8bc1\u660e\u7684\u662f Eckart \u4e0e Young \u5728 1936 \u5e74\u300c\u63d0\u51fa\u4f46\u672a\u8bc1\u660e\u300d\u7684\u4e00\u6761\u5b9a\u7406 [3]\u3002\u4ee5 Frobenius \u8303\u6570&#xff08;\u628a\u77e9\u9635\u6240\u6709\u5143\u7d20\u7684\u5e73\u65b9\u548c\u5f00\u65b9&#xff09;\u4e3a\u8bef\u5dee\u53e3\u5f84&#xff0c;\u7ed3\u8bba\u662f&#xff1a;<\/p>\n<ul>\n<li>\n<p>\u5728\u6240\u6709\u79e9\u4e0d\u8d85\u8fc7 r \u7684\u77e9\u9635\u91cc&#xff0c;A_r \u7684\u8bef\u5dee\u6700\u5c0f&#xff1b;<\/p>\n<\/li>\n<li>\n<p>\u6700\u5c0f\u8bef\u5dee\u6709\u95ed\u5f0f&#xff1a;<\/p>\n<p>\u2016A \u2212 A_r\u2016F &#061; \u221a(\u03c3{r&#043;1}\u00b2 &#043; \u03c3_{r&#043;2}\u00b2 &#043; \u2026 &#043; \u03c3_min\u00b2)<\/p>\n<\/li>\n<\/ul>\n<p>\u4e5f\u5c31\u662f\u8bf4&#xff0c;\u300c\u4fdd\u7559\u51e0\u5c42\u300d\u8fd9\u4e2a\u51b3\u5b9a\u53ef\u4ee5\u4e8b\u5148\u7b97\u6e05\u695a\u4ee3\u4ef7&#xff1a;\u4e22\u6389\u7684\u90a3\u90e8\u5206\u8bef\u5dee&#xff0c;\u6070\u597d\u7b49\u4e8e\u6240\u6709\u88ab\u4e22\u6389\u7684\u5947\u5f02\u503c\u7684\u5e73\u65b9\u548c\u518d\u5f00\u65b9\u3002\u8fd9\u6761\u6052\u7b49\u5f0f\u7684\u8bc1\u660e\u601d\u8def\u53ea\u7528\u4e00\u6761\u6027\u8d28\u2014\u2014Frobenius \u8303\u6570\u5728\u6b63\u4ea4\u53d8\u6362\u4e0b\u4e0d\u53d8&#xff1a;\u628a A \u4e0e\u4efb\u610f\u5019\u9009\u77e9\u9635 B \u540c\u65f6\u7528\u4e24\u7ec4\u65b9\u5411\u5939\u4f4f&#xff0c;\u95ee\u9898\u5c31\u5316\u6210\u5bf9\u89d2\u77e9\u9635\u4e0e\u67d0\u4e2a\u4e00\u822c\u77e9\u9635\u7684\u9010\u5143\u6bd4\u8f83&#xff0c;\u4efb\u4f55\u975e\u5bf9\u89d2\u5143\u90fd\u53ea\u4f1a\u8ba9\u5e73\u65b9\u548c\u53d8\u5927&#xff0c;\u5bf9\u89d2\u5143\u53d6 \u03c3\u1d62 \u65f6\u6700\u5c0f [1]\u3002<\/p>\n<p>\u628a\u5b83\u653e\u5230\u4e24\u4e2a\u5177\u4f53\u77e9\u9635\u4e0a\u590d\u7b97&#xff0c;\u4e24\u6761\u8def\u5f84&#xff08;\u76f4\u63a5\u7b97\u5dee\u77e9\u9635\u7684\u8303\u6570&#xff0c;\u4e0e\u6309\u95ed\u5f0f\u628a\u5c3e\u90e8\u5947\u5f02\u503c\u5e73\u65b9\u548c\u5f00\u65b9&#xff09;\u7ed9\u51fa\u7684\u6570\u5b57\u5e94\u5f53\u4e00\u81f4&#xff1a;<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/09\/20260923125705-6ab3cca13748a.png\" alt=\"\u4e24\u4e2a\u6574\u6570\u77e9\u9635\u7684\u622a\u65ad\u8bef\u5dee\u968f\u4fdd\u7559\u5c42\u6570\u4e0b\u964d\" \/><\/p>\n<p>\u56fe 3&#xff1a;\u4e24\u4e2a\u6574\u6570\u77e9\u9635\u7684\u622a\u65ad\u8bef\u5dee&#xff08;\u7eb5\u8f74\u4e3a\u4ee5 10 \u4e3a\u5e95\u7684\u5bf9\u6570&#xff09;<\/p>\n<p>\u56fe 3 \u7684\u7eb5\u8f74\u662f\u8bef\u5dee\u7684\u4ee5 10 \u4e3a\u5e95\u7684\u5bf9\u6570\u3002\u4e4b\u6240\u4ee5\u8981\u53d6\u5bf9\u6570&#xff0c;\u662f\u56e0\u4e3a\u4e24\u6761\u66f2\u7ebf\u7684\u91cf\u7a0b\u5dee\u5f97\u592a\u8fdc&#xff1a;3\u00d73 \u90a3\u4e2a\u77e9\u9635&#xff08;\u4e0b\u65b9\u90a3\u6761&#xff09;\u5728 r &#061; 1 \u65f6\u8bef\u5dee\u662f 0.897030554588&#xff0c;\u6b63\u597d\u7b49\u4e8e \u221a(0.875161\u00b2 &#043; 0.196867\u00b2)&#xff1b;r &#061; 2 \u65f6\u964d\u5230 0.196866521117&#xff0c;\u4e5f\u5c31\u662f\u7b2c\u4e09\u4e2a\u5947\u5f02\u503c\u672c\u8eab&#xff1b;r &#061; 3 \u65f6\u4e24\u6761\u8def\u5f84\u5206\u522b\u7ed9\u51fa 4.446\u00d710\u207b\u00b9\u2075 \u4e0e 0\u2014\u2014\u90fd\u5df2\u7ecf\u662f\u96f6&#xff0c;\u5dee\u522b\u53ea\u662f\u6d6e\u70b9\u820d\u5165\u7684\u65b9\u5411\u30026\u00d74 \u90a3\u4e2a\u77e9\u9635&#xff08;\u4e0a\u65b9\u90a3\u6761&#xff09;\u5728 r &#061; 1 \u65f6\u8bef\u5dee\u662f 1.735790466056&#xff0c;\u800c r &#061; 2 \u65f6\u76f4\u63a5\u843d\u5230 7.5\u00d710\u207b\u00b9\u2075\u2014\u2014\u56fe\u4e0a\u662f\u90a3\u6761\u7b14\u76f4\u4e0b\u5760\u7684\u865a\u7ebf\u3002\u4e24\u4e2a\u77e9\u9635\u5728\u300c\u8be5\u4e22\u7684\u90fd\u4e22\u6389\u4e4b\u540e\u300d\u90fd\u505c\u5728 10\u207b\u00b9\u2074 \u5230 10\u207b\u00b9\u2075 \u8fd9\u4e00\u5e26&#xff0c;\u90a3\u4e0d\u662f\u96f6&#xff0c;\u800c\u662f\u53cc\u7cbe\u5ea6\u5728\u8fd9\u4e2a\u89c4\u6a21\u4e0a\u7684\u820d\u5165\u5730\u677f&#xff1b;0.197 \u4e0e 10\u207b\u00b9\u2074 \u4e4b\u95f4\u5dee\u4e86\u5341\u56db\u4e2a\u6570\u91cf\u7ea7&#xff0c;\u53ea\u6709\u5bf9\u6570\u523b\u5ea6\u80fd\u540c\u65f6\u5bb9\u4e0b\u5b83\u4eec\u3002\u56db\u79cd\u4fdd\u7559\u5c42\u6570\u4e0b&#xff0c;\u4e24\u6761\u8def\u5f84&#xff08;\u76f4\u63a5\u7b97\u5dee\u77e9\u9635\u7684\u8303\u6570\u3001\u6309\u95ed\u5f0f\u628a\u5c3e\u90e8\u5e73\u65b9\u548c\u5f00\u65b9&#xff09;\u7684\u6700\u5927\u5dee\u5c31\u662f\u8fd9 4.446\u00d710\u207b\u00b9\u2075&#xff0c;\u51fa\u73b0\u5728 3\u00d73 \u77e9\u9635\u7684 r &#061; 3 \u90a3\u4e00\u70b9\u4e0a\u3002<\/p>\n<p>\u6709\u4e00\u4e2a\u5bf9\u7167\u80fd\u8bf4\u660e\u300c\u6700\u4f18\u300d\u8fd9\u4e24\u4e2a\u5b57\u7684\u5206\u91cf\u30023\u00d73 \u90a3\u4e2a\u77e9\u9635\u5982\u679c\u4e0d\u7528\u5947\u5f02\u503c\u5206\u89e3&#xff0c;\u800c\u662f\u968f\u624b\u6311\u4e5d\u4e2a\u300c\u53ea\u7559\u4e00\u4e2a\u5143\u7d20\u300d\u7684\u79e9\u4e00\u77e9\u9635\u91cc\u6700\u597d\u7684\u4e00\u4e2a&#xff0c;\u8bef\u5dee\u662f 14.282857&#xff1b;\u622a\u65ad\u5947\u5f02\u503c\u5206\u89e3\u7ed9\u51fa\u7684 0.897031 \u6bd4\u5b83\u5c0f\u4e86\u5c06\u8fd1\u5341\u516d\u500d\u3002\u800c 0.897031 \u8fd9\u4e2a\u6570\u8fd8\u4e0d\u662f\u300c\u78b0\u5de7\u4e0d\u9519\u300d\u2014\u2014\u6309\u4e0a\u9762\u7684\u95ed\u5f0f&#xff0c;\u5b83\u5c31\u662f\u7406\u8bba\u4e0b\u754c\u3002<\/p>\n<p>\u4e0b\u9762\u8fd9\u6bb5\u4ee3\u7801\u628a\u4e0a\u9762\u8fd9\u4e9b\u6570\u5b57\u5f53\u573a\u7b97\u4e00\u904d\u3002\u5b83\u53ea\u7528\u6807\u51c6\u5e93&#xff0c;\u4e0d\u542b\u4efb\u4f55\u968f\u673a\u6570&#xff0c;\u91cd\u590d\u8fd0\u884c\u7ed3\u679c\u5b8c\u5168\u4e00\u81f4&#xff1a;<\/p>\n<p><span class=\"token comment\"># \u6700\u5c0f\u53ef\u8fd0\u884c\u793a\u4f8b&#xff1a;\u5355\u8fb9 Jacobi \u8fed\u4ee3\u6c42\u5947\u5f02\u503c&#xff0c;\u5e76\u9010\u9879\u9a8c\u8bc1\u622a\u65ad\u8bef\u5dee\u6052\u7b49\u5f0f<\/span><br \/>\n<span class=\"token keyword\">import<\/span> math<\/p>\n<p><span class=\"token keyword\">def<\/span> <span class=\"token function\">svd_jacobi<\/span><span class=\"token punctuation\">(<\/span>A<span class=\"token punctuation\">,<\/span> tol<span class=\"token operator\">&#061;<\/span><span class=\"token number\">1e-15<\/span><span class=\"token punctuation\">,<\/span> max_sweeps<span class=\"token operator\">&#061;<\/span><span class=\"token number\">100<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">:<\/span><br \/>\n    <span class=\"token triple-quoted-string string\">&#034;&#034;&#034;\u8fd4\u56de U\u3001S\u3001V&#xff0c;\u6ee1\u8db3 A \u7ea6\u7b49\u4e8e U \u4e58 diag(S) \u4e58 V \u7684\u8f6c\u7f6e&#xff08;A \u4e3a m \u4e58 n&#xff0c;m \u4e0d\u5c0f\u4e8e n&#xff09;\u3002&#034;&#034;&#034;<\/span><br \/>\n    m<span class=\"token punctuation\">,<\/span> n <span class=\"token operator\">&#061;<\/span> <span class=\"token builtin\">len<\/span><span class=\"token punctuation\">(<\/span>A<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token builtin\">len<\/span><span class=\"token punctuation\">(<\/span>A<span class=\"token punctuation\">[<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span><br \/>\n    B <span class=\"token operator\">&#061;<\/span> <span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">[<\/span><span class=\"token builtin\">float<\/span><span class=\"token punctuation\">(<\/span>v<span class=\"token punctuation\">)<\/span> <span class=\"token keyword\">for<\/span> v <span class=\"token keyword\">in<\/span> row<span class=\"token punctuation\">]<\/span> <span class=\"token keyword\">for<\/span> row <span class=\"token keyword\">in<\/span> A<span class=\"token punctuation\">]<\/span><br \/>\n    V <span class=\"token operator\">&#061;<\/span> <span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">[<\/span><span class=\"token number\">1.0<\/span> <span class=\"token keyword\">if<\/span> i <span class=\"token operator\">&#061;&#061;<\/span> j <span class=\"token keyword\">else<\/span> <span class=\"token number\">0.0<\/span> <span class=\"token keyword\">for<\/span> j <span class=\"token keyword\">in<\/span> <span class=\"token builtin\">range<\/span><span class=\"token punctuation\">(<\/span>n<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">]<\/span> <span class=\"token keyword\">for<\/span> i <span class=\"token keyword\">in<\/span> <span class=\"token builtin\">range<\/span><span class=\"token punctuation\">(<\/span>n<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">]<\/span><br \/>\n    <span class=\"token keyword\">for<\/span> _ <span class=\"token keyword\">in<\/span> <span class=\"token builtin\">range<\/span><span class=\"token punctuation\">(<\/span>max_sweeps<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">:<\/span><br \/>\n        rotated <span class=\"token operator\">&#061;<\/span> <span class=\"token boolean\">False<\/span><br \/>\n        <span class=\"token keyword\">for<\/span> p <span class=\"token keyword\">in<\/span> <span class=\"token builtin\">range<\/span><span class=\"token punctuation\">(<\/span>n <span class=\"token operator\">&#8211;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">:<\/span><br \/>\n            <span class=\"token keyword\">for<\/span> q <span class=\"token keyword\">in<\/span> <span class=\"token builtin\">range<\/span><span class=\"token punctuation\">(<\/span>p <span class=\"token operator\">&#043;<\/span> <span class=\"token number\">1<\/span><span class=\"token punctuation\">,<\/span> n<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">:<\/span><br \/>\n                alpha <span class=\"token operator\">&#061;<\/span> <span class=\"token builtin\">sum<\/span><span class=\"token punctuation\">(<\/span>B<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>p<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">**<\/span> <span class=\"token number\">2<\/span> <span class=\"token keyword\">for<\/span> i <span class=\"token keyword\">in<\/span> <span class=\"token builtin\">range<\/span><span class=\"token punctuation\">(<\/span>m<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><br \/>\n                beta <span class=\"token operator\">&#061;<\/span> <span class=\"token builtin\">sum<\/span><span class=\"token punctuation\">(<\/span>B<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>q<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">**<\/span> <span class=\"token number\">2<\/span> <span class=\"token keyword\">for<\/span> i <span class=\"token keyword\">in<\/span> <span class=\"token builtin\">range<\/span><span class=\"token punctuation\">(<\/span>m<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><br \/>\n                gamma <span class=\"token operator\">&#061;<\/span> <span class=\"token builtin\">sum<\/span><span class=\"token punctuation\">(<\/span>B<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>p<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">*<\/span> B<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>q<span class=\"token punctuation\">]<\/span> <span class=\"token keyword\">for<\/span> i <span class=\"token keyword\">in<\/span> <span class=\"token builtin\">range<\/span><span class=\"token punctuation\">(<\/span>m<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><br \/>\n                <span class=\"token keyword\">if<\/span> gamma <span class=\"token operator\">&#061;&#061;<\/span> <span class=\"token number\">0.0<\/span> <span class=\"token keyword\">or<\/span> <span class=\"token builtin\">abs<\/span><span class=\"token punctuation\">(<\/span>gamma<span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&lt;&#061;<\/span> tol <span class=\"token operator\">*<\/span> math<span class=\"token punctuation\">.<\/span>sqrt<span class=\"token punctuation\">(<\/span>alpha <span class=\"token operator\">*<\/span> beta<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">:<\/span><br \/>\n                    <span class=\"token keyword\">continue<\/span><br \/>\n                zeta <span class=\"token operator\">&#061;<\/span> <span class=\"token punctuation\">(<\/span>beta <span class=\"token operator\">&#8211;<\/span> alpha<span class=\"token punctuation\">)<\/span> <span class=\"token operator\">\/<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token number\">2.0<\/span> <span class=\"token operator\">*<\/span> gamma<span class=\"token punctuation\">)<\/span><br \/>\n                t <span class=\"token operator\">&#061;<\/span> math<span class=\"token punctuation\">.<\/span>copysign<span class=\"token punctuation\">(<\/span><span class=\"token number\">1.0<\/span><span class=\"token punctuation\">,<\/span> zeta<span class=\"token punctuation\">)<\/span> <span class=\"token operator\">\/<\/span> <span class=\"token punctuation\">(<\/span><span class=\"token builtin\">abs<\/span><span class=\"token punctuation\">(<\/span>zeta<span class=\"token punctuation\">)<\/span> <span class=\"token operator\">&#043;<\/span> math<span class=\"token punctuation\">.<\/span>sqrt<span class=\"token punctuation\">(<\/span><span class=\"token number\">1.0<\/span> <span class=\"token operator\">&#043;<\/span> zeta <span class=\"token operator\">*<\/span> zeta<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><br \/>\n                c <span class=\"token operator\">&#061;<\/span> <span class=\"token number\">1.0<\/span> <span class=\"token operator\">\/<\/span> math<span class=\"token punctuation\">.<\/span>sqrt<span class=\"token punctuation\">(<\/span><span class=\"token number\">1.0<\/span> <span class=\"token operator\">&#043;<\/span> t <span class=\"token operator\">*<\/span> t<span class=\"token punctuation\">)<\/span><br \/>\n                s <span class=\"token operator\">&#061;<\/span> c <span class=\"token operator\">*<\/span> t<br \/>\n                <span class=\"token keyword\">for<\/span> i <span class=\"token keyword\">in<\/span> <span class=\"token builtin\">range<\/span><span class=\"token punctuation\">(<\/span>m<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">:<\/span><br \/>\n                    bp<span class=\"token punctuation\">,<\/span> bq <span class=\"token operator\">&#061;<\/span> B<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>p<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">,<\/span> B<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>q<span class=\"token punctuation\">]<\/span><br \/>\n                    B<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>p<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> c <span class=\"token operator\">*<\/span> bp <span class=\"token operator\">&#8211;<\/span> s <span class=\"token operator\">*<\/span> bq<br \/>\n                    B<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>q<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> s <span class=\"token operator\">*<\/span> bp <span class=\"token operator\">&#043;<\/span> c <span class=\"token operator\">*<\/span> bq<br \/>\n                <span class=\"token keyword\">for<\/span> i <span class=\"token keyword\">in<\/span> <span class=\"token builtin\">range<\/span><span class=\"token punctuation\">(<\/span>n<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">:<\/span><br \/>\n                    vp<span class=\"token punctuation\">,<\/span> vq <span class=\"token operator\">&#061;<\/span> V<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>p<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">,<\/span> V<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>q<span class=\"token punctuation\">]<\/span><br \/>\n                    V<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>p<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> c <span class=\"token operator\">*<\/span> vp <span class=\"token operator\">&#8211;<\/span> s <span class=\"token operator\">*<\/span> vq<br \/>\n                    V<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>q<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#061;<\/span> s <span class=\"token operator\">*<\/span> vp <span class=\"token operator\">&#043;<\/span> c <span class=\"token operator\">*<\/span> vq<br \/>\n                rotated <span class=\"token operator\">&#061;<\/span> <span class=\"token boolean\">True<\/span><br \/>\n        <span class=\"token keyword\">if<\/span> <span class=\"token keyword\">not<\/span> rotated<span class=\"token punctuation\">:<\/span><br \/>\n            <span class=\"token keyword\">break<\/span><br \/>\n    norms <span class=\"token operator\">&#061;<\/span> <span class=\"token punctuation\">[<\/span>math<span class=\"token punctuation\">.<\/span>sqrt<span class=\"token punctuation\">(<\/span><span class=\"token builtin\">sum<\/span><span class=\"token punctuation\">(<\/span>B<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">**<\/span> <span class=\"token number\">2<\/span> <span class=\"token keyword\">for<\/span> i <span class=\"token keyword\">in<\/span> <span class=\"token builtin\">range<\/span><span class=\"token punctuation\">(<\/span>m<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token keyword\">for<\/span> j <span class=\"token keyword\">in<\/span> <span class=\"token builtin\">range<\/span><span class=\"token punctuation\">(<\/span>n<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">]<\/span><br \/>\n    order <span class=\"token operator\">&#061;<\/span> <span class=\"token builtin\">sorted<\/span><span class=\"token punctuation\">(<\/span><span class=\"token builtin\">range<\/span><span class=\"token punctuation\">(<\/span>n<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">,<\/span> key<span class=\"token operator\">&#061;<\/span><span class=\"token keyword\">lambda<\/span> j<span class=\"token punctuation\">:<\/span> <span class=\"token operator\">&#8211;<\/span>norms<span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span><br \/>\n    S <span class=\"token operator\">&#061;<\/span> <span class=\"token punctuation\">[<\/span>norms<span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token keyword\">for<\/span> j <span class=\"token keyword\">in<\/span> order<span class=\"token punctuation\">]<\/span><br \/>\n    U <span class=\"token operator\">&#061;<\/span> <span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">[<\/span>B<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">\/<\/span> norms<span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token keyword\">for<\/span> j <span class=\"token keyword\">in<\/span> order<span class=\"token punctuation\">]<\/span> <span class=\"token keyword\">for<\/span> i <span class=\"token keyword\">in<\/span> <span class=\"token builtin\">range<\/span><span class=\"token punctuation\">(<\/span>m<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">]<\/span><br \/>\n    Vs <span class=\"token operator\">&#061;<\/span> <span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">[<\/span>V<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token keyword\">for<\/span> j <span class=\"token keyword\">in<\/span> order<span class=\"token punctuation\">]<\/span> <span class=\"token keyword\">for<\/span> i <span class=\"token keyword\">in<\/span> <span class=\"token builtin\">range<\/span><span class=\"token punctuation\">(<\/span>n<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">]<\/span><br \/>\n    <span class=\"token keyword\">return<\/span> U<span class=\"token punctuation\">,<\/span> S<span class=\"token punctuation\">,<\/span> Vs<\/p>\n<p><span class=\"token keyword\">def<\/span> <span class=\"token function\">frobenius<\/span><span class=\"token punctuation\">(<\/span>M<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">:<\/span><br \/>\n    <span class=\"token keyword\">return<\/span> math<span class=\"token punctuation\">.<\/span>sqrt<span class=\"token punctuation\">(<\/span><span class=\"token builtin\">sum<\/span><span class=\"token punctuation\">(<\/span>v <span class=\"token operator\">*<\/span> v <span class=\"token keyword\">for<\/span> row <span class=\"token keyword\">in<\/span> M <span class=\"token keyword\">for<\/span> v <span class=\"token keyword\">in<\/span> row<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><\/p>\n<p><span class=\"token keyword\">def<\/span> <span class=\"token function\">best_rank_r<\/span><span class=\"token punctuation\">(<\/span>U<span class=\"token punctuation\">,<\/span> S<span class=\"token punctuation\">,<\/span> V<span class=\"token punctuation\">,<\/span> r<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">:<\/span><br \/>\n    m<span class=\"token punctuation\">,<\/span> n <span class=\"token operator\">&#061;<\/span> <span class=\"token builtin\">len<\/span><span class=\"token punctuation\">(<\/span>U<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token builtin\">len<\/span><span class=\"token punctuation\">(<\/span>S<span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token keyword\">return<\/span> <span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">[<\/span><span class=\"token builtin\">sum<\/span><span class=\"token punctuation\">(<\/span>U<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>k<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">*<\/span> S<span class=\"token punctuation\">[<\/span>k<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">*<\/span> V<span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>k<span class=\"token punctuation\">]<\/span> <span class=\"token keyword\">for<\/span> k <span class=\"token keyword\">in<\/span> <span class=\"token builtin\">range<\/span><span class=\"token punctuation\">(<\/span>r<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span> <span class=\"token keyword\">for<\/span> j <span class=\"token keyword\">in<\/span> <span class=\"token builtin\">range<\/span><span class=\"token punctuation\">(<\/span>n<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">]<\/span> <span class=\"token keyword\">for<\/span> i <span class=\"token keyword\">in<\/span> <span class=\"token builtin\">range<\/span><span class=\"token punctuation\">(<\/span>m<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">]<\/span><\/p>\n<p>A <span class=\"token operator\">&#061;<\/span> <span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">[<\/span><span class=\"token number\">1.0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">2.0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">3.0<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token punctuation\">[<\/span><span class=\"token number\">4.0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">5.0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">6.0<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token punctuation\">[<\/span><span class=\"token number\">7.0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">8.0<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token number\">10.0<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">]<\/span><br \/>\nU<span class=\"token punctuation\">,<\/span> S<span class=\"token punctuation\">,<\/span> V <span class=\"token operator\">&#061;<\/span> svd_jacobi<span class=\"token punctuation\">(<\/span>A<span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token keyword\">print<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;\u5947\u5f02\u503c:&#034;<\/span><span class=\"token punctuation\">,<\/span> <span class=\"token punctuation\">[<\/span><span class=\"token string\">&#034;%.12f&#034;<\/span> <span class=\"token operator\">%<\/span> s <span class=\"token keyword\">for<\/span> s <span class=\"token keyword\">in<\/span> S<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token keyword\">print<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;\u6761\u4ef6\u6570: %.12f&#034;<\/span> <span class=\"token operator\">%<\/span> <span class=\"token punctuation\">(<\/span>S<span class=\"token punctuation\">[<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">]<\/span> <span class=\"token operator\">\/<\/span> S<span class=\"token punctuation\">[<\/span><span class=\"token number\">2<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token keyword\">print<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;\u884c\u5217\u5f0f\u7edd\u5bf9\u503c: 3.0   \u5947\u5f02\u503c\u4e58\u79ef: %.12f&#034;<\/span> <span class=\"token operator\">%<\/span> <span class=\"token punctuation\">(<\/span>S<span class=\"token punctuation\">[<\/span><span class=\"token number\">0<\/span><span class=\"token punctuation\">]<\/span> <span class=\"token operator\">*<\/span> S<span class=\"token punctuation\">[<\/span><span class=\"token number\">1<\/span><span class=\"token punctuation\">]<\/span> <span class=\"token operator\">*<\/span> S<span class=\"token punctuation\">[<\/span><span class=\"token number\">2<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><br \/>\n<span class=\"token keyword\">for<\/span> r <span class=\"token keyword\">in<\/span> <span class=\"token builtin\">range<\/span><span class=\"token punctuation\">(<\/span><span class=\"token number\">4<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">:<\/span><br \/>\n    Ar <span class=\"token operator\">&#061;<\/span> best_rank_r<span class=\"token punctuation\">(<\/span>U<span class=\"token punctuation\">,<\/span> S<span class=\"token punctuation\">,<\/span> V<span class=\"token punctuation\">,<\/span> r<span class=\"token punctuation\">)<\/span><br \/>\n    direct <span class=\"token operator\">&#061;<\/span> frobenius<span class=\"token punctuation\">(<\/span><span class=\"token punctuation\">[<\/span><span class=\"token punctuation\">[<\/span>A<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token operator\">&#8211;<\/span> Ar<span class=\"token punctuation\">[<\/span>i<span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">[<\/span>j<span class=\"token punctuation\">]<\/span> <span class=\"token keyword\">for<\/span> j <span class=\"token keyword\">in<\/span> <span class=\"token builtin\">range<\/span><span class=\"token punctuation\">(<\/span><span class=\"token number\">3<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">]<\/span> <span class=\"token keyword\">for<\/span> i <span class=\"token keyword\">in<\/span> <span class=\"token builtin\">range<\/span><span class=\"token punctuation\">(<\/span><span class=\"token number\">3<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span><br \/>\n    tail <span class=\"token operator\">&#061;<\/span> math<span class=\"token punctuation\">.<\/span>sqrt<span class=\"token punctuation\">(<\/span><span class=\"token builtin\">sum<\/span><span class=\"token punctuation\">(<\/span>s <span class=\"token operator\">*<\/span> s <span class=\"token keyword\">for<\/span> s <span class=\"token keyword\">in<\/span> S<span class=\"token punctuation\">[<\/span>r<span class=\"token punctuation\">:<\/span><span class=\"token punctuation\">]<\/span><span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><br \/>\n    <span class=\"token keyword\">print<\/span><span class=\"token punctuation\">(<\/span><span class=\"token string\">&#034;r&#061;%d \u76f4\u63a5\u7b97\u51fa&#061;%.12f  \u5c3e\u90e8\u5e73\u65b9\u548c\u5f00\u65b9&#061;%.12f&#034;<\/span> <span class=\"token operator\">%<\/span> <span class=\"token punctuation\">(<\/span>r<span class=\"token punctuation\">,<\/span> direct<span class=\"token punctuation\">,<\/span> tail<span class=\"token punctuation\">)<\/span><span class=\"token punctuation\">)<\/span><\/p>\n<p>\u5728 Python 3.9 \u4e0e 3.12 \u4e24\u4e2a\u7248\u672c\u4e0b\u8fd0\u884c&#xff0c;\u8f93\u51fa\u9010\u5b57\u8282\u76f8\u540c&#xff1a;<\/p>\n<p>\u5947\u5f02\u503c: [&#039;17.412505166809&#039;, &#039;0.875161350110&#039;, &#039;0.196866521117&#039;]<br \/>\n\u6761\u4ef6\u6570: 88.448279920699<br \/>\n\u884c\u5217\u5f0f\u7edd\u5bf9\u503c: 3.0   \u5947\u5f02\u503c\u4e58\u79ef: 3.000000000000<br \/>\nr&#061;0 \u76f4\u63a5\u7b97\u51fa&#061;17.435595774163  \u5c3e\u90e8\u5e73\u65b9\u548c\u5f00\u65b9&#061;17.435595774163<br \/>\nr&#061;1 \u76f4\u63a5\u7b97\u51fa&#061;0.897030554588  \u5c3e\u90e8\u5e73\u65b9\u548c\u5f00\u65b9&#061;0.897030554588<br \/>\nr&#061;2 \u76f4\u63a5\u7b97\u51fa&#061;0.196866521117  \u5c3e\u90e8\u5e73\u65b9\u548c\u5f00\u65b9&#061;0.196866521117<br \/>\nr&#061;3 \u76f4\u63a5\u7b97\u51fa&#061;0.000000000000  \u5c3e\u90e8\u5e73\u65b9\u548c\u5f00\u65b9&#061;0.000000000000<\/p>\n<p>\u6bcf\u4e00\u884c\u90fd\u662f\u4e24\u6761\u8def\u5f84\u5e76\u6392&#xff1a;\u5de6\u8fb9\u628a A \u4e0e\u91cd\u5efa\u51fa\u6765\u7684 A_r \u76f8\u51cf\u3001\u518d\u7b97\u5168\u90e8\u5143\u7d20\u7684\u5e73\u65b9\u548c\u5f00\u65b9&#xff1b;\u53f3\u8fb9\u53ea\u628a\u7b2c r \u5c42\u4e4b\u540e\u7684\u5947\u5f02\u503c\u5e73\u65b9\u76f8\u52a0\u518d\u5f00\u65b9\u3002\u56db\u79cd\u4fdd\u7559\u5c42\u6570\u4e0b\u4e24\u5217\u6570\u5b57\u5b8c\u5168\u76f8\u540c&#xff0c;\u8fd9\u5c31\u662f\u7b2c\u4e94\u8282\u90a3\u6761\u6052\u7b49\u5f0f\u5728\u5177\u4f53\u77e9\u9635\u4e0a\u7684\u6837\u5b50\u3002<\/p>\n<h3>\u516d\u3001\u6362\u4e00\u4e2a\u4efb\u52a1&#xff1a;\u540c\u4e00\u4e2a\u77e9\u9635\u53bb\u89e3\u65b9\u7a0b&#xff0c;\u6761\u4ef6\u6570\u628a\u8bef\u5dee\u653e\u5927\u591a\u5c11<\/h3>\n<p>\u4f4e\u79e9\u903c\u8fd1\u95ee\u7684\u662f\u300c\u54ea\u4e2a\u77e9\u9635\u79bb\u5b83\u6700\u8fd1\u300d&#xff0c;\u73b0\u5728\u6362\u4e00\u4e2a\u95ee\u9898&#xff1a;\u628a\u8fd9\u4e2a\u77e9\u9635\u5f53\u4f5c\u7ebf\u6027\u65b9\u7a0b\u7ec4 Ax &#061; b \u7684\u7cfb\u6570\u77e9\u9635&#xff0c;\u5b83\u7684\u8c31\u7ed3\u6784\u4f1a\u5e72\u4ec0\u4e48\u3002<\/p>\n<p>\u7b54\u6848\u8fd8\u662f\u90a3\u4e32\u5947\u5f02\u503c&#xff0c;\u53ea\u4e0d\u8fc7\u8fd9\u6b21\u5173\u5fc3\u7684\u662f\u6bd4\u503c\u3002\u6761\u4ef6\u6570\u5b9a\u4e49\u4e3a<\/p>\n<p>\u03ba(A) &#061; \u03c3_max \/ \u03c3_min<\/p>\n<p>\u5b83\u5927&#xff0c;\u8bf4\u660e\u692d\u7403\u88ab\u62c9\u5f97\u53c8\u957f\u53c8\u6241&#xff1a;\u67d0\u4e9b\u65b9\u5411\u7684\u8f93\u5165\u88ab\u653e\u5927\u5f88\u591a&#xff0c;\u53e6\u4e00\u4e9b\u88ab\u538b\u5f97\u5f88\u5c0f\u3002\u8fd9\u4ef6\u4e8b\u5bf9\u89e3\u65b9\u7a0b\u7684\u5f71\u54cd\u662f\u76f4\u63a5\u7684\u3002\u8bbe\u53f3\u7aef\u6709\u4e00\u4e2a\u6270\u52a8 \u03b4b&#xff0c;\u89e3\u5c31\u4ece x \u53d8\u6210 x &#043; \u03b4x\u3002\u628a\u4e24\u4ef6\u4e8b\u5206\u522b\u5199\u6210<\/p>\n<p>x &#061; A\u207b\u00b9b        \u4e0e        \u03b4x &#061; A\u207b\u00b9\u03b4b<\/p>\n<p>\u5404\u81ea\u53d6\u8303\u6570\u518d\u76f8\u9664&#xff0c;\u4e2d\u95f4\u63d2\u5165 \u2016A\u2016\u00b7\u2016A\u2016\u207b\u00b9 \u8fd9\u4e2a\u7b49\u4e8e 1 \u7684\u56e0\u5b50&#xff0c;\u5c31\u5f97\u5230<\/p>\n<p>\u2016\u03b4x\u2016 \/ \u2016x\u2016  \u2264  \u03ba(A) \u00b7 \u2016\u03b4b\u2016 \/ \u2016b\u2016<\/p>\n<p>\u5de6\u8fb9\u662f\u89e3\u7684\u76f8\u5bf9\u8bef\u5dee&#xff0c;\u53f3\u8fb9\u662f\u8f93\u5165\u76f8\u5bf9\u8bef\u5dee\u4e58\u4e0a\u6761\u4ef6\u6570\u3002\u8fd9\u6761\u63a8\u5bfc\u53ea\u7528\u8303\u6570\u7684\u4e24\u6761\u6027\u8d28&#xff08;\u2016A\u207b\u00b9\u03b4b\u2016 \u2264 \u2016A\u207b\u00b9\u2016\u00b7\u2016\u03b4b\u2016 \u4e0e \u2016b\u2016 \u2264 \u2016A\u2016\u00b7\u2016x\u2016&#xff09;&#xff0c;\u4e0d\u4f9d\u8d56\u4efb\u4f55\u5177\u4f53\u7b97\u6cd5&#xff0c;\u56e0\u6b64\u5bf9\u4efb\u4f55\u6c42\u89e3\u65b9\u6cd5\u90fd\u6210\u7acb\u3002\u5b83\u662f\u4e00\u6761\u6700\u574f\u60c5\u51b5\u7684\u754c&#xff1a;\u5b83\u8bf4\u7684\u662f\u300c\u653e\u5927\u500d\u6570\u4e0d\u4f1a\u8d85\u8fc7 \u03ba\u300d&#xff0c;\u4e0d\u662f\u300c\u4e00\u5b9a\u4f1a\u653e\u5927 \u03ba \u500d\u300d\u3002\u6270\u52a8\u65b9\u5411\u4e0e\u89e3\u5411\u91cf\u7684\u51e0\u4f55\u5173\u7cfb\u51b3\u5b9a\u4e86\u5b9e\u9645\u653e\u5927\u591a\u5c11\u3002<\/p>\n<p>\u8fd9\u53e5\u8bdd\u53ef\u4ee5\u91cf\u51fa\u6765\u3002\u53d6 10 \u4e2a\u7b49\u8ddd\u70b9\u30019 \u6b21\u591a\u9879\u5f0f\u7684\u62df\u5408\u95ee\u9898&#xff08;\u4e0b\u4e00\u8282\u4f1a\u7528\u5230\u540c\u4e00\u6279\u70b9&#xff09;&#xff0c;\u8bbe\u8ba1\u77e9\u9635\u662f\u4e00\u4e2a 10\u00d710 \u7684\u65b9\u9635&#xff0c;\u5b83\u7684\u6761\u4ef6\u6570\u662f 2.568136\u00d710\u2077\u3002\u5f80\u53f3\u7aef\u52a0\u4e00\u4e2a\u76f8\u5bf9\u5927\u5c0f 10\u207b\u00b9\u2070 \u7684\u6270\u52a8&#xff1a;<\/p>\n<ul>\n<li>\u5982\u679c\u6270\u52a8\u65b9\u5411\u662f\u300c\u6700\u654f\u611f\u300d\u7684\u90a3\u4e2a\u65b9\u5411&#xff0c;\u89e3\u7684\u76f8\u5bf9\u8bef\u5dee\u662f 6.089035\u00d710\u207b\u2074&#xff0c;\u653e\u5927\u4e86 6.089035\u00d710\u2076 \u500d&#xff1b;<\/li>\n<li>\u5982\u679c\u6270\u52a8\u65b9\u5411\u6362\u6210\u300c\u6700\u4e0d\u654f\u611f\u300d\u7684\u90a3\u4e2a\u65b9\u5411&#xff0c;\u89e3\u7684\u76f8\u5bf9\u8bef\u5dee\u53ea\u6709 2.370996\u00d710\u207b\u00b9\u00b9&#xff0c;\u653e\u5927\u500d\u6570 0.237099\u2014\u2014\u4e0d\u4f46\u6ca1\u653e\u5927&#xff0c;\u53cd\u800c\u7f29\u5c0f\u4e86\u3002<\/li>\n<\/ul>\n<p>\u628a\u6270\u52a8\u5e45\u5ea6\u4ece 10\u207b\u00b9\u2070 \u4e00\u8def\u52a0\u5230 10\u207b\u2074&#xff0c;\u8fd9\u4e24\u4e2a\u653e\u5927\u500d\u6570\u4e00\u4f4d\u4e0d\u53d8&#xff08;\u6270\u52a8\u662f\u7cbe\u786e\u7ebf\u6027\u7684&#xff09;&#xff0c;\u800c\u5b83\u4eec\u90fd\u843d\u5728\u6761\u4ef6\u6570 2.568136\u00d710\u2077 \u8fd9\u6761\u754c\u4e4b\u5185\u3002\u6240\u4ee5\u6761\u4ef6\u6570\u7684\u6b63\u786e\u8bfb\u6cd5\u662f&#xff1a;\u5b83\u662f\u4e00\u4e2a\u627f\u8bfa\u4e0a\u9650\u7684\u6307\u6807\u3002\u770b\u5230 \u03ba &#061; 10\u2077&#xff0c;\u80fd\u786e\u5b9a\u7684\u53ea\u662f\u300c\u6700\u574f\u60c5\u51b5\u4e0b\u89e3\u4f1a\u4e22\u6389 7 \u4f4d\u6709\u6548\u6570\u5b57\u300d&#xff1b;\u81f3\u4e8e\u624b\u4e0a\u7684\u8fd9\u4e00\u4efd\u6570\u636e\u5b9e\u9645\u4e22\u6389\u51e0\u4f4d&#xff0c;\u8981\u770b\u8bef\u5dee\u843d\u5728\u54ea\u4e2a\u65b9\u5411\u4e0a\u3002<\/p>\n<h3>\u4e03\u3001\u4e3a\u4ec0\u4e48\u4e0d\u8be5\u76f4\u63a5\u89e3\u6b63\u89c4\u65b9\u7a0b<\/h3>\n<p>\u6700\u5c0f\u4e8c\u4e58\u95ee\u9898\u7684\u6807\u51c6\u89e3\u6cd5\u662f\u89e3\u6b63\u89c4\u65b9\u7a0b&#xff1a;<\/p>\n<p>A\u1d40A x &#061; A\u1d40b<\/p>\n<p>\u5b83\u7684\u63a8\u5bfc\u5f88\u6f02\u4eae&#xff1a;\u628a\u6b8b\u5dee\u5e73\u65b9\u548c \u2016Ax \u2212 b\u2016\u00b2 \u5bf9 x \u6c42\u5bfc\u3001\u4ee4\u5bfc\u6570\u4e3a\u96f6&#xff0c;\u5c31\u5f97\u5230\u8fd9\u7ec4\u65b9\u7a0b\u3002\u53ef\u60dc\u5b83\u7684\u6570\u503c\u6027\u8d28\u5f88\u5dee&#xff0c;\u539f\u56e0\u53ef\u4ee5\u4e00\u884c\u5199\u5b8c\u2014\u2014A\u1d40A \u7684\u5947\u5f02\u503c\u662f A \u7684\u5947\u5f02\u503c\u7684\u5e73\u65b9\u3002\u628a A &#061; U\u03a3V\u1d40 \u4ee3\u8fdb\u53bb&#xff1a;<\/p>\n<p>A\u1d40A &#061; (U\u03a3V\u1d40)\u1d40(U\u03a3V\u1d40) &#061; V\u03a3\u1d40U\u1d40U\u03a3V\u1d40 &#061; V\u03a3\u00b2V\u1d40<\/p>\n<p>\u4e2d\u95f4\u90a3\u6b65 U\u1d40U &#061; I \u662f\u6b63\u4ea4\u77e9\u9635\u7684\u6027\u8d28\u3002\u4e8e\u662f A\u1d40A \u7684\u5947\u5f02\u503c\u5c31\u662f \u03c3\u2081\u00b2, \u03c3\u2082\u00b2, \u2026, \u03c3_min\u00b2&#xff0c;\u6761\u4ef6\u6570\u968f\u4e4b\u53d8\u6210<\/p>\n<p>\u03ba(A\u1d40A) &#061; \u03c3_max\u00b2 \/ \u03c3_min\u00b2 &#061; \u03ba(A)\u00b2<\/p>\n<p>\u6761\u4ef6\u6570\u88ab\u5e73\u65b9&#xff0c;\u610f\u5473\u7740\u89e3\u7684\u76f8\u5bf9\u8bef\u5dee\u754c\u4ece \u03ba\u00b7\u03b5 \u53d8\u6210 \u03ba\u00b2\u00b7\u03b5&#xff08;\u03b5 \u662f\u673a\u5668\u7cbe\u5ea6&#xff09;\u3002\u5728\u53cc\u7cbe\u5ea6\u4e0b \u03b5 \u2248 2.2\u00d710\u207b\u00b9\u2076&#xff1a;\u5982\u679c \u03ba \u662f 10\u2076&#xff0c;\u76f4\u63a5\u5bf9 A \u505a\u6b63\u4ea4\u53d8\u6362\u65f6\u754c\u662f 2.2\u00d710\u207b\u00b9\u2070&#xff0c;\u8fd8\u80fd\u4fdd\u4f4f\u4e5d\u5230\u5341\u4f4d\u6709\u6548\u6570\u5b57&#xff1b;\u663e\u5f0f\u4e58\u51fa A\u1d40A \u4e4b\u540e\u754c\u53d8\u6210 10\u00b9\u00b2\u00b7\u03b5 \u2248 2.2\u00d710\u207b\u2074&#xff0c;\u53ea\u5269\u4e09\u5230\u56db\u4f4d\u3002\u800c\u4e14 A\u1d40A \u4e00\u65e6\u5728\u6d6e\u70b9\u91cc\u88ab\u4e58\u51fa\u6765&#xff0c;\u90a3\u4e9b\u5728 A \u91cc\u53ea\u662f\u300c\u5f88\u5c0f\u300d\u7684\u5947\u5f02\u503c\u4f1a\u5728\u5e73\u65b9\u540e\u6c89\u5230\u820d\u5165\u566a\u58f0\u4ee5\u4e0b&#xff0c;\u4fe1\u606f\u5728\u90a3\u4e00\u6b65\u5c31\u5df2\u7ecf\u4e22\u4e86\u2014\u20141965 \u5e74\u90a3\u7bc7\u6587\u7ae0\u5728\u8ba8\u8bba\u4f2a\u9006\u65f6\u6b63\u662f\u8fd9\u4e48\u8bf4\u7684&#xff1a;\u7528\u666e\u901a\u6d6e\u70b9\u7b97\u672f\u8ba1\u7b97 A\u1d40A&#xff0c;\u4f1a\u4e25\u91cd\u635f\u5bb3\u8f83\u5c0f\u7684\u5947\u5f02\u503c\u4ee5\u53ca\u5bf9\u5e94\u7684\u65b9\u5411 [1]\u3002<\/p>\n<p>\u66ff\u4ee3\u8def\u7ebf\u662f\u7528\u6b63\u4ea4\u53d8\u6362\u3002\u628a A \u5206\u89e3\u6210 A &#061; QR&#xff08;Q \u7684\u5217\u6b63\u4ea4&#xff0c;R \u4e0a\u4e09\u89d2&#xff09;&#xff0c;\u6b8b\u5dee\u5e73\u65b9\u548c\u5c31\u53d8\u6210 \u2016QRx \u2212 b\u2016\u00b2 &#061; \u2016Rx \u2212 Q\u1d40b\u2016\u00b2&#xff08;\u6b63\u4ea4\u53d8\u6362\u4e0d\u6539\u53d8\u957f\u5ea6&#xff09;&#xff0c;\u4e8e\u662f\u6700\u5c0f\u4e8c\u4e58\u95ee\u9898\u5316\u6210\u4e00\u4e2a\u4e09\u89d2\u5f62\u65b9\u7a0b\u7ec4&#xff0c;\u56de\u4ee3\u5373\u53ef\u3002\u8fd9\u6761\u8def\u7ebf\u7684\u6761\u4ef6\u6570\u7531 R \u51b3\u5b9a&#xff0c;\u800c R \u7684\u5947\u5f02\u503c\u4e0e A \u7684\u76f8\u540c&#xff0c;\u6240\u4ee5\u653e\u5927\u500d\u6570\u662f \u03ba \u800c\u4e0d\u662f \u03ba\u00b2\u3002\u7528 Householder \u53cd\u5c04\u628a\u77e9\u9635\u9010\u6b65\u5316\u6210\u4e09\u89d2\u5f62\u5f62\u5f0f&#xff0c;\u662f 1965 \u5e74\u5c31\u5199\u6e05\u695a\u7684\u505a\u6cd5 [4]&#xff1b;\u4e94\u5e74\u540e&#xff0c;\u628a\u5947\u5f02\u503c\u5206\u89e3\u672c\u8eab\u62ff\u6765\u89e3\u6700\u5c0f\u4e8c\u4e58\u3001\u5e76\u628a\u5b83\u5199\u6210\u53ef\u590d\u7528\u8fc7\u7a0b\u7684\u65b9\u6848\u4e5f\u51fa\u73b0\u4e86 [5]\u3002<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/09\/20260923125706-6ab3cca20304f.png\" alt=\"\u8bbe\u8ba1\u77e9\u9635\u4e0e\u5b83\u7684\u6b63\u89c4\u65b9\u7a0b\u77e9\u9635\u7684\u6761\u4ef6\u6570\u968f\u591a\u9879\u5f0f\u6b21\u6570\u589e\u957f\" \/><\/p>\n<p>\u56fe 4&#xff1a;\u8bbe\u8ba1\u77e9\u9635\u4e0e\u6b63\u89c4\u65b9\u7a0b\u77e9\u9635\u7684\u6761\u4ef6\u6570&#xff08;\u7eb5\u8f74\u4e3a\u4ee5 10 \u4e3a\u5e95\u7684\u5bf9\u6570&#xff09;<\/p>\n<p>\u56fe 4 \u628a\u8fd9\u6761\u300c\u5e73\u65b9\u300d\u5173\u7cfb\u753b\u51fa\u6765&#xff1a;\u6a2a\u8f74\u662f\u591a\u9879\u5f0f\u7684\u6b21\u6570&#xff0c;\u7eb5\u8f74\u662f\u6761\u4ef6\u6570\u7684\u4ee5 10 \u4e3a\u5e95\u7684\u5bf9\u6570&#xff0c;\u4e0b\u9762\u90a3\u6761\u662f\u8bbe\u8ba1\u77e9\u9635&#xff0c;\u4e0a\u9762\u90a3\u6761\u662f\u5b83\u7684\u6b63\u89c4\u65b9\u7a0b\u77e9\u9635\u3002\u4e24\u6761\u66f2\u7ebf\u51e0\u4e4e\u5904\u5904\u76f8\u5dee\u4e00\u500d\u2014\u2014\u5728\u5bf9\u6570\u8f74\u4e0a\u5dee\u4e00\u500d&#xff0c;\u5c31\u662f\u771f\u503c\u4e0a\u5dee\u4e00\u4e2a\u5e73\u65b9\u3002\u540c\u4e00\u6279 11 \u4e2a\u7b49\u8ddd\u70b9\u4e0a&#xff0c;1 \u6b21\u591a\u9879\u5f0f\u7684\u8bbe\u8ba1\u77e9\u9635\u6761\u4ef6\u6570\u662f 4.020340&#xff0c;10 \u6b21\u65f6\u6da8\u5230 115575200&#xff1b;\u5bf9\u5e94\u7684\u6b63\u89c4\u65b9\u7a0b\u77e9\u9635\u4ece 16.16313 \u6da8\u5230 1.491006\u00d710\u00b9\u2076&#xff0c;\u540e\u8005\u5df2\u7ecf\u8d85\u51fa\u53cc\u7cbe\u5ea6\u80fd\u53ef\u9760\u8868\u793a\u7684\u8303\u56f4&#xff08;1\/\u03b5 \u2248 4.5\u00d710\u00b9\u2075&#xff09;\u3002<\/p>\n<h3>\u516b\u3001\u4e00\u4e2a\u75c5\u6001\u62df\u5408\u7b97\u4f8b&#xff1a;\u4e09\u6761\u8def\u7ebf\u7ed9\u51fa\u4e09\u7ec4\u7cfb\u6570<\/h3>\n<p>\u628a\u4e0a\u9762\u8fd9\u4e9b\u91cf\u653e\u5230\u4e00\u4e2a\u5177\u4f53\u95ee\u9898\u4e0a\u3002\u53d6 11 \u4e2a\u7b49\u8ddd\u70b9 x &#061; 0, 0.1, 0.2, \u2026, 1.0&#xff0c;\u7528 0 \u5230 9 \u6b21\u7684\u591a\u9879\u5f0f\u53bb\u62df\u5408\u4e00\u6761\u7ed9\u5b9a\u7684\u66f2\u7ebf&#xff1b;\u4e3a\u4e86\u8ba9\u95ee\u9898\u4e0d\u662f\u300c\u63d2\u503c\u300d\u800c\u662f\u771f\u6b63\u7684\u6700\u5c0f\u4e8c\u4e58&#xff0c;\u7ed9\u6bcf\u4e2a\u70b9\u7684\u51fd\u6570\u503c\u52a0\u4e00\u4e2a\u56fa\u5b9a\u7684\u5c0f\u6270\u52a8&#xff1a;<\/p>\n<p>f(x) &#061; 1 \/ (1 &#043; x)<br \/>\ny_i &#061; f(x_i) &#043; \u03b4_i,   \u03b4_i &#061; (\u22121)^i \u00b7 (i &#043; 1) \/ 100000<\/p>\n<p>\u8fd9\u4e2a\u6270\u52a8\u6700\u5927\u53ea\u6709 1.1\u00d710\u207b\u2074&#xff0c;\u8089\u773c\u770b\u66f2\u7ebf\u5b8c\u5168\u4e0d\u53d8\u3002\u8bbe\u8ba1\u77e9\u9635 V \u662f 11\u00d710 \u7684\u5355\u9879\u5f0f\u77e9\u9635&#xff08;\u7b2c i \u884c\u662f 1, x_i, x_i\u00b2, \u2026, x_i\u2079&#xff09;&#xff0c;\u5b83\u7684\u5947\u5f02\u503c\u4ece 4.694490 \u4e00\u8def\u8870\u51cf\u5230 5.167242\u00d710\u207b\u2077&#xff0c;\u6761\u4ef6\u6570<\/p>\n<p>\u03ba(V) &#061; 9.085098\u00d710\u2076<\/p>\n<p>\u800c\u5b83\u7684\u6b63\u89c4\u65b9\u7a0b\u77e9\u9635 V\u1d40V \u7684\u6761\u4ef6\u6570\u662f<\/p>\n<p>\u03ba(V\u1d40V) &#061; 8.257747\u00d710\u00b9\u00b3<\/p>\n<p>\u4e24\u8005\u4e4b\u6bd4\u662f 9.089332\u00d710\u2076&#xff0c;\u4e0e \u03ba(V) \u540c\u91cf\u7ea7\u2014\u2014\u7b2c\u4e03\u8282\u90a3\u6761 \u03ba\u00b2 \u7684\u5173\u7cfb\u5728\u771f\u5b9e\u77e9\u9635\u4e0a\u5c31\u662f\u8fd9\u6837\u843d\u5730\u7684\u3002<\/p>\n<p>\u73b0\u5728\u7528\u4e09\u6761\u8def\u7ebf\u89e3\u540c\u4e00\u4e2a\u6700\u5c0f\u4e8c\u4e58\u95ee\u9898\u3002\u4e3a\u4e86\u80fd\u5224\u65ad\u8c01\u5bf9\u8c01\u9519&#xff0c;\u53e6\u5916\u7528\u7cbe\u786e\u6709\u7406\u6570\u7b97\u672f&#xff08;Python \u7684 Fraction&#xff0c;\u6ca1\u6709\u4efb\u4f55\u6d6e\u70b9\u820d\u5165&#xff09;\u628a\u540c\u4e00\u4e2a\u95ee\u9898\u7cbe\u786e\u6c42\u89e3\u4e00\u904d&#xff0c;\u4f5c\u4e3a\u53c2\u8003\u89e3&#xff1a;<\/p>\n<table>\n<tr>\u6c42\u89e3\u8def\u7ebf\u7cfb\u6570\u6700\u5927\u5e45\u5ea6\u7cfb\u6570\u4e0e\u53c2\u8003\u89e3\u7684\u5dee\u6b8b\u5dee\u8303\u6570<\/tr>\n<tbody>\n<tr>\n<td>\u7cbe\u786e\u53c2\u8003\u89e3&#xff08;\u6709\u7406\u6570\u7b97\u672f&#xff09;<\/td>\n<td>51.712921<\/td>\n<td>0<\/td>\n<td>1.429520\u00d710\u207b\u2074<\/td>\n<\/tr>\n<tr>\n<td>\u6b63\u89c4\u65b9\u7a0b&#xff08;\u663e\u5f0f\u6784\u9020 V\u1d40V&#xff09;<\/td>\n<td>51.747553<\/td>\n<td>5.797176\u00d710\u207b\u00b2<\/td>\n<td>1.429520\u00d710\u207b\u2074<\/td>\n<\/tr>\n<tr>\n<td>Householder QR<\/td>\n<td>51.712921<\/td>\n<td>1.395630\u00d710\u207b\u2078<\/td>\n<td>1.429520\u00d710\u207b\u2074<\/td>\n<\/tr>\n<tr>\n<td>\u57fa\u4e8e\u5206\u89e3\u7684\u4f2a\u9006<\/td>\n<td>51.712921<\/td>\n<td>2.281649\u00d710\u207b\u2079<\/td>\n<td>1.429520\u00d710\u207b\u2074<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u8fd9\u5f20\u8868\u6700\u503c\u5f97\u770b\u7684\u662f\u6700\u540e\u4e24\u5217\u7684\u5173\u7cfb\u3002\u56db\u884c\u7684\u6b8b\u5dee\u8303\u6570\u90fd\u7b49\u4e8e 1.429520\u00d710\u207b\u2074\u2014\u2014\u6253\u5370\u5230\u4e03\u4f4d\u6709\u6548\u6570\u5b57\u5b8c\u5168\u4e00\u6837\u3002\u5982\u679c\u53ea\u76ef\u7740\u6b8b\u5dee\u5224\u65ad\u89e3\u6cd5\u597d\u574f&#xff0c;\u4f1a\u5f97\u51fa\u300c\u4e09\u6761\u8def\u7ebf\u6ca1\u6709\u533a\u522b\u300d\u7684\u7ed3\u8bba\u3002\u53ef\u7cfb\u6570\u4e0e\u53c2\u8003\u89e3\u7684\u5dee\u5b8c\u5168\u662f\u53e6\u4e00\u56de\u4e8b&#xff1a;\u6b63\u89c4\u65b9\u7a0b\u89e3\u7684\u5dee\u662f 5.797176\u00d710\u207b\u00b2&#xff0c;QR \u662f 1.395630\u00d710\u207b\u2078&#xff0c;\u4f2a\u9006\u662f 2.281649\u00d710\u207b\u2079\u3002\u6b63\u89c4\u65b9\u7a0b\u7684\u7cfb\u6570\u8bef\u5dee\u662f QR \u7684 4153805 \u500d&#xff0c;\u800c \u03ba(V) \u662f 9.085098\u00d710\u2076\u2014\u2014\u4e24\u8005\u7684\u6570\u91cf\u7ea7\u5bf9\u5f97\u4e0a\u3002\u8fd9\u5c31\u662f\u6761\u4ef6\u6570\u5e73\u65b9\u7684\u4ee3\u4ef7&#xff1a;\u6b8b\u5dee\u770b\u8d77\u6765\u4e00\u6837\u597d&#xff0c;\u89e3\u5374\u5df2\u7ecf\u9519\u5230\u7b2c\u4e94\u4f4d\u3002<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/09\/20260923125706-6ab3cca2bb9f4.png\" alt=\"\u6b63\u89c4\u65b9\u7a0b\u3001Householder QR \u4e0e\u4f2a\u9006\u5728\u56db\u4e2a\u7ef4\u5ea6\u4e0a\u7684\u5bf9\u7167\" \/><\/p>\n<p>\u56fe 5&#xff1a;\u6b63\u89c4\u65b9\u7a0b\u3001Householder QR \u4e0e\u4f2a\u9006\u5728\u56db\u4e2a\u7ef4\u5ea6\u4e0a\u7684\u5bf9\u7167<\/p>\n<p>\u56fe 5 \u628a\u4e09\u6761\u8def\u7ebf\u5e76\u6392\u653e\u5728\u56db\u4e2a\u7ef4\u5ea6\u4e0a\u3002\u5de6\u8fb9\u4e00\u5217\u662f\u6b63\u89c4\u65b9\u7a0b&#xff0c;\u4e2d\u95f4\u662f QR&#xff0c;\u53f3\u8fb9\u662f\u4f2a\u9006\u3002\u56db\u884c\u5206\u522b\u662f\u7cfb\u6570\u6700\u5927\u5e45\u5ea6\u3001\u7cfb\u6570\u8bef\u5dee\u3001\u6b8b\u5dee\u8303\u6570&#xff0c;\u4ee5\u53ca\u6c42\u89e3\u8fc7\u7a0b\u4e2d\u5b9e\u9645\u8d77\u4f5c\u7528\u7684\u6761\u4ef6\u6570\u3002\u53ef\u4ee5\u770b\u5230\u5dee\u522b\u96c6\u4e2d\u5728\u7cfb\u6570\u8bef\u5dee\u90a3\u4e00\u884c&#xff0c;\u800c\u6b8b\u5dee\u90a3\u4e00\u884c\u4e09\u5217\u51e0\u4e4e\u76f8\u540c\u2014\u2014\u8fd9\u6b63\u662f\u8fd9\u5f20\u8868\u5bb9\u6613\u9a97\u4eba\u7684\u5730\u65b9\u3002<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/09\/20260923125707-6ab3cca3b1452.png\" alt=\"\u5341\u4e2a\u7cfb\u6570\u4e0a\u4e09\u79cd\u89e3\u6cd5\u7684\u8bef\u5dee\" \/><\/p>\n<p>\u56fe 6&#xff1a;\u5341\u4e2a\u7cfb\u6570\u4e0a\u4e09\u79cd\u89e3\u6cd5\u7684\u8bef\u5dee&#xff08;\u7eb5\u8f74\u4e3a\u4ee5 10 \u4e3a\u5e95\u7684\u5bf9\u6570&#xff09;<\/p>\n<p>\u56fe 6 \u628a\u8bef\u5dee\u62c6\u5230\u6bcf\u4e2a\u7cfb\u6570\u4e0a\u770b\u3002\u6a2a\u8f74\u662f\u7cfb\u6570\u4e0b\u6807&#xff08;x\u2070 \u5230 x\u2079&#xff09;&#xff0c;\u7eb5\u8f74\u662f\u8fd9\u4e00\u4e2a\u7cfb\u6570\u4e0a\u4e0e\u53c2\u8003\u89e3\u4e4b\u5dee\u7684\u5bf9\u6570\u3002\u7ea2\u8272\u90a3\u7ec4\u662f\u6b63\u89c4\u65b9\u7a0b&#xff0c;\u84dd\u8272\u662f QR&#xff0c;\u7eff\u8272\u662f\u6309 10\u207b\u2076 \u9608\u503c\u622a\u65ad\u540e\u7684\u4f2a\u9006\u3002\u524d\u4e24\u7ec4\u4e4b\u95f4\u7684\u5dee\u8ddd\u5728\u5341\u4e2a\u7cfb\u6570\u4e0a\u90fd\u5b58\u5728&#xff0c;\u7a33\u5b9a\u5728\u4e94\u5230\u4e03\u4e2a\u6570\u91cf\u7ea7&#xff1b;\u7eff\u8272\u90a3\u7ec4\u5219\u6574\u4f53\u843d\u5728 5.9\u00d710\u207b\u2077 \u5230 50.1 \u4e4b\u95f4\u2014\u2014\u6700\u9ad8\u7684\u90a3\u4e00\u70b9\u5df2\u7ecf\u4e0e\u89e3\u672c\u8eab\u7684\u91cf\u7ea7\u76f8\u5f53&#xff08;\u53c2\u8003\u89e3\u7684\u7cfb\u6570\u6700\u5927\u5e45\u5ea6\u662f 51.712921&#xff09;\u3002\u90a3\u4e0d\u662f\u820d\u5165\u8bef\u5dee&#xff0c;\u800c\u662f\u4e3b\u52a8\u4e22\u6389\u65b9\u5411\u6362\u6765\u7684\u4ee3\u4ef7&#xff0c;\u7b2c\u4e5d\u8282\u4f1a\u628a\u8fd9\u7b14\u8d26\u7b97\u6e05\u695a\u3002<\/p>\n<h3>\u4e5d\u3001\u4f2a\u9006\u4e0e\u622a\u65ad&#xff1a;\u628a\u4e0d\u53ef\u9760\u7684\u65b9\u5411\u4e22\u6389\u4e4b\u540e<\/h3>\n<p>\u5982\u679c A \u4e0d\u662f\u65b9\u9635\u3001\u6216\u8005\u4e0d\u6ee1\u79e9&#xff0c;A\u207b\u00b9 \u6839\u672c\u4e0d\u5b58\u5728&#xff0c;\u4f46\u6700\u5c0f\u4e8c\u4e58\u95ee\u9898\u4ecd\u7136\u53ef\u4ee5\u56de\u7b54\u3002\u628a A &#061; U\u03a3V\u1d40 \u4ee3\u8fdb\u6b63\u89c4\u65b9\u7a0b&#xff0c;\u53ef\u4ee5\u5f97\u5230\u4e00\u4e2a\u5f62\u5f0f\u4e0a\u5f88\u5e72\u51c0\u7684\u89e3&#xff1a;<\/p>\n<p>A\u207a &#061; V \u03a3\u207a U\u1d40<\/p>\n<p>\u5176\u4e2d \u03a3\u207a \u662f\u628a \u03a3 \u91cc\u6bcf\u4e2a\u975e\u96f6\u5947\u5f02\u503c\u53d6\u5012\u6570\u3001\u5176\u4f59\u4fdd\u6301\u96f6\u5f97\u5230\u7684\u7ed3\u679c\u3002\u8fd9\u4e2a A\u207a \u53eb\u4f2a\u9006&#xff0c;\u7528\u5b83\u7b97\u51fa\u7684 x &#061; A\u207ab \u5c31\u662f\u6700\u5c0f\u4e8c\u4e58\u89e3 [1]\u3002\u4e3a\u4ec0\u4e48\u5b83\u662f\u6700\u5c0f\u8303\u6570\u7684\u90a3\u4e2a&#xff1f;\u56e0\u4e3a\u628a\u89e3\u5199\u6210 A\u207ab \u65f6&#xff0c;\u7528\u5230\u7684\u53ea\u6709 U \u7684\u524d\u82e5\u5e72\u5217&#xff0c;\u4e5f\u5c31\u662f A\u1d40 \u7684\u503c\u7a7a\u95f4&#xff1b;\u6700\u5c0f\u4e8c\u4e58\u95ee\u9898\u7684\u89e3\u96c6\u662f\u4e00\u4e2a\u4eff\u5c04\u96c6\u5408&#xff0c;\u800c\u96c6\u5408\u91cc 2 \u8303\u6570\u6700\u5c0f\u7684\u90a3\u4e2a\u70b9\u6070\u597d\u843d\u5728 A\u1d40 \u503c\u7a7a\u95f4\u91cc&#xff08;\u96f6\u7a7a\u95f4\u7684\u6b63\u4ea4\u8865\u4e0a&#xff09;\u3002\u6240\u4ee5\u300c\u4f2a\u9006\u300d\u8fd9\u4e2a\u540d\u5b57\u91cc\u5176\u5b9e\u85cf\u4e86\u4e24\u4e2a\u6761\u4ef6&#xff1a;\u6700\u5c0f\u4e8c\u4e58&#xff0c;\u4ee5\u53ca\u6700\u5c0f\u8303\u6570\u3002<\/p>\n<p>\u4f2a\u9006\u771f\u6b63\u7684\u5b9e\u7528\u4ef7\u503c\u5728\u622a\u65ad\u3002\u6240\u8c13\u622a\u65ad&#xff0c;\u5c31\u662f\u5728\u53d6\u5012\u6570\u4e4b\u524d\u5148\u505a\u4e00\u4e2a\u51b3\u5b9a&#xff1a;\u54ea\u4e9b\u5947\u5f02\u503c\u7b97\u300c\u591f\u5927\u300d&#xff0c;\u54ea\u4e9b\u7b97\u300c\u4e0d\u53ef\u9760\u300d\u3002\u53ea\u4fdd\u7559 \u03c3\u1d62 &gt; \u03c4\u00b7\u03c3_max \u7684\u90a3\u4e9b\u65b9\u5411&#xff0c;\u5176\u4f59\u5f53\u4f5c\u96f6\u2014\u2014\u4e0e\u7b2c\u56db\u8282\u5b9a\u4e49\u6570\u503c\u79e9\u7528\u7684\u662f\u540c\u4e00\u4e2a\u9608\u503c\u3002<\/p>\n<p>\u8fd9\u4e2a\u51b3\u5b9a\u503c\u591a\u5c11\u94b1&#xff0c;\u524d\u9762\u7684\u7b97\u4f8b\u53ef\u4ee5\u91cf\u51fa\u6765\u3002\u7b2c\u516b\u8282\u7684\u62df\u5408\u95ee\u9898\u5728\u4e24\u79cd\u9608\u503c\u4e0b&#xff1a;<\/p>\n<table>\n<tr>\u9608\u503c \u03c4\u4fdd\u7559\u7684\u65b9\u5411\u6570\u7cfb\u6570\u6700\u5927\u5e45\u5ea6\u7cfb\u6570\u8303\u6570\u6b8b\u5dee\u8303\u6570<\/tr>\n<tbody>\n<tr>\n<td>10\u207b\u2078<\/td>\n<td>10<\/td>\n<td>51.712921<\/td>\n<td>81.76810<\/td>\n<td>1.429520\u00d710\u207b\u2074<\/td>\n<\/tr>\n<tr>\n<td>10\u207b\u2076<\/td>\n<td>9<\/td>\n<td>4.424275<\/td>\n<td>8.194994<\/td>\n<td>1.490051\u00d710\u207b\u2074<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u77e9\u9635\u4e00\u4e2a\u5b57\u6ca1\u6539\u3001\u6570\u636e\u4e00\u4e2a\u5b57\u6ca1\u6539&#xff0c;\u53ea\u628a\u9608\u503c\u6539\u4e86\u4e24\u4e2a\u6570\u91cf\u7ea7&#xff1a;\u7cfb\u6570\u6700\u5927\u5e45\u5ea6\u4ece 51.712921 \u6389\u5230 4.424275&#xff0c;\u7cfb\u6570\u8303\u6570\u4ece 81.76810 \u6389\u5230 8.194994&#xff0c;\u800c\u6b8b\u5dee\u53ea\u4ece 1.429520\u00d710\u207b\u2074 \u6da8\u5230 1.490051\u00d710\u207b\u2074\u2014\u2014\u5927\u7ea6 4%\u3002\u4e22\u6389\u6700\u540e\u4e00\u4e2a\u65b9\u5411&#xff0c;\u6362\u6765\u7684\u662f\u4e00\u4e2a\u300c\u5c0f\u5f97\u591a\u300d\u7684\u89e3&#xff0c;\u4ee3\u4ef7\u662f\u62df\u5408\u8bef\u5dee\u7565\u5fae\u53d8\u5927\u3002<\/p>\n<p>\u5bf9\u8fd9\u4ef6\u4e8b\u6709\u4e24\u79cd\u8bfb\u6cd5&#xff0c;\u90fd\u6210\u7acb&#xff0c;\u4f46\u5fc5\u987b\u77e5\u9053\u81ea\u5df1\u5728\u7528\u54ea\u4e00\u79cd\u3002\u7b2c\u4e00\u79cd\u662f\u628a\u5b83\u8bfb\u6210\u6b63\u5219\u5316&#xff1a;\u4e0d\u7a33\u5b9a\u7684\u65b9\u5411\u672c\u6765\u5c31\u88ab\u566a\u58f0\u4e3b\u5bfc&#xff0c;\u4e22\u6389\u5b83\u7b49\u4e8e\u653e\u5f03\u300c\u7cbe\u786e\u590d\u73b0\u566a\u58f0\u300d&#xff0c;\u8fd9\u6b63\u662f 1965 \u5e74\u90a3\u7bc7\u6587\u7ae0\u8bf4\u4f2a\u9006\u80fd\u6291\u5236\u865a\u5047\u632f\u8361\u4e0e\u76f8\u6d88\u7684\u610f\u601d [1]\u3002\u7b2c\u4e8c\u79cd\u662f\u628a\u5b83\u8bfb\u6210\u4e3b\u52a8\u5f15\u5165\u7684\u6a21\u578b\u8bef\u5dee&#xff1a;1965 \u5e74\u90a3\u7bc7\u6587\u7ae0\u5728\u540c\u4e00\u9875\u5199\u5f97\u5f88\u6e05\u695a\u2014\u2014\u5ffd\u7565 \u03c3_{r&#043;1} \u5230 \u03c3_n \u8fd9\u4e9b\u65b9\u5411&#xff0c;\u7b49\u4ef7\u4e8e\u7ed9 A \u52a0\u4e86\u4e00\u4e2a\u8303\u6570\u4e3a \u221a(\u03c3_{r&#043;1}\u00b2 &#043; \u2026 &#043; \u03c3_n\u00b2) \u7684\u6270\u52a8 [1]\u3002\u6240\u4ee5\u300c\u622a\u65ad\u4e4b\u540e\u89e3\u66f4\u51c6\u300d\u8fd9\u53e5\u8bdd\u6ca1\u6709\u666e\u9002\u7248\u672c&#xff1b;\u80fd\u8bf4\u7684\u662f&#xff1a;\u622a\u65ad\u7528\u4e00\u4e2a\u5df2\u77e5\u5927\u5c0f\u7684\u504f\u5dee&#xff0c;\u6362\u6389\u4e86\u4e0d\u53d7\u63a7\u7684\u8bef\u5dee\u653e\u5927\u3002<\/p>\n<p>\u8fd9\u4e5f\u662f\u4e3a\u4ec0\u4e48\u62a5\u544a\u6570\u503c\u79e9\u5fc5\u987b\u540c\u65f6\u62a5\u544a\u9608\u503c\u3002\u540c\u4e00\u7ec4\u6570\u636e\u5728 \u03c4 &#061; 10\u207b\u2078 \u4e0b\u6570\u503c\u79e9\u662f 10&#xff0c;\u5728 \u03c4 &#061; 10\u207b\u2076 \u4e0b\u662f 9&#xff0c;\u4e24\u4e2a\u7b54\u6848\u90fd\u5bf9&#xff0c;\u56e0\u4e3a\u53e3\u5f84\u4e0d\u540c&#xff1b;\u53ea\u62a5\u300c\u79e9 &#061; 10\u300d\u800c\u4e0d\u8bf4\u9608\u503c&#xff0c;\u522b\u4eba\u5c31\u65e0\u6cd5\u590d\u6838\u8fd9\u4e2a\u6570\u5b57\u662f\u600e\u4e48\u6765\u7684\u3002<\/p>\n<h3>\u5341\u3001\u8fb9\u754c&#xff1a;\u8303\u6570\u7684\u9009\u62e9&#xff0c;\u4ee5\u53ca\u8fd9\u5957\u5206\u89e3\u4e0d\u5229\u7528\u7684\u4e1c\u897f<\/h3>\n<p>\u7b2c\u4e00\u6761\u8fb9\u754c&#xff1a;\u6362\u4e00\u4e2a\u8303\u6570&#xff0c;\u95ed\u5f0f\u5c31\u6362\u4e86\u3002 \u7b2c\u4e94\u8282\u90a3\u6761\u300c\u8bef\u5dee\u7b49\u4e8e\u5c3e\u90e8\u5947\u5f02\u503c\u5e73\u65b9\u548c\u5f00\u65b9\u300d\u662f\u5728 Frobenius \u8303\u6570\u53e3\u5f84\u4e0b\u6210\u7acb\u7684\u3002\u628a\u8bef\u5dee\u53e3\u5f84\u6362\u6210\u522b\u7684\u9149\u4e0d\u53d8\u8303\u6570&#xff08;\u5728\u6b63\u4ea4\u53d8\u6362\u4e0b\u4fdd\u6301\u4e0d\u53d8\u7684\u8303\u6570&#xff0c;\u6bd4\u5982\u8c31\u8303\u6570\u2014\u2014\u6700\u5927\u7684\u5947\u5f02\u503c&#xff09;&#xff0c;\u622a\u65ad\u4ecd\u662f\u6700\u4f18\u7684&#xff0c;\u4f46\u8bef\u5dee\u4e0d\u518d\u6709\u90a3\u6761\u95ed\u5f0f&#xff1a;\u8c31\u8303\u6570\u53e3\u5f84\u4e0b&#xff0c;\u79e9 r \u622a\u65ad\u7684\u8bef\u5dee\u7b49\u4e8e\u6700\u5927\u7684\u90a3\u4e2a\u88ab\u4e22\u6389\u7684\u5947\u5f02\u503c \u03c3_{r&#043;1}\u3002\u628a\u7ed3\u8bba\u4ece Frobenius \u63a8\u5230\u4e00\u822c\u9149\u4e0d\u53d8\u8303\u6570\u7684\u5de5\u4f5c\u51fa\u73b0\u5728 1960 \u5e74\u7684\u4e00\u7bc7\u6587\u7ae0\u91cc 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9.085098\u00d710\u2076&#xff0c;\u5df2\u7ecf\u80fd\u8ba9\u4eba\u5728\u5b8c\u5168\u770b\u4e0d\u51fa\u5f02\u5e38\u7684\u60c5\u51b5\u4e0b\u4e22\u6389\u4e94\u4f4d\u6709\u6548\u6570\u5b57\u3002\u5224\u65ad\u4e00\u4e2a\u6700\u5c0f\u4e8c\u4e58\u95ee\u9898\u8981\u4e0d\u8981\u7d27&#xff0c;\u5148\u7b97\u6761\u4ef6\u6570\u901a\u5e38\u6bd4\u5148\u770b\u6b8b\u5dee\u6709\u7528\u5f97\u591a\u3002<\/p>\n<h3>\u53c2\u8003\u6587\u732e<\/h3>\n<p>[1] G. Golub, W. Kahan. Calculating the Singular Values and Pseudo-Inverse of a Matrix. Journal of the Society for Industrial and Applied Mathematics Series B Numerical Analysis, 2(2):205-224, 1965. DOI: 10.1137\/0702016<br \/>\n[2] Carl Eckart, Gale Young. The Approximation of One Matrix by Another of Lower Rank. Psychometrika, 1(3):211-218, 1936. DOI: 10.1007\/BF02288367<br \/>\n[3] Richard M. Johnson. On a Theorem Stated by Eckart and Young. Psychometrika, 28(3):259-263, 1963. DOI: 10.1007\/BF02289573<br \/>\n[4] Peter Businger, Gene H. Golub. Linear least squares solutions by householder transformations. Numerische Mathematik, 7(3):269-276, 1965. DOI: 10.1007\/BF01436084<br \/>\n[5] G. H. Golub, C. Reinsch. Singular value decomposition and least squares solutions. Numerische Mathematik, 14(5):403-420, 1970. DOI: 10.1007\/BF02163027<br \/>\n[6] L. Mirsky. SYMMETRIC GAUGE FUNCTIONS AND UNITARILY INVARIANT NORMS. The Quarterly Journal of Mathematics, 11(1):50-59, 1960. 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